aa r X i v : . [ m a t h . L O ] J u l ON SEQUENTS OF Σ FORMULAS
ANDRE KORNELL
Abstract.
We investigate the position that foundational theories should be mod-elled on ordinary computability. In this context, we investigate the metamathemat-ics of Σ formulas. We consider theories whose axioms are implications between Σformulas. We show that arbitrarily strong such theories prove their own correctness.
Motivation.
The standard distinction between the metalanguage and the object language inmetamathematics entails that we never consider the metamathematics of the mathematicaluniverse as a whole. Instead, we study each structure from an external vantage point: anextension or expansion of that structure, or both. The necessity of the metalanguage/object-language distinction is of course a consequence of Tarski’s undefinability theorem. The basicpremise of our approach is to restrict the logic of the foundational system to circumventTarski’s theorem. Our approach follows most clearly the work of Feferman. Our discussionof procedures is very much in the spirit of his Operational Set Theory [5]. We also adoptthe position that classical reasoning is valid for initial segments of the set theoretic universe,but not for the universe as a whole. However, we reject even intuitionistic reasoning forthe set theoretic universe as a whole, as it is sufficient for Tarski’s theorem. Instead, weconsider a formula meaningful just in case it is equivalent to some procedure halting; hence,we term the approach “positivistic”. In the context of set theory, these are essentially the Σ formulas. It is an elementary fact that the truth predicate for Σ sentences is itself Σ , so wemay investigate the metamathematics of this fragment without an external vantage point,and therefore, for the mathematical universe as a whole. Results.
We define a postivistic theory to consist of sequents of Σ formulas. We definea positivistic proof to be a sequence of Σ formulas, with each formula obtained from thepreceding formula according to an axiom, which may be applied deeply (definition 5.1).We obtain a complete list of logical axioms for positivistic reasoning using a term modelconstruction (corollary 6.3). We show that there is a positivistic theory for pure sets thatproves Tarski’s semantic axioms for the truth predicate, that proves that the conclusion ofany axiom of that theory is true if its assumption is true, and that interprets ZFC ; likewisethere is such a positivistic theory for natural numbers that interprets
PRA (corollary 9.8).Observing that the powerset operation is not definable by a Σ formula (proposition 8.1), wepropose two completeness axioms for the universe of pure sets, and we show that they areequivalent (theorem 10.3). These completeness axioms imply that every set is countable.Following Weaver [13], we define an assertibility predicate for intuitionistic formulas in thelanguage of set theory (definition 11.1), we show that the assertibility predicate satisfiesTarski’s semantic axioms for the positivistic connectives, and in particular, we show that a Σ sentence is true if and only if it is assertible (theorem 11.11). bout these notes. These notes are not the final draft of a paper; I am posting this draftprematurely for practical reasons. Some details need more attention; some ideas should be expressedmore concisely. The foundational requirements of some of the later arguments need to be determinedwith care. I have included a minimal bibliography, that will need to be substantially extended.I first started on this line of thought as a student in a course of Burgess. I cite his discussion[3] of the distinction between accepting each derivation of a system and accepting that systemas whole. Three authors inadvertently spurred me to finally put pen to paper. First, Weaver’sincisive discussion [13] of foundational positions in terms of a choice of objects, constructions andlogic, and of sets in terms of surveyability, expresses a very similar approach to the one that I hadheld vaguely, and am now expressing here. I diverge from Weaver in my position that the choiceof logic is essentially forced by the choice of objects and constructions. I regret that I was notable to include anything about surveyablility in this draft. Second, I was struck by Schweber’sdefinition [7] of computational reducibility for uncountable objects. The computational propertiesof a structure do not depend on how that structure is forced to be countable; one simple, perhapssimplistic, explanation for this phenomenon is that the structure simply is countable. Third, I wasintrigued by Hamkins’s embeddability result [6], which suggests to me some hope of explaining thelinear ordering of natural set theories by consistency strength.I am not a logician by training. Some effort went into proving results that are widely known inthe logic community. I do not have a good understanding of what in these notes is new. I hadobtained the complete list of logical axioms for positivistic proof before I encountered Beklemishev’swork on positive deep inference [2]. The application of positive deep inference to predicate logicmay nevertheless be new. I had obtained the formalization of the naive argument for the validityof bounded proofs before I encountered Pudl´ak’s work on finitistic consistency proofs [8]. Pudl´ak’sresult uses a different measure of size, for both the derivations he considers and the proof itself,and he permits classical logic, in both the derivations he considers and in the proof itself; I donot yet fully understand the relationship between the two results. My impression is that at leastthe extreme naturality of the proof given here is novel. Takahashi developed a theory of finitarymathematics [12] that identifies finitary methods with the Σ definable functions, as does ourapproach. Takahashi’s approach is less general in that it studies a single formal system, whichnecessarily describes the hereditarily finite sets; furthermore, this system is not positivistic becauseit includes bounded universal generalization; see section 8. I expect to encounter other examplesof previous work that includes some of the results here, partially or wholly. I am even less familiarwith the literature in the philosophy of mathematics.Our exposition draws heavily on Rathjen’s development of PRS [9], the theory of primitiverecursive set functions, and Feferman’s proof of cut-elimination [4] for an infinitary sequent calculusthat includes countable conjunctions and disjunctions. We apply the Rauszer-Sabalski lemma [10]in our completeness argument.I thank Lev Beklemishev, Joel David Hamkins, Joost Joosten, Alex Kruckman, Michael Rathjen,and Stephen Simpson, for responding patiently to my invariably elementary questions about prooftheory. I thank Nik Weaver for discussion before and after this project. I am grateful to AndrewMarks for his comments on an earlier, messier presentation of these results. I apologize to anypeople whom I may have left out, in haste.Figures summarizing various deductive systems appear at the end of these notes. I use a colon(:) after each quantifier as a visual separator. The range of the quantifier extends to the next condi-tional implication symbol ( ⇒ ), unless the conditional implication symbol is enclosed by parentheses. . summary The positivistic approach to foundations, introduced here, includes no assumption on whatmathematical objects may exist. Rather, it is the position that a mathematical propositionis meaningful if and only if it is verifiable by a mathematical procedure, whatever objectsmay exist and whatever procedures may be possible. The positivistic approach is then akind of extension of the positivism of the Vienna Circle to mathematical objects, replacingthe human agent with an ideal agent.In our motivating example of a mathematical universe, the objects are natural numbersand the procedures are recursive partial function presentations. The positivistic position isthen that the computably verifiable predicates are the only meaningful ones. This exampleexhibits many characteristic features of the positivistic approach; in particular one obtains aself-contained universe with a truth predicate given by the universal procedure, and havingthe nature of a potential, rather than completed totality.The restriction to computably verifiable predicates is a feature of finitism; we might saythat finitism is Turing computability plus positivism. We vary the notion of computation,while retaining the positivism.In the examples that we consider, the mathematical universe consists of pure sets. Weconsider model universes H κ of sets of hereditary cardinality less than a regular cardinal κ . The computable propositions are given by Σ formulas (essentially Σ formulas) in thelanguage of set theory, at times augmented by various function symbols. The axioms aregiven by implications between Σ formulas (essentially Π sentences). Each axiom expressesthe intuition that while the antecedent formula is true of a given tuple of objects, theconsequent formula is also true of that tuple of objects. Each axiom permits inference byreplacing the antecedent by the consequent, even a substitution instance of the antecedent bythe corresponding substitution instance of the consequent, and even when that substitutioninstance of the antecedent occurs as a subformula. This is called deep inference; it is validhere because we confine negation to atomic formulas.We obtain a list of logical axioms that is complete for positivistic reasoning. The the-ory of primitive recursive arithemtic is sufficient to prove the reduction of intuitionistic andclassical derivations to positivistic proofs. With the additional axiom of iterated reflectionthrough the ordinals, infinitary intuitionistic derivations can also be reduced to positivisticproofs. Defining the assertiblity of an intuitionistic sentence to be its derivability in infini-tary intuitionistic logic, from the axioms of a fixed theory and the true Σ sentences withparameters, we find that assertibility, like the truth predicate, respects the meaning of thepositivistic logical connectives.Thus, the positivistic approach does not reject the existence of any object, nor the feasi-bility of any procedure, nor the validity of any classical deduction, nor even the justifiabilityof any Π axiom. The impact of taking the positivistic position is the rejection of axioms onthe basis of their logical complexity. Among the axioms of ZFC , the powerset axiom may berejected on this basis. If we do not assume the feasibility of the powerset operation, limitingourselves to Σ formulas in the langague of set theory, then we cannot express the powersetaxiom as a Π formula.Furthermore, an inclusive conception of the universe of pure sets suggests that all puresets are countable. The forcing method potentially produces a bijection between any giveninfinite set and the set of natural numbers; this bijection should belong to the universe of ll possible pure sets. The forcing method is not simply a central tool in our understandingof set theory, it is a special case of the principle of completeness for infinitary logic, itself anatural part of the inclusive conception.Following the positivistic approach, we are thus led to rejecting the powerset axiom in favorof a universe of pure sets that is complete and self-contained in a number of ways. It is amathematical universe that contains its own metamathematics; we avoid Tarski’s theorem byrejecting the notion that every proposition must have a negation. Instead, G¨odel’s theoremsand reflections principles come to the foreground.Any given theory, that is, a class of axioms given by some unary predicate has a correctnessprinciple, which asserts that if the premise of some axiom is true, then its conclusion is alsotrue. Such a correctness axiom proves its own correctness: it can be used to show thatif the premise of any axiom is true, then its conclusion is also true. Thus, the resultingtheory proves that each of its inferences is correct! Is this in contradiction with G¨odel’ssecond incompeleteness theorem? No, because the induction rule is not admissible for sucha theory. We have only obtained, for every natural number n , the proof that if the first lineof an n -line proof is the true atomic sentence ⊤ , then the second line of the proof is true,then the third line of the proof is true, then ... , then the n -th line of that proof is true,so the proof does not prove the false atomic sentence ⊥ . The reader will surely agree thatthough such a theory does not prove its own consistency, it very nearly does so.2. potentialism The positivistic approach may be considered a form of mathematical potentialism. Poten-tialism is, roughly, the position that the mathematical universe is not a completed totality,and that its objects cannot be taken to all exist simultaneously. Potentialism has its roots inclassical inquiry into the nature of the infinite. In the context of set theory, the motivationfor potentialism may be summarized as follows: First, the paradoxes of naive set theorysuggest that there is something alien about the totality of all sets and the totality of allordinals. Second, the intuitive framework underlying large cardinal axioms suggests that theuniverse of sets is not vertically completed. Third, set-theoretic pluralism arising from theforcing method suggests that the universe is not horizontally completed either.The positivistic approach also presents an account of truth in mathematics. The majorobstacle to the naive conception of truth is the lair paradox, formalized as Tarski’s un-definability theorem. Tarski’s undefinability theorem is the proposition that any systemthat includes classical first-order logic, a truth predicate, and minimal axioms about finitis-tic mathematics, is necessarily inconsistent. Consequently, foundational theories typicallylack a truth predicate, leading to the necessary metamathematical distinction between theobject-language and the metalanguage.Thus, there is a sense of incompleteness both in the standard account of the mathematicaluniverse, and in the standard account of truth. This incompleteness can be approachedusing hierarchies or modalities. The motivation of this paper is to describe the mathematicaluniverse as a whole. For this purpose, we give up classical first-order logic in exchange fora universal truth predicate. We must even give up intuitionistic first-order logic, becauseit is sufficient for the proof of Tarski’s undefinability theorem. This paper accepts only Σ sentences as meaningful propositions about the mathematical universe. he true motivation of this paper is toward an account of infinitary data. The inter-pretation of mathematics in set theory reduces mathematical objects to their informationcontent, with pure sets in the role of datums. This is the only significance of sets in thispaper; we look at pure sets and formulas because these constitute the established foundationof infinitary mathematics, and the established subject of metamathematical research. In aresearch vacuum, I would prefer to study sets of ordinals, i. e., well-ordered sequences ofbit values, and procedures on these sequences. Where these two models of infinitary datadiverge for choice reasons, I trust the latter.3. objects and procedures The positivistic viewpoint includes no position on what mathematical objects may exist orwhat mathematical procedures are possible; it is a position on what predicates about theseobjects and procedures are meaningful. We use the term “procedure” in a wide sense, toinclude any abstract way of obtaining an object from given objects. We include nondeter-ministic procedures such as Vitali’s “construction” of a nonmeasurable set of real numbers;in particular, we include the procedure that produces an arbitrary mathematical object.The reader is invited to take a position on what mathematical objects may exist and whatmathematical procedures are possible, out of conviction, or for the sake of argument. Yourchoice of objects may reasonably be termed your mathematical ontology, and your choiceof procedures may, more controversially, be termed your mathematical epistemology. Thislatter term may be justified by the notion that the phrase “mathematical procedure” doesnot refer to a mechanical method, but to a way of deriving abstract entities. This choice ofmathematical ontology and mathematical epistemology determines the mathematical uni-verse.Our use of word “epistemology” captures the essence of the positivistic viewpoint: propo-sitions are mathematical objects, and truth is a mathematical procedure, in the sense above.Metamathematical objects are mathematical objects, and any way of assigning truth topropositions must be part of the mathematical universe, if indeed the mathematical universeincludes all mathematical objects that may exist and all possible mathematical procedures.Crucially, mathematical procedures must themselves be counted among mathematical ob-jects.We take a proposition to be a nullary procedure, i. e., a procedure from no given objects.We call a proposition true when it produces a specified “truth” object, which we may taketo be the 0-tuple. More generally, an n -ary predicate is just an n -ary procedure; when an n -ary predicate produces the truth object from n given objects, we say that it is true ofthose objects. We assume that each object has a canonical name, i. e., a nullary procedurethat produces just that object, so we can express the truth of an n -ary predicate of n givenobjects as a proposition.We can implement each positive Boolean connective by composing the given propositionswith a specific procedure. Conjunction is implement as the binary procedure that producesthe truth object when both given objects are the truth object. Disjunction is implementedas the binary procedure that produces the truth object when either given object is the truthobject. Falsehood is implemented as the the nullary procedure that produces no objects.The usual diagonalization argument shows that there is no procedure that produces anegation of the given proposition, i. e., a proposition that produces the truth object just in ase the given proposition does not. More generally, there is no procedure that produces animplication of two given propositions. Though we may be willing to infer one propositionfrom another, the validity of this inference cannot generally be expressed as a proposition,and so cannot be a truth about the mathematical universe.A theory, vaguely speaking, consists of the inferences one is willing to make; we formalize itessentially as a binary predicate. There is no general notion of truth for theories, since thereis no general notion of truth for a single inferences. Furthermore, there is no general notionof falsehood. We would like to say that a false theory is one that infers a false propositionfrom a true proposition, but we have no falsehood predicate. We are reduced to doubtingtheories that draw dubious conclusions.4. formulas To simplify the presentation we break with the very general discussion of the previoussection by supposing a universe of pure sets. Many of the results that follow hold in moregeneral settings, but clearly enunciating the necessary assumptions muddles the presenta-tion. The interpretation of the mathematical universe in its class of pure sets is ubiquitousin the study of foundations, and it is useful to us for precisely this reason. We justify thisinterpretability assumption by remarking that pure sets appear to serve adequately as in-finitary datums. Thus, we assume that the objects of the mathematical universe are puresets.We model our notion of procedure on ordinary computability. Specifically we imagine aregister machine, with each register holding a pure set that varies over the course of theprocedure. Appropriately, we label each register with a variable symbol; while we considerfinitary logic, it is enough to imagine countably many registers, with one register and onevariable symbol for each natural number. Within the universe of pure sets, a natural numberis naturally a finite von Neumann ordinal.We express procedures in first-order logic. Each notion of procedure is given by thecardinal characteristics of its logic, and the functions and predicates in its vocabulary. Ourvocabulary will consist of the binary predicate symbols =, =, ∈ , and , and various functionsymbols, generally definable by a L ωω (= , = , ∈ , ) formula. We will primarily focus on thelogics L ωω , and L ∞ ω . In a universe that does not satisfy the axiom of choice, we will write L Ω ω to refer to the class of infinitary formulas in which the conjunctions and disjunctionsare well-ordered.The notation L ωω ( S ) refers to formulas in a given vocabulary S . When the vocabulary S is not specified, it is understood to be the vocabulary of our set theory, usually consisting ofjust the predicate symbols =, =, ∈ , and , but possibly including the vocabulary of PRS ,or even the symbol ℘ for the powerset construction .A formula is in Σ( L ωω ( S )) if it is build up from atomic formulas, including the logicalsymbols ⊤ and ⊥ , using binary conjunction, binary disjunction, bounded universal quan-tification, and unbounded existential quantification. We accept bounded universal quan-tification, and reject unbounded universal quantification, because we assume that each setis surveyable, but the universe as a whole is not. (Surveyability is a term introduced byWeaver [13] that for our purposes is essentially synonymous to definiteness, as of a totality;I sometimes prefer this term for its procedural intuitions.) We include the negated predicatesymbols = and , but exclude the negation connective, because we assume that it is possible o verify that given pure sets are distinct, or that one is not an element of the other, butthat it is generally impossible to verify that a given formula is not true. Indeed, our attitudeis that it is generally meaningless to speak of a given formula not being true.Similarly, we say that a formula is in I( L ωω ( S )) if it is built up using conjunction, disjunc-tion, unbounded existential quantification, unbounded universal quantification, and impli-cation; the letter I suggests intuitionistic logic, as in Gentzen’s LI , the letter I commonlymistaken for J . We regard the formulas of I( L ωω ) as prima facie meaningless.We would like the meaning of K( L ωω ( S )) to depend on context; the letter K suggestsclassical logic, as in Gentzen’s LK . In the context of the G¨odel-Gentzen translation, it isnatural to say that a formula is in K( L ωω ( S )) if it is built up using conjunction, unboundeduniversal quantification, and implication. However, for proof theoretic arguments it is moreconvenient to represent disjunction, unbounded existential implication, and negation as sep-arate symbols. Furthermore, in the context of Friedman’s translation, it is convenient towork with the same class of formulas as in the context of G¨odel-Gentzen translation, butto parse them differently. For definiteness, we will say that a formula is in K( L ωω ( S )) if itis built up using conjunction, disjunction, unbounded universal quantification, unboundedexistential quantification, and negation.Finally, we write Σ , I , and K in place of Σ, I and K, to indicate that parameters arepermitted. Note that because truth is definable for Σ ( L ∞ ω ) sentences, the formulas in Σ ( L ∞ ω ) do not yields a strictly stronger notion of computability than the apparently weaker Σ ( L ωω ). Note also that within the universe of hereditarily finite sets, L ∞ ω and L ωω essentiallycoincide, with the latter formally restricted to binary conjunction and disjunction.We write φ vt and φ ( v/t ) for the result of substituting the term t for the variable v . Wewrite φ ( v , . . . , v n ) for a formula whose free variables are among v , . . . , v n , and we write φ ( t , . . . , t n ) as a shorthand for φ ( v /t , . . . , v n /t n ). We write a for the parametric symbolnaming the object a , and we write φ [ a ] as a shorthand for φ ( a ). We sometimes placecorners around a string, e. g., p ∀ s ∈ a : s = a q , to denote that string, but more often weblur this important distinction, writing φ ( x/t ) where we should write φ ( p x q /t ), and writing“the symbol π ” where we should write “the symbol p π q ”. We may abbreviate the phrase“substitution instance” by the word “instance”.5. positivistic proof If we intuit a procedure q to have at least the outputs of procedure p on every tupleof inputs, we write p ⇒ q , and we feel justified in replacing any occurrence of p , as asubprocedure in a proposition, by q . In the context of a universe of pure sets, we only directlyconsider procedures given by Σ formulas, so if we intuit that ψ ( x , . . . , x n ) is true whenever φ ( x , . . . , x n ) is true for every tuple of values, we write φ ( x , . . . , x n ) ⇒ ψ ( x , . . . , x n ). Suchan implication between Σ formulas is called a conditional.Replacing a subprocedure φ by ψ amounts to replacing a subformula of the form φ ( t , . . . t n )by a subformula of the form ψ ( t , . . . , t n ), where the terms t , . . . , t n may include boundvariables. If the subformula φ ( t , . . . t n ) occurs below a bounded universal quantifier, thenwe may think of the subprocedure φ occuring within a kind of for loop. An inference on asubformula, such as this, is termed “deep inference” in the literature. theory consists of conditionals φ ( x , . . . , x n ) ⇒ ψ ( x , . . . , x n ), which are called theaxioms of the theory. Formally, we distinguish between intentional theories and exten-sional theories. An intentional theory is formula τ ( x ), which is satisfied by its axioms,whereas an extensional theory is just the set T of its axioms. We say that a structuremodels an extension theory T just in case that structure models the universal closure φ ( x , . . . , x n ) ⇒ ψ ( x , . . . , x n ) of every element of T , in the ordinary sense. The notionof “a model” is more subtle for intentional theories. We say that a structure models anintentional theory τ just in case it models the universal closure of the correctness principleof τ , which we define in section 9. Definition 5.1.
Let φ ( v , . . . , v n ) ⇒ ψ ( v , . . . , v n ) be a conditional, that is, an implica-tion between Σ ( L ωω ) formulas. An application of φ ⇒ ψ , is a pair of Σ ( L ωω ) formulas( χ , χ ) such that χ is obtained from χ by replacing a subformula of χ that is of the form φ ( t , . . . , t n ) by ψ ( t , . . . , t n ).An extensional theory T is a set of conditionals, and its elements are called its axioms.An intentional theory τ is a Σ ( L ωω ) formula with a single free variable. An axiom of τ is aconditional φ ⇒ ψ that satisfies this formula. A finitary positivistic proof of χ ⇒ χ ′ in τ is a finite sequence of Σ ( L ωω ) formulas ( χ , . . . , χ n ) such that χ = χ , χ n = χ ′ , and eachstep ( χ i , χ i +1 ) is an application of some axiom.Is the appearance of free variables in the axioms and formulas of a positivistic proofcompatible with the positivistic rejection of universal claims? This question is addressed insection 12. Briefly, the free variables that appear in the formulas that make up a positivisticproof are assumed to have fixed values for the entire course of the proof; they may be treatedas syntactic variables. The free variables that appear in axioms do have a universal sense;however, axioms express inference rules rather than propositions. The absolute justifiabilityof a given inference is not a positivistically meaningful proposition.6. the logical axioms Our informal remarks have been in or about some hypothetical universe. For concreteness,the following more formal remarks will be about model universes under standard foundations,i. e.,
ZFC , possibly augmented by large cardinal axioms. We begin by obtaining a completelist of logical axioms for positivisitic proof. Thus, we consider all models of any vocabularycontaining distinguished binary predicates =, =, ∈ , and , such that = and = denotecomplementary relations, ∈ and denote complementary relations, and = denotes genuineequality.We initially obtain our list of logical axioms for positivistic proof by constructing a com-pleteness argument in the standard way. This argument may be viewed as taking place in ZFC , where we are considering model universes.First, we obtain a sequent calculus, whose sequents are conditionals φ ⇒ ψ . Fix anextensional theory T , and a conditional χ ⇒ χ ′ that cannot be proved from T using rulesof inference to be added retroactively. Define a preorder on all Σ formulas with φ ≤ ψ justin case φ ⇒ ψ is provable from T . We obtain a term model of T that satisfies χ but not χ ′ from a prime filter on the preorder that contains χ , but not χ ′ . This argument is entirelystandard apart from the difficulty that our preorder is not complemented, because we do nothave negation. To ensure the existence of a prime filter that respects the infinitary joins and eets corresponding to quantified formulas, we add distributive axioms that are unnecessaryin the classical case. We arrive at the conditional calculus in figure 1. Lemma 6.1 ( ZFC ) . Let S be a countable vocabulary consisting of function symbols andpredicate symbols of any finite arity, including = , = , ∈ and . Let T be a set of Σ( L ωω ( S )) conditionals. Let χ ⇒ χ ′ be any Σ( L ωω ( S )) conditional. Then, either χ ⇒ χ ′ is derivablefrom T using the system given in figure 1, or there is a model M of T where χ ⇒ χ ′ isfalse for some tuple of elements.Proof. Define φ ≤ ψ whenever φ ⇒ ψ is derivable from T . Rules (-2) and (-1) imply that ≤ is a preorder on Σ( L ωω ( S )). Rules (1)-(9) imply that, modulo equivalence, (Σ( L ωω ( S )) , ≤ )is a bounded distributive lattice.As usual for a term model construction, we distinguish between free variables, a , b , c ,...,and bound variables, x , y , z ,.... Rules (10), (11), and (0), imply that ∃ x : φ is the joinof { φ xt : t is a term } . Rules (13), (14), and (0), imply that ∀ x ∈ s : φ is the meet of { t s ∨ φ xt : t is a term } . Rules (12) and (15) imply that these infinitary meets and joins aredistributive.Assume χ χ ′ . The Rauszer-Sabalski lemma[10] guarantees the existence of prime filterthat preserves the meets and joins described above. As usual, we define the universe of M to be the set of all terms in the vocabulary S using only the free variables, and we defineterms t , . . . , t n to satisfy the formula φ ( a , . . . , a n ) just in case φ ( a /t , . . . , a n /t n ) is theprime filter. Our choice of filter guarantees that this definition satisfies Tarski’s definition oftruth. Rule (0) implies that M is a model of T , while the predicate symbols =, =, ∈ , and are treated as being entirely nonlogical.Rules (16)-(19) imply that = and = denote complementary relations on M , and likewisefor ∈ and . Rules (20)-(22) imply that = denotes an equivalence relation on M . Finally,rule (23) implies that our definition of satisfaction respects this equivalence relation. Weconclude that M/ = is a model of T . (cid:3) We add logical axioms as they become convenient to eliminate the nonaxiomatic rules ofinference given in figure 1.
Theorem 6.2 ( ZFC ) . Let S be a countable vocabulary consisting of function symbols andpredicate symbols of any finite arity. Let T be a set of Σ( L ωω ( S )) . Let χ ⇒ χ ′ be any Σ( L ωω ( S )) conditional. Then, either χ ⇒ χ ′ has a finitary positivistic proof using theaxioms of T and the logical axioms given in figure 2, or there is a model M of T where χ ⇒ χ ′ is false for some tuple of elements.Proof. Proof is by induction on the length of the derivation produced by the precedinglemma. If the terminal conditional is obtained by the axiomatic rule (-2), it has a proofin zero steps. If the terminal conditional is obtained by the rule (-1), then it has a proofobtained by concatenating the proofs of the input conditionals.If the terminal conditional is obtained by the rule (0), then it has a proof obtained bysubstituting t for a in every formula of the proof of the input conditional. Indeed, if χ i ⇒ χ i +1 is an application of some axiom φ ⇒ ψ , then ( χ i ) at ⇒ ( χ i +1 ) at is also an applicationof this axiom. Specifically, if a subformula of the form φ bs is replaced by ψ bs to infer χ i +1 from χ i , then a subformula of the form ( φ bs ) at is replaced by ( ψ bs ) at to infer ( χ i +1 ) at from ( χ i ) at .Note that the terms in s may have bound variables, and that a may be among the variables f b , but the term t has no bound variables, and the double substitutions in ( ψ bs ) at may beresolved as single substitutions.If the terminal conditional is obtained using an axiomatic rule (1)-(23), then it has aproof obtained by applying the corresponding axiom. If the terminal conditional is obtainedby the rule (5), then it has a proof obtained by applying axiom (5), and the running theproofs of the input conditionals below the new conjunction symbol. Similarly, if the terminalconditional is obtained by rule (8), then it has a proof obtained by running the proofs of theinput conditionals below the highest disjunction symbol, and then applying axiom (8).If the terminal conditional is obtained by the rule (11), then it has a proof obtained essen-tially by running the proof of the input conditional below the highest existential quantifier,and then applying axiom (11). Fix a proof χ ⇒ · · · ⇒ χ n of the input conditional φ ya ⇒ ψ . Let x be a variable that does not occur in this proof; then the sequence ∃ x : ( χ ) ax ⇒ ∃ x : ( χ ) ax ⇒ · · · ⇒ ∃ x : ( χ n ) ax is also a proof. The same observation may fail for y in place of x , because a may be inthe scope of a quantifier on y in some χ i . We now observe that ∃ x : ( χ n ) ax ⇒ ψ is anapplication of axiom (11), since neither x nor a occurs in χ n = ψ . We also have a proof of ∃ y : φ ⇒ ∃ x : ( χ ) ax : ∃ y : φ = ∃ y : ( φ ya ) ay = ∃ y : ( χ ) ay = ∃ y : (( χ ) ax ) xy ⇒ ∃ y : ∃ x : ( χ ) ax ⇒ ∃ x : ( χ ) ax We apply axiom (10) in the first inference; we apply axiom (11) in the second inference,since y is not free in ∃ x : ( χ ) ax .If the terminal conditional is obtained by the axiomatic rule (13), then it has a proof usingaxiom (13) and other axioms. If the terminal conditional is obtained by the rule (14), thenit has a proof obtained essentially by applying axiom (14), and then running the proof ofthe input conditional below the highest conjunction. Fix a proof χ ⇒ · · · ⇒ χ n of ψ ⇒ a s ∨ φ ya . Let x be a variable that does not occur in this proof; then the sequence ∀ x ∈ s : ( χ ) ax ∧ x ∈ s ⇒ ∀ x ∈ s : ( χ ) ax ∧ x ∈ s ⇒ · · · ⇒ ∀ x ∈ s : ( χ n ) ax ∧ x ∈ s is also a proof. Since neither x nor a occur in χ = ψ , the conditional ψ ⇒ ∀ x ∈ s : ( χ ) ax ∧ x ∈ s is an application of axiom (14). We also have a proof of ∀ x ∈ s : ( χ n ) ax ∧ x ∈ s ⇒ ∀ y ∈ s : φ : ∀ x ∈ s : ( χ n ) ax ∧ x ∈ s ⇒ ∀ y ∈ s : y ∈ s ∧ [ ∀ x ∈ s : ( χ n ) ax ∧ x ∈ s ] ⇒ ∀ y ∈ s : ( χ n ) ay ∧ y ∈ s ⇒ ∀ y ∈ s : (( y s ∨ φ ) ∧ y ∈ s ) ⇒ ∀ y ∈ s : φ (cid:3) We modify the list of logical axioms in figure 2 to remove the distinction between free andbound variables.
Corollary 6.3 ( ZFC ) . The above theorem holds also for the list of logical axioms in figure3, where we make no distinction between free and bound variables, i. e., each variable ispermitted to appear in a free or a bound position.Proof.
In the application of an axiom φ ⇒ ψ , we replace a substitution instance of φ by a substitution instance of ψ , so in figure 2 we may replace schematic term variables byschematic variable variables, and in fact, by simple variables. In our proof of completeness e have implicitly assumed a bijective correspondence between the free variables and thebound variables, as formally variables may occur as both free and bound in even a singleaxiom of the given theory T . In identifying corresponding pairs of variables, we must placesome restriction on axioms (11), (12), (14), (15), and (23) to preserve their validity. Itremains to show that we may find a proof of χ ⇒ χ ′ even with these restrictions. Since therestrictions are relevant only when the same variable appears both free and bound in thesame axiom, it is enough to find proofs of χ ⇔ ` χ whenever ` χ obtained from χ by replacingeach bound variable with a new variable, and likewise for χ ′ . This follows by induction oncomplexity. Let ` x be a new variable. ∃ x : φ ⇒ ∃ x : ∃ ` x : φ x ` x ⇒ ∃ ` x : φ x ` x ⇒ ∃ ` x : ∃ x : ( φ x ` x ) ` xx ⇒ ∃ ` x : ∃ x : φ ⇒ ∃ x : φ ∀ x ∈ z : φ ⇒ ∀ ` x ∈ z : ( ∀ x ∈ z : φ ) ∧ ` x ∈ z ⇒ ∀ ` x ∈ z : φ x ` x ⇒ ∀ x ∈ z : ( ∀ ` x ∈ z : φ x ` x ) ∧ x ∈ z ⇒ ∀ x ∈ z : ( φ x ` x ) ` xx ⇒ ∀ x ∈ z : φ (cid:3) The completeness theorem can also be proved by reducing classical logic to positivisticlogic via the cut-elimination theorem. This approach has the advantage that it justifies theuse of classical logic in a finitary universe, where the completeness theorem itself does nothold. The proof does not appear in these notes, but we include a description of the system L Σ ωω ( τ ), which forms an intermediate step in the argument.7. the base theory The positivistic approach rejects axioms exclusively on the basis of their logical complex-ity. Nevertheless, we will refer to a “base theory” that is intended to include uncontroversialaxioms about basic set-theoretic constructions. We shy away from fully specifying the basetheory, to a lesser degree out of principle, and to a greater degree out of practical necessity.The principle just mentioned is that the positivistic approach is not tied to any particulartheory. The practical consideration is that it is easy to misjudge the foundational assump-tions necessary for a given argument, particularly the quantity of induction. I have not hada chance to comb over the arguments to be certain of their foundational assumptions, but Ihope that the reader can agree that the arguments are indeed basic.At a minimum, our base theory includes
PRS , the theory of primitive recursive set func-tions, introduced by Rathjen [9] as an adaptation of
PRA to the set-theoretic setting. Prim-itive recursive set functions on pure sets are a natural generalization of primitive recursivefunctions on the natural numbers, first extensively studied by Jensen and Karp. I like a vari-ant of this formalization that takes pairing, union, projection, and ∆ separation as initialconstructions, and introduces new constructions using composition, image, and recursion;these axioms are given in figure 7.Generalizing the theorems for PRA , Rathjen showed that a Σ formula in the language ofset theory defines a primitive recursive set function if and only if KP − with Σ foundationproves that the formula is a function class; KP − is Kripke-Platek set theory with the foun-dation schema replaced by the much weaker set foundation axiom, i. e. regularity axiom.Rathjen also showed that a Π sentence in the language of set theory is a theorem of PRS if nd only if it is a theorem of KP − with Σ foundation. Instances of the Σ foundation schemaare not Π , so the latter theory is not positivistic. However, it is possible to present PRS as a positivistic theory in the language of set theory by coding the primitive recursive setfunctions by Σ formulas, and adding axioms expressing their totality. This fact is significantto us, because it shows that primitive recursive set functions are procedures as long as themembership relation is given by a procedure.Our base theory includes two natural axioms that do not follow from
PRS . The first is ∆ collection: ∀ x ∈ X : ∃ y : φ ⇒ ∃ Y : ∀ x ∈ X : ∃ y ∈ Y : φ. The second is the well-ordering principle: every set is equinumerous to a von Neumannordinal. The ∆ collection schema may be justified on the basis that the set X is an actualtotality, which may be surveyed, with witnesses for φ selected. The well-ordering principlemay be justified on the basis that while we talk about pure sets, the actual objects of ouruniverse are well-ordered sequences of bit-values, i. e., sets of ordinals; thus, not only can aset be well-ordered, but on the “machine level” a set is represented by a well-ordered list ofits elements. We will rarely consider universes that do not satisfy the well-ordering principle.With the addition of the ∆ collection schema, we see that our base theory includes KP − .It certainly does not include all of KP , because most instances of the full foundation schemaare not expressible by a positivistic conditional. However, the base theory is sufficient forseveral of the most basic consequences of KP . Specifically, the base theory is sufficient for theproofs of the Σ reflection schema, the Σ collection schema, and the ∆ separation rule, whichare given in Barwise [1, s. I.4]. The Σ reflection schema expresses that each Σ formula φ isequivalent to ∃ x : φ ( x ) , where φ ( x ) is obtained from φ by replacing every unbounded existentialquantifier ∃ v by ∃ v ∈ x . The Σ collection schema is just like the ∆ collection schema givenabove, but for all Σ formulas φ . The ∆ separation rule is the rule that if we derive that Σformulas φ ⊤ and φ ⊥ are complementary, then we can derive that each set has a subset ofelements satisfying φ ⊤ : ⊤ ⇒ φ ⊤ ∨ φ ⊥ φ ⊤ ∧ φ ⊥ ⇒ ⊥⊤ ⇒ ∃ y : ( ∀ z ∈ y : z ∈ x ) ∧ ( ∀ z ∈ x : φ ⊥ ∨ z ∈ y ) ∧ ( ∀ z ∈ x : φ ⊤ ∨ z y )The ∆ separation principle as stated in Barwise section I.4 [1] is not expressible by a positivis-tic conditional, but its proof shows that the the above rule is admissible in every extensionof the base theory. The ∆ separation rule allows induction for ∆ formulas: the followingrule is admissible for every extension of the base theory: ⊤ ⇒ φ ⊤ ∨ φ ⊥ φ ⊤ ∧ φ ⊥ ⇒ ⊥∃ x : φ ⊤ ⇒ ∃ x : φ ⊤ ∧ ∀ y ∈ x : ( φ ⊥ ) xy The axiom of infinity, expressing the existence of a limit ordinal, is not formally an ax-iom of our base theory, because we wish to include the universe of hereditarily finite sets H ω . In a universe that does contain infinite sets, it is natural to investigate infinitary logic,on the intuition that it is appropriate to the ideal agent for that universe. Consequently,cut-elimination for infinitary logic will be our main tool in a number of the arguments thatfollow. Feferman obtained a proof-theoretic proof of cut-elimination for an infinitary systemthat permits countable conjunctions and disjunctions. The proof appears highly construc-tive, and I expect it to go through in the extension of PRS just described, with only minor odifications. It follows a typical framework of a syntactic proof of cut-elimination, by in-dunction on cut-rank, with a subinduction on derivation height, and with subsubinductionon the surreal sum of heights of subdervations, when the cut-rule is encountered. In sec-tion 10, we will apply cut-elimination for a variant of Feferman’s system that includes moreinitial sequents; I believe this variant requires only minor additions to Feferman’s argument.In section 11 we will apply cut-elimination for the intuitionistic version of Feferman’s system.Feferman explicitly writes that his development does not “go into” non-classical fragments;however, I believe that Feferman’s argument can be adapted to intuitionistic logic in theusual starightforward way.In the unexpected situation that Feferman’s argument does not go through in the basetheory we have described, we will extend PRS still further, to accommodate it. This ex-tension should not significantly undermine the credibility of our base theory, because Fe-ferman’s procedure is certainly intuitively possible. At the very least, Feferman’s argumentestablishes the consistency of cut-elimination with our base theory; the universe H ω ofhereditarily countable sets is our canonical model for the base theory with the axiom of in-finity. Thus, countable cut-elimination is effectively an axiom of our base theory. There arecut-elimination results for systems that permit arbitrary infinitary conjunctions and disjunc-tions, such as those in Takeuti’s book. However, we will work with variants of Feferman’ssystem, and we will generally apply cut-elimination for these variants under the additionalassumption that every set is countable.It is natural to extend our base theory with reflection principles. The validity principlefor the base theory is the standard example: it expresses that the conclusion of any proofwhose assumptions are true is itself true. A set-theoretic variant expresses that each set is anelement of some transitive model of the base theory. Such a transitive model is automaticallya model of KP . In fact, we obtain a conservation result for KP , since every Π formula in thelanguage of set theory that is true in each of these transitive models applies to the universeas a whole.If we stretch our relaxed attitude toward axioms to its extreme, we might accept all theconditionals modelled by some model universe as axioms, and investigate the classical first-order sentences that follow from these axioms.8. three model universes We examine three model universes, models of the base theory: the levels H κ for κ = ω , for κ = ω , and for κ an inaccessible cardinal. We might identify these models universes withfinitist, predicativist, and realist foundations.8.1. H ω . The theory that we obtain by adding the axiom that each set if finite, i. e.,equinumerous to a natural number, is equivalent to
PRA , the theory of primitive recursivearithmetic, but this takes some work to prove. The Ackermann coding is a simple bijectionbetween hereditary sets and natural numbers that is a primitive recursive set function. Wemay formulate
PRA itself as positivistic theory; the axioms of postivistic logic make noassumptions about the binary relation ∈ , so we may simply replace it with < . The axiomsof PRA may be naturally formulated using conditionals; for example we may render ∆ induction as φ (0 , z ) ∧ ∀ x < y : ˜ φ ( x, z ) ∨ φ ( x + 1 , z ) ⇒ ∀ x < y : φ ( x ) , or each ∆ formula φ , with the negation ˜ φ obtained by switching dual connectives and pred-icate symbols. The Parsons-Friedman theorem [11, s. IX.3] implies that every conditionalprovable in WKL is positivistically provable from the axioms of PRA .The original formulation of
PRA is essentially positivistic, in the sense that while it isnot prima facie positivistic, it can be easily adjusted to be positivistic. The reason it isnot, strictly speaking, positivistic is that the free variables in a deduction of
PRA cannot beunderstood as having the same values throughout the proof. It is clear that the inductionvariable in the premise of the induction rule must be read as being implicitly universallyquantified in order to justify the application of the induction rule. Thus, a deduction of
PRA consists of Π sentences, rather than Σ sentences. However, each deduction of
PRA can beread as a positivistic proof if it is read backwards: if the terminal formula fails, then one ofthe predecessors used to derive it fails, and so on, until the set of possibly failing formulasis found to be empty, i. e., a contradiction is derived.We remark that Takahashi [12] identified Σ sentences in the language of set theory as themeaningful sentences in the context of finitistic foundations based on hereditarily finite sets.However, Takahashi’s deductive system is not positivistic, again because its free variables donot maintain fixed values over the course of deduction. Specifically, a formula of the form ∀ x ∈ y : φ can be derived from a deduction that assumes x ∈ y and proves φ . This boundedvariant of the universal generalization rule is a covert appeal to the validity of the system.While these appeals to validity are ultimately innocuous, it is surely a basic principle ofmodern mathematical logic that no system should assume its own validity.8.2. H ω . We obtain a basic theory of hereditarily countable sets by adding the axiom ofinfinity, concretely the existence of a least nonzero ordinal that is not a successor, and theaxiom that every set set equinumerous to this ordinal, or to a smaller ordinal. The positionthat every set is countable is well established; I suppose it is associated most strongly withpredicativism, and related denials of the powerset axiom. I am drawn to the conclusion thatindeed, if there are infinite sets, then they are all countable, but I am not persuaded bypredicativist appeals, possibly because I do not understand them.My intuition of the positivistic approach is that, though it is conservative with regard tosemantics, it is liberal with regard to ontology. Whatever the objects in the universe, theyare taken to exist in themselves, rather than be generated by a mental process. Furthermore,although the positivistic approach has the formal features of potentialism, I do not imaginethe emergence of objects that seems to underly potentialist intuitions. When I imaginea mathematical universe, I imagine its objects as existing fully; the meaninglessness of auniversal proposition does not arise from a dynamic growth of the universe, but rather froman inability to survey all of its objects. Ultimately, this departure from the narrative ofpotentialism may be entirely aesthetic, i. e., of no practical consequence.I will argue that the full universe of sets looks very much like H ω , but the reader isencouraged to keep a mental distinction between H ω , and the full universe of sets. The modeluniverse H ω is an initial segment of the “ambient” transitive model of ZFC in which we havebeen investigating the notion of a model universe. The position that every set is countable,for which I will argue, is not an exclusive position that rejects the many uncountable setsthat are used in ordinary mathematics; rather, it is an inclusive position that accepts theexistence of a bijection between the set of natural numbers and its powerset in this transitivemodel. s a model universe, H ω has a number of attractive features. It is the only model universeof the form H κ that satisfies the two completeness principles given in section 10; infinitarylogic is well-behaved, and forcing for transitive models is a provable principle via Rasiowa-Sikorski lemma. Furthermore, by the L´evy absoluteness principle, every Σ( L ωω (= , = , ∈ , ))conditional valid in the ambient model of ZFC is also valid in H ω , so within the ambientmodel, we cannot express that the ambient model contains uncountable sets using Σ( L ωω (= , = , ∈ , )) conditionals.8.3. H κ . If κ is an inaccessible cardinal, then H κ is a model of ZFC . However, the idealagent for H κ may not accept the validity of ZFC in that universe, because a number of itsaxioms have high logical complexity. It is of course possible to formulate any theory as a Π theory, or even as a Π theory by adding nonlogical symbols, essentially Skolem functions,with the implicit claim that these nonlogical symbols denote possible procedures. However,if we take our notion of procedure to be the one given by the language of set theory, i. e., Σ ( L κω (= , = , ∈ , )), then there is no powerset procedure in H κ : Proposition 8.1 ( ZFC ) . Let κ be a strongly inaccessible cardinal. The powerset operationis not definable in H κ by a Σ ( L κω (= , = , ∈ , )) formula.Proof. Suppose that there is a Σ ( L κω (= , = , ∈ , )) formula φ ( x, y ) such that H κ | = φ [ a, b ] iff b is the powerset of a , for all a, b ∈ H κ . The formula φ is in H λ + for some regular λ < κ .The transitive set H κ = V κ satisfies the powerset axiom, so certainly H κ | = ∃ y : φ ( λ, y ).Truth for Σ ( L κω (= , = , ∈ , )) formulas is definable by a Σ( L ωω (= , = , ∈ , )) formula, so H κ | = T [ ∃ y : φ ( λ, y )]. Since H λ + reflects Σ formulas, we have that H λ + | = T [ ∃ y : φ ( λ, y )], so H λ + | = ∃ y : φ ( λ, y ). We conclude that there is an element p ∈ H λ + such that H λ | = φ [ λ, p ], andtherefore H κ | = φ [ λ, p ]. By assumption, p is the powerset of λ , contradicting that p is anelement of H λ + . (cid:3) If we view the powerset operation as given by a procedure, we may add it to our vocabulary,together with other such operations. Our general method for adding procedures withoutadding objects is to fix a coding of elements outside the model by elements of the model; wethen expand the vocabulary of that model by a truth predicate, interpreted as composing thegenuine truth predicate with that coding. In this way, we may include the metamathematicsof a model of
ZFC into that model of
ZFC . Note that we are not simply adding a truthpredicate in the usual way, because our truth predicate may be applied to formulas thatcontain that truth predicate, but at the expense restricting our reasoning to positivistic logic.If we code K ( L ωω (= , = , ∈ , )) sentences with parameters in the model as Σ ( L ωω (= , = , ∈ , ))sentences, by relativizing them to that model, then the added truth predicate will apply tothese sentences also. Thus, we may retain classical logic for all the usual sentences of thelanguage of set theory, while expanding the model with a self-applicable truth predicate.9. truth, correctness, and validity It is a basic fact that the truth of Σ sentences is definable by a Σ( L ωω ) formula; thisobservation applies to Σ ( L ∞ ω ) sentences just as it does to Σ ( L ωω ) sentences. Thus, thereis a truth predicate in any model universe. For simplicity, we formalize the theory of truthfor a vocabulary that contains no function symbols. We may generalize this developmentreadily by adjusting our definition of verification to include a computation of values for theterms appearing in the given formula. e distinguish between various notions of rightness in the following way. A Σ ( L ∞ ω )sentence it true if and only if it has a verification in the sense below. As we have emphasized,truth is an internal notion that can be formalized in a given mathematical universe; incontrast, the remaining notions of rightness are expressed by contionals, which may beproved, or taken as axioms, or verified in a model universe , but which cannot be in themselvestrue. The correctness of an intentional theory τ is the reflection principle “if φ is true and φ ⇒ φ is an application of an axiom of τ , then φ is true”. By contrast, the validity ofan intentional theory τ is the reflection principle “if χ is true and χ ⇒ χ is provablein τ , the χ is true”. The correctness of τ affirms the axioms of τ , whereas the validity of τ affirms theorems of τ . When the induction rule is not admissible, the former does notprove the later. Our use of the word “validity” is faithful its philosophical meaning: if theassumption of a proof is true then its conclusion is also true; however, it clashes with thestandard metamathematical meaning. We will use the phrase “logical validity” for the latermeaning. Definition 9.1 (base theory) . We define a verification V to be a set of Σ ( L ωω ) sentencesthat is closed in the following sense:(1) If φ ∧ ψ is in V , then φ and ψ are in V .(2) If φ ∨ ψ are in V , then φ is in V or ψ is in V .(3) If ∃ x : φ ( x ) is in V , then a sentence of the form φ ( a ) is in V .(4) If ∀ x ∈ b : φ ( x ) is in V , then for each element a ∈ b , the formula φ ( a ) is in V .(5) If a ∈ b is in V , then a ∈ b . (Likewise for the other predicate symbols.)Let φ be a Σ ( L ωω ) sentence. We define a verification of φ to be a verification that contains φ . We define φ to be true iff it has a verification; we write T ( φ ).There is a straightforward generalization of this definition that includes sentences withfunction symbols from a finite vocabulary: a verification should include equalities that com-pute the values of terms. It is more difficult to generalize this definition to an infinitevocabulary, due to the parenthetical remark in condition (5); any definition uses only a fi-nite subset of the vocabulary. In the case of PRS , the algorithms implicit in the primitiverecursive function symbols may be unravelled by an appropriate definition of verification.Alternatively, an equation in
PRS may be verified by verifying the corresponding Σ sentencein the language of set theory.Our base theory is sufficient for verifying the standard semantic properties of the truthpredicate: Proposition 9.2 (base theory) . For all Σ ( L Ω ω ) sentences φ and ψ ,(1) T ( φ ∧ ψ ) ⇔ T ( φ ) ∧ T ( ψ ) (2) T ( φ ∨ ψ ) ⇔ T ( φ ) ∨ T ( ψ ) (3) T ( ∃ v : ψ ( v )) ⇔ ∃ a : T ( ψ ( a )) (4) T ( ∀ v ∈ b : φ ( v )) ⇔ ∀ a ∈ b : T ( φ ( a )) (5) T ( a ∈ b ) ⇔ a ∈ b (Likewise for the other predicate symbols.)Proof. Immediate from the definition. (cid:3)
Lemma 9.3 (base theory) . Let χ and χ be Σ ( L ωω ) sentences. Assume χ ⇒ χ is aninstance of a logical axiom, and χ is true. Conclude χ is true. roof. There is a separate argument for each logical axiom schema. As an example, wesuppose that χ ⇒ χ is an instance of the schema x ∈ z ∧ ∀ x ∈ z : φ ⇒ φ . Thus, χ ⇒ χ is equal to a ∈ c ∧ ∀ x ∈ c : φ ⇒ φ xa for some Σ ( L ωω ) formula φ with a singlefree variable x , and objects a and c . Let V be a verification for a ∈ c ∧ ∀ x ∈ c : φ . The set V contains a ∈ c , so a ∈ c . The set V also contains ∀ v ∈ c : φ , so it also contains φ xb for all b ∈ c . Thus, V contains φ xa , so V is a verification for φ xa .As another example, we suppose that χ ⇒ χ is an instance of the schema x = y ∧ φ vx ⇒ φ vy . Thus, χ ⇒ χ is equal to a = b ∧ φ va ⇒ φ vb for some Σ ( L ωω ) formula φ with a singlefree variable v , and objects a and b . Let V be a verification of a = b ∧ φ va . Then V contains a = b , so a and b are the same object. It follows that the symbols a and b are equal, so theformulas φ va and φ vb are also equal. Thus V is a verification of both φ va and φ vb . (cid:3) Lemma 9.4 (base theory) . Let χ and χ be Σ ( L ωω ) formulas. Assume that χ ⇒ χ isan application of a logical axiom, and that χ is true. Conclude χ is true.Proof. Let φ ⇒ ψ be the axiom that we are applying. Examining the proof of the preceedinglemma, we find a primitive recursive function that takes verifications of instances of φ toinstances of ψ . Let V be a verification of χ . Let V be the result of including, for eachinstance of φ in V , the verification for the corresponding instance of ψ , and adjoining, foreach formula having an instance of φ as a subformula, the result of applying φ ⇒ ψ .Examining each element of V we find that it is a verification. (cid:3) Now we show the validity of positivistic logic.
Proposition 9.5 (base theory) . Let χ and χ ′ be Σ ( L ωω ) formulas. Assume that χ ⇒ χ ′ has a positivistic proof using only logical axioms, and that χ is true. It follows that χ ′ istrue.Proof. Let V be a verification of χ . The procedure taking V to V in the proof of thepreceding lemma is primitive recursive, so there is a function that assigns a verification toeach sentence of the proof. We establish that each value of this function is a verificationusing ∆ induction. (cid:3) Each step of a proof using only the logical axioms is uniquely an application of a logicalaxiom, except for trivial applications of logical axiom (21). However, where that is not thecase, we may ask that the axiom being applied and the site of its application be specified ateach step, as part of the definition of the proof.
Lemma 9.6 (base theory) . Let φ ( v , . . . v n ) be a Σ ( L ωω ) formula. The following is a theo-rem of the base theory: “Let a , . . . , a n be objects. Assume φ ( a , . . . , a n ) is true. Conclude φ ( a , . . . , a n ) .” The converse is also a theorem. Our base theory does not have any axioms that include parameters, so the parameters thatappear in our formulas are essentially free variables. Thus, the assumption that φ ( a , . . . , a n is true should be expressed with an explicit description of φ , rather than using a parametersymbol. Later, we will supplement the base theory with all true Σ sentences, on the intuitionthat since truths may be verified, truths may be assumed, and our present caveat will bemoot. We avoid supplementing the base theory now, to keep it countable. (Of course, wemay add parameters just for the finite sets, which would make for a more aestheatic theory,but which would also slightly complicat our arguments.) roof of lemma. We apply induction on syntax to the the semantic properties of the Truthpredicate. The construction of the proofs of the two theorems are primitive recursive. (cid:3)
Theorem 9.7 (base theory) . Let τ be a theory extending the base theory, and let φ ⇒ ψ beone of its theorems. The following is a theorem of τ : “Let χ and χ be Σ ( L ωω ) sentences.Assume that χ is true, and that χ ⇒ χ is an application of φ ⇒ ψ . Conclude that χ is true.”Proof. We argue in τ : “Let χ and χ be Σ ( L ωω ) sentences. Assume that χ is true, andthat χ ⇒ χ is an instance of φ ⇒ ψ . By definition of instance, there are objects a , . . . , a n such that χ is φ ( a , . . . , a n ), and χ is ψ ( a , . . . , a n ). Since φ ( a , . . . , a n ) is true, φ ( a , . . . , a n ), so ψ ( a , . . . , a n ). We conclude that ψ ( a , . . . , a n ) is true.” This argumentincludes the given proof of φ ⇒ ψ , and it includes the truth schema established in thepreceding lemma.We argue again in τ : “Let χ and χ be Σ ( L ωω ) sentences. Assume that χ is a true,and that χ ⇒ χ is an application of φ ⇒ ψ . Let V be a witness of χ . Let V be theresult of including, for each instance of φ in V , a verification of the corresponding instanceof ψ , and adjoining, for each formula having an instance of φ as a subformula, the result ofapplying φ ⇒ ψ . We may check that V is a verification by examining its elements. Weconclude that χ is true.” We insert our previous argument in τ when we we include, foreach instance of φ in V , a verification of the corresponding instance of ψ . That argumentdoes not give us a primitive recursive procedure for obtaining verifications of instances of ψ .This argument obtains the verification V by applying Σ collection. (cid:3) The second part of the proof demonstrates that the correctness of a theory τ is equivalentto “Let χ ⇒ χ be a substitution instance of an axiom of τ . Assume χ is true. Conclude χ is true.” Corollary 9.8 (base theory) . Let τ be a Σ ( L ωω ) theory extending the base theory, and let ˜ τ be τ together with the conditional expressing the correctness of τ . The theory ˜ τ proves thecorrectness of ˜ τ .Proof. The theory ˜ τ proves the correctness of τ by definition. The theory ˜ τ proves thecorrectness of the new axiom by the above theorem, because the new axiom is trivially oneof its theorems. (cid:3) The above corollary is at the same time reassuring and alarming. It is reassuring that wecan prove the correctness of ˜ τ without additional assumptions, but what of G¨odel’s secondincompleteness theorem? G¨odel’s second incompleteness theorem is most certainly provablein our base theory. The kernel of its proof is so elementary, that it applies to any systemthat allows even the most basic finitary procedures.The usual proofs of G¨odel’s first and second incompleteness theorems need to be modifiedonly slightly to fit the positivistic framework; the G¨odel sentence, which is nonpositivistic,must be replaced by its logical complement. The heart of the matter is that diagonalization isonly possible for positivistic predicates. Diagonalization yields a sentence σ that is logicallyequivalent to P ( σ ⇒ ⊥ ).There is a proof from σ to P ( ⊤ ⇒ ⊥ ): “Assume σ . It follows that P ( ⊤ ⇒ σ ), because σ is a positivistic arithmetical sentence, and that P ( σ ⇒ ⊥ ), by construction of σ . Concludethat P ( ⊤ ⇒ ⊥ ).” Assume consistency, i. e., P ( P ( ⊤ ⇒ ⊥ ) ⇒ ⊥ ). We have just shown ( σ ⇒ P ( ⊤ ⇒ ⊥ )). It follows that P ( σ ⇒ ⊥ ), i. e, that σ . Since σ is a positivisticarithmetical sentence, we infer P ( ⊤ ⇒ σ ), so we may conclude P ( ⊤ ⇒ ⊥ ) as desired.The resolution of this apparent contradiction is that we have shown the correctness of ˜ τ ,and not its validity. Thus, we do not have a proof of consistency. We cannot apply ourargument for the validity of the theory of positivistic logic, because there is no primitiverecursive function producing witnesses. We cannot appeal to Σ induction because it isn’tamong our axioms; it cannot be among our axioms because of its high logical complexity. Itis generally not admissible.Thus, ˜ τ cannot prove that every proof is valid; it can only prove that every step is correct.This, however essentially guarantees that every proof that we write down is valid: for everynatural n , we have a simple proof of the validity of proofs of fewer than n steps, linearly in n . This is a very natural variant of Pudl´ak’s result [8] that proofs of finitary consistencyexist with size linear in the size of the proofs considered.In the finitary setting we may take τ to consist of the theorems of PRA . Thus, ˜ τ is obtainedby adding the validity of PRA . The theory ˜ τ is equivalent to PRA together with the validityof
PRA . The theory
PRA proves Parson’s theorem, and it proves the reduction of
WKL to IΣ ; so, the theory ˜ τ proves its own correctness and the validity of WKL . When I adoptthe mindset of a finitist, I find the theory ˜ τ just described to be quite acceptable, though itdoes go a hair further than PRA itself, which, after Tait’s analysis, is often identified withfinitism. 10. the universe of pure sets
We have so far considered model universes within a cumulative hierarchy described by theaxioms of
ZFC . We now consider the genuine universe of pure sets: the totality of all potentialpure sets. This universe of all pure sets may be like one of the models we have considered;it may consist of hereditarily finite sets, or of hereditarily countable sets, or it may permitthe powerset construction. Since the axioms of
ZFC are not all Π , we might suppose thatour investigation of these model universes occurred within some transitive model of ZFC .Our speculation about the nature of the full universe is guided by two principles. First,the truth of any predicate should be determined by examining membership following somelogical scheme. Thus, the totality of procedures of the full universe should be essentiallyexhausted by Σ formulas, with no additional nonlogical symbols. Each Σ ( L ∞ ω ) formula isequivalent to a Σ ( L ωω ) formula, so our notion of procedure does not depend on this choiceof logic.Our second principle is that the universe should be complete with respect to the setsthat may exist. The simplest interpretation of this principle is the completeness of some L ∞ ω system, e. g., LK ∞ ω . A special case of this principle is more deeply compelling: weask that every consistent description of a unary predicate on a transitive set is satisfied bysome subset of that transitive set. This is analogous to constructibility, except instead ofexhibiting the predicate that defines the new set, we describe it indirectly. Definition 10.1 (base theory) . The set completeness principle: Let A be a transitive set.Let S be a unary predicate symbol. Let T be a K ( L ∞ ω (= , ∈ , S )) theory with parametersfrom A , that includes the following axioms.(1) The equality axioms:(a) ∀ x : x = x b) ∀ x : ∀ y : x = y ↔ y = x (c) ∀ x : ∀ y : ∀ z : x = y ∧ y = z → x = z (d) ∀ x : ∀ y : x = y ∧ S ( x ) → S ( y )(2) The atomic axioms:(a) a ∈ b , for a ∈ b (b) ¬ a ∈ b , for a b (c) a = b , for a = b (d) ¬ a = b , for a = b (3) The content axiom: ∀ x : W a ∈ A x = a Conclude that either T is inconsistent for LK ∞ ω (figure 4), or there is a subset B ⊆ A suchthat ( A, = , ∈ , B ) is a model of T .For an arbitrary structure: the equality axioms are the substitution axioms for atomicformulas, together with the equivalence relation axioms for equality; the atomic axioms arethe true atomic sentences, and the true negated atomic sentences; and, the content axiomis the universally closed disjunction expressing that every element is named by some closedterm. The inconsistency of T for LK ∞ ω means the derivability of V T ⊢ ∅ in LK ∞ ω . Definition 10.2 (base theory) . The model completeness principle: Let T be a K( L ∞ ω ( S ))a theory whose vocabulary S consists of finitary function and predicate symbols. Concludethat either T is inconsistent for LK ∞ ω , or T has a model.The model completeness principle is not as well motivated by the notion of a full universe ofsets, but it may be motivated by the notion of a full universe of structures. The very presenceof infinite sets in mathematical discourse is arguably to ensure that any consistent finitarytheory has model. Notably, the model completeness principle determines the cardinalitystructure of the universe: infinite sets exist, and every set is countable.Even the weaker set completeness principle implies that every set is countable. Unlikethe model completeness principle, the set completeness principle holds in the model universe H ω of hereditarily finite sets. In the presence of infinite sets, the set completeness principleproduces generic sets. Let P be a forcing partial order, let A be its transitive closure, andlet C be a nonempty collection of dense subsets of P . Forcing semantics shows that thefollowing axioms are consistent with the previously given axioms:(4) ¬ S ( a ), for a P (5) S ( p ) → S ( q ), for p, q ∈ P such that p ≤ q (6) S ( p ) ∧ S ( q ) → W r ≤ p,q S ( r ), for p, q ∈ P (7) W p ∈ D S ( p ), for D in C In the usual way, we obtain a surjection from ω onto any given set. Theorem 10.3 (base theory) . The model completeness principle is equivalent to the setcompleteness principle with the axiom of infinity.
The proof is unexpectedly long for such an expected result. The main difficulty is that thederivability of a sequent is a priori not decidable. Thus, we cannot form sets of derivablesequents. We cannot appeal to Barwise completeness, because the universe is not assumedto be a transitive set in a model of
ZFC . Proof.
The forward direction is simpler. The model completeness principle implies the setcompleteness principle because the equality axioms, the content axiom, and the atomic xioms ensure that any model is canonically isomorphic to an expansion of ( A, = , ∈ ). Themodel completeness principle implies the axiom of infinity because our base theory is morethan sufficient to show that Robinson arithmetic Q is consistent, and this theory cannot havea finite model as the successor function is an injection that is not a surjection.The backward direction is more involved. We fix a set T of K( L ∞ ω ( S )) sentences, for someset S of finitary function and predicate symbols; we will show that T is inconsistent or ithas a model. Without loss of generality we may assume that S consists entirely of predicatesymbols, by replacing function symbols by predicate symbols, as usual. Thus, we assumethat S = { P , P , . . . } is a countable set of predicate symbols. Let A be the set of hereditarilyfinite sets. We can interpret T as a set of K ( L ∞ ω (= , ∈ , S )) sentences, with parameters from A , by interpreting each atomic formula P n ( x , . . . , x m ) as the formula “there is an objectthat is an ( m + 1)-tuple ( n, x , . . . , x m ) that satisfies S ”.We define T to be the K ( L ∞ ω (= , ∈ , S )) theory with parameters from A , obtained byadding the equality axioms, the content axiom, and the atomic axioms to these interpretedaxioms of T . If the vocabulary S includes a binary equality relation symbol, we treat thisrelation symbol as distinct from the equality symbol in T ; thus it is an equivalence relationthat may fail to satisfy substitution for some K ( L ωω (= , ∈ , S )) formulas. We code tuplesas partial functions whose domain is a finite ordinal; we exclude 0-tuples. Note that everyelement of A that is a tuple is uniquely such, that no tuple is a natural number, and that A has infinitely many elements that are neither tuples nor natural numbers. By the setcompleteness principle, the theory T either is inconsistent or is modelled by some subset of A . By construction of T , any such model may be interpreted as a model of T ; since anyequality symbol in the vocabulary of T is treated as a nonlogical symbol, we expect T to havean infinite model; since the set completeness principle implies that every set is countable,we can expect T to have a countable model. It remains to argue that if T is inconsistent,then T is inconsistent. We assume that T is inconsistent, and we proceed by repeatedlymodifying T in minor way, and checking that the result is still be inconsistent.We define T by enlarging the vocabulary of T to include an n -ary n -tuple formation func-tion symbol for each positive integer n , and then interpreting the axioms of T as we have donein the definition of T , but using these new function symbols to translate the formulas of T ;we include atomic axioms and substitution axioms with these new function symbols. Becausewe are working with the infinitary deductive system LK ∞ ω , the atomic axioms, together withthe content axiom, are enough to prove every true K ( L ∞ ω (= , ∈ , S, h·i , h· , ·i , h· , · , ·i , . . . )) sen-tence, and in particular that the new function symbols do form tuples. Thus T derives everyaxiom of T , so T is inconsistent.We define T by excluding all parameters that name tuples in A , and removing the axiomswhere they occur from T , rewriting the content axiom to express that every object is equalto some closed term. This step may be achieved simply by replacing each parameter thatnames a tuple in A by the closed term expressing that tuple in terms of parameters thatdo not name tuples in A . This second presentation shows that we may obtain a deductionof the inconsistency of T , by taking a deduction of the inconsistency of T , and performingsuch a replacement throughout. We note that every element of A is the value of a uniqueterm whose parameter symbols do not name tuples in A .We define T by excluding the membership relation symbol ∈ , and removing the atomicand substitution axioms for ∈ from T . We obtain a deduction witnessing the inconsistency f T by applying the interpolation theorem, with sentences containing ∈ on the left, andsentences containing S on the right. The usual proof of Craig’s interpolation theorem fromthe cut-elimination theorem applies. Recall that we rewrite all the sentences with onlyatomic formulas negated, and then we obtain a cut-free derivation where negation is appliedimmediately, by pushing applications of the negation rules in a cut-free derivation towardsinitial sequents. We then obtain interpolants recursively over the proof tree; negation is onlyapplied in the familiar case where we have sequents of literals, and after negation is applied,interpolants are simply combined in the obvious way. The interpolant we obtain is a sentencethat contains neither ∈ nor S , i. e., whose only only predicate symbol is equality (as wellas ⊤ and ⊥ ). The sentences appearing on the left are true sentences about the structure( A, = , ∈ ), so the interpolant is also true about this structure. Thus, the interpolant may bededuced using just the atomic axioms and substitution axioms for =, and therefore, from T . Bringing the sentences on the right of our sequent back to the left, we conclude that T is inconsistent.We will apply the cut-elimination theorem to the deduction of a contradiction in T .There is a standard variant of cut-elimination that makes accommodations for equality; itimplements the equality axioms as initial sequents. This is the approach taken by Fefermanin his syntactic proof of infinitary cut-elimination [4]. The deductive system without equalityis thus extended by two rules: the first deduces a sequent of the form ∅ ⊢ t = t from nothing;the second deduces sequents such as s = s , t = t , t = t , P ( s , t ) ⊢ P ( s , t ) fromnothing, with the set of equalities in the antecedent called an equality chain. I believe thatFeferman intended to include the the reverse inequalities s i +1 = s i in his definition of anequality chain, as there is otherwise no way to derive the sequent x = y ⊢ y = x withoutusing cut. We will apply a stronger variant of cut-elimination that implements the trueinequalities between closed terms as initial sequents. Specifically, for any pair of distinctclosed terms s () and s ′ (), we allow s () = s ′ () ⊢ ∅ as an initial sequent. To eliminate the cut rule, we also allow initial sequents whose an-tecedents consist of equality chains connecting distinct closed term s () and s ′ (), and whoseconsequent are empty. These new additions require only a minor addition to Feferman’sproof, of the same kind as is made for the other initial sequents.Thus, we have a deduction in this extended system of the sequent ∀ x : _ s () x = s () , ^ Atomic , ^ Translations ⊢ ∅ where the disjunction is taken over all closed terms s (), and Atomic is the set of all atomicaxioms, and Translations is the set of translations of the axioms of T . Since the parametersnaming atomic tuples are not in T , the atomic axioms are equalities between equal closedterms, and inequalities between distinct closed terms. Both kinds of sentences follow fromour initial sequents, so by the cut rule, we have a deduction of ∀ x : _ s () x = s () , ^ Translations ⊢ ∅ . Applying the cut-elimination theorem to this derivation, we obtain a deduction that is cut-free. We may assume that any disjunction W x = s () becomes universally closed immediatelyafter it forms on the left. e now show by induction over the fixed cut-free derivation that each sequent in thederivation has a derivation even when the content axiom ∀ x : W x = s () is excluded fromits antecedent. The only nontrivial steps are where the content axiom is formed by the leftdisjunction and left universal quantification rules; there is nothing to do if it is formed by theleft weakening rule. Assume that the content axiom is just formed by successive applicationsof the left disjunction rule and the left universal quantification rule, producing some sequent ∀ x : W x = s () , Γ ⊢ ∆ in the fixed cut-free derivation. Write Γ for Γ with all instances ofthe content axiom removed. By the induction hypothesis, the sequent t ( x , . . . , x n ) = s () , Γ ⊢ ∆has a derivation for some term t ( x , . . . , x n ) and all closed terms s (). The formulas in Γ and∆ are equalities and substitution instances of subformulas of translations of axioms of T .Since Γ and ∆ are finite, and infinitely many parameters are excluded from translations ofaxioms of T , there is a parameter a that does not appear in any formula in Γ or ∆. Thus,we have derivations of the following sequents: t ( x , . . . , x n ) = a, Γ ⊢ ∆ ∃ y : t ( x , . . . , x n ) = y, Γ ⊢ ∆Γ ⊢ ∆By induction, we obtain a derivation of V Translations ⊢ ∅ , and therefore a cut-free derivationof this sequent. Since equality does not occur in any translation of an axiom of T , thesubformula property implies that this sequent may be derived in the equality-free system.We define T to be the theory whose axioms are translations of the axioms of T ; we haveshown that its inconsistency sequent V Translations ⊢ ∅ has a cut-free derivation. Everyatomic subformula in the translation of an axiom of T is of the form S (( n, x , . . . , x m ))for some integers n and m , so every atomic subformula of any formula in any sequent inthe cut-free derivation is a substitution instance of S (( n, x , . . . , x m )). We now follow theintuition that we may treat any function symbols appearing below a top-most functionsymbol as distinct from each top-most function symbol, and that these distinct new functionsymbols may name constant functions; so, we may replace any nonvariable term below atop-most function symbol by the same new constant symbol. Formally, we simply replaceevery nonvariable term appearing as the argument of a top-most function symbol after thefirst position with the same new variable, and check that the cut-free deduction is still acut-free deduction.We now have a cut-free deduction of inconsistency in T with the property that everyatomic subformula is of the form S (( n, v , . . . v m )) for some natural number n and variables v , . . . , v m . We may simply replace each such formula by P n ( v , . . . , v n ) to obtain a proof ofincoconsistency in T . (cid:3) The principle that every set is countable is not as restrictive as it appears. Under
ZFC ,every recursively enumerable K( L ωω (= , ∈ )) theory that has a transitive model, has a count-able transitive model. Thus, under various standard large cardinal assumptions, there arecountable transitive models of ZFC satisfying various standard large cardinal assumptions.So, our base theory with the model completeness principle is consistent with the existenceof such transitive models. he same sort of argument may be made for the axiom of constructibility. If a recursivelyenumerable K( L ωω (= , ∈ )) theory has a transitive model, it has a countable transitive model,so by the Shoenfield absoluteness theorem, it has a countable transitive model in L . Theset of hereditarily countable sets in L models our base theory, and the model completenessprinciple, and the axiom of constructibility. I do not view the axiom of constructibility to beas natural an axiom for the full universe of pure sets as the completeness principles discussed,but it does preserve some features of the standard picture of the universe. The axiom ofconstructibility stratifies the universe of sets into a cumulative hierarchy, and I suppose thatit expresses a form of transfinite predicativism. In this setting, it is natural to consider “largeordinal” axioms of the form “ L α is a model of ZFC + · · · ”.The reasoning here echos Hamkins’s arguments for principles expressing an extensibleconcept of set, including that every universe should be a transitive model in a universesatisfying the axiom of constructibility. My impression is that Hamkins advocates for amultiverse view of set theory, whereas these notes advocate for a universe view. However, amultiverse view in which well-foundedness is absolute may be reconciled with the universeview expressed here by saying that the former focuses on the many transitive models of ZF ,whereas the latter focuses on the set-theoretic universe as a whole.11. assertibility The L ωω completeness theorem implies that any theorem, that can be derived from the ax-ioms of a positivistic theory T via classical logic, also has a positivistic proof from that theory.Furthermore, the syntactic proof of the cut-elimination theorem yields a primitive recursiveprocedure that transforms any given classical derivation of a conditional into a positivisticproof. However, as is well known, the elimination of cuts can produce a superexponentialgrowth in the length of a proof.The cut-elimination theorem for LK ωω can be proved in PRA , so it is certainly a theoremof our base theory. Thus, we may first prove the existence of a derivation in classical logicby exhibiting such a derivation, and then infer the existence of a positivistic proof. However,the existence of a proof does not establish the proved conditional without an additionallyaccepted validity principle. For a theory that proves its own correctness, the acceptance ofsuch a validity principle amounts to an instance of the Σ -induction rule on ω . Thus, a minuteextension of the base theory shows that the theorems proved from the axioms of the basetheory using classical logic are valid. The same observation holds for the nominally weakerintuitionistic logic, of course. However, intuitionistic derivability is the more interestingnotion because it is constructive.We introduce a new notion of assertibility as a form of semantics for intuitionistic formulas.It is inspired by the notion of assertibility in Weaver’s Truth and Assertibility , but there aretwo significant differences. First, this notion of assertibility is defined relative to a theory τ . Second, this notion of assertibility does not satisfy the full capture schema φ ⇒ A ( φ ),though it does satisfy the other axioms. Both of these differences may be rationalizedconceptually. The positivistic approach recognizes only Σ formulas as meaningful; it does notrecognize any conditionals as absolutely justifiable, only justifiable relative to given axioms.I have no intuition that there is any complete objective notion of “conclusive demonstration”[13, p. 89]. Weaver justifies the capture law, that A implies the assertibility of A , on theprinciple that “all truths can be known” [13, p. 109]. We do not depart from this principle; e simply have a stricter conception of what sentences may be truths. Indeed, for any Σ sentence φ , we will have that φ implies the assertibility of φ . Definition 11.1 (base theory + every set is countable) . Let τ be an intentional Σ ( L ωω )theory extending the base theory. For a sentence φ ∈ I ( L ∞ ω ) we say that φ is assertiblefrom τ , and write A τ ( φ ), in case there is a set A of axioms of τ (“axioms”), and a set of true Σ ( L ∞ ω ) sentences E (“observations”) such that V A, V E ⊢ φ is derivable in the infinitaryintuitionistic sequent calculus LI ∞ ω (figure 5). Proposition 11.2 (base theory + every set is countable) . Let φ ⇒ ψ be an axiom of τ .Conclude that φ ⇒ ψ is assertible.Proof. Immediate from the definition. (cid:3)
Proposition 11.3 (base theory + every set is countable) . Let φ , . . . , φ n be assertible I ( L ∞ ω ) sentences. Assume that the sequent φ , . . . , φ n ⊢ ψ is derivable in LI ( L ∞ ω ) for some I ( L ∞ ω ) sentence ψ . Conclude that ψ is also assertible.Proof. Combine the axioms, and observations for φ , . . . , φ n . (cid:3) We will show that assertible Σ ( L ωω ) sentences are true, assuming the iterated validity ofthe theory τ . Essentially, we will appeal to a validity principle each time we combine proofsof sentences φ ( a ) for a ∈ b into a proof of ∀ x ∈ b : φ ( x ). For each theory τ , it is naturalto consider the stronger theory τ ′ obtained by adding the validity of τ to τ . This processmay be naturally iterated through the ordinals, by defining τ ( α ) to be ( τ ( α − ) ′ , when α is asuccessor ordinal, and τ ( α ) = S β ∈ α τ ( β ) , when α is a limit ordinal. Since we are working withintentional theories, this description is informal. Furthermore, our formalization diverges forthis description because the absence of induction complicates the normally trivial observationthat higher-indexed theories are stronger. Definition 11.4 (base theory) . Let τ be a Σ ( L ωω ) theory. For each ordinal β , we define τ ( β ) to be τ together with the validity of τ ( α ) , for each α ∈ β .The most concise way to formalize this definition is as a fixed point formula τ ( α, φ ⇒ ψ );we write τ ( α ) ( φ ⇒ ψ ) for τ ( α, φ ⇒ ψ ). For each ordinal α , the formula τ ( α ) ( φ ⇒ ψ ) isgenerally strictly Σ ( L ωω ), but of course the function taking each ordinal α to τ ( α ) is primitiverecursive. It is immediate that any theorem of τ ( α ) is a theorem of τ ( β ) , whenever α ∈ β .Naturally, we define τ (Ω) to be τ together with the validity of τ ( α ) , for each ordinal α . Definition 11.5 (base theory) . Let the theory of observations σ be the theory of true Σ ( L ωω ) sentences, i. e., the theory whose axioms are conditionals ⊤ ⇒ ε for true Σ ( L ωω )sentences ε .It is natural to consider a Σ( L ωω ) theory τ in combination with the theory σ . If theprovability of a conditional in τ justifies that conditional on the basis of reason, then theprovability of a conditional in τ + σ justifies that conditional on the basis of reason andfact. A finitary agent in an infinitary universe cannot verify sentences directly, but if theyrecognize the validity of a parameter-free theory τ , they may quickly prove the validity of τ + σ . In a sense, the finitary agent proves that an infinitary agent may take the true Σ ( L ωω )to be axioms. emma 11.6 (base theory + every set is countable) . Let τ be a Σ ( L ωω ) theory extendingthe base theory. Let A be a set of axioms of τ ; let E be a set of true Σ ( L ∞ ω ) sentences;and let φ . . . φ n , ψ be Σ ( L ωω ) formulas. Assume that the sequent V A, V E, φ , . . . , φ n ⊢ ψ isderivable in LI ∞ ω . Conclude that the conditional φ ∧ . . . ∧ φ n ⇒ ψ is provable in ( τ + σ ) (Ω) .Proof. We first observe that the true Σ ( L ∞ ω ) formulas are derivable from a simple subclassof such formulas: the atomic axioms and the content axioms. The atomic axioms are simplythe true atomic sentences; recall that we are treating = as relation symbol rather than as anabbreviation. The content axioms are the universally closed disjunctions: ∀ x ∈ b : W a ∈ b x = a . To simplify the presentation, we render each content axiom as the derivably equivalentformula ∀ x : W { x b } ∪ { x = a | a ∈ b } .We assume the existence of a derivation V A, V E, φ , . . . , φ n ⊢ ψ . It follows that there is aderivation of V A, V E atomic , V E content , φ , . . . , φ n ⊢ ψ , for some set E atomic of atomic axioms,and some set E content of content axioms. By the cut-elimination theorem, there is a cut-freederivation of this sequent. We inductively construct, for each sequent in the derivation, aproof from the conjunction of the Σ ( L ωω ) formulas in the sequent’s antecedent to the Σ ( L ωω )formula in its succedent. By induction, any formula in the succedent of a sequent in thefixed derivation must be Σ ( L ωω ). If the succedent is empty, we instead construct a proof to ⊥ . Note that our use of induction is justified because the argument is constructive.The following cases are trivial: the axiom rule, the structural rule, the weakening rules,the truth rule ( ⊢ ⊤ ) and the falsehood rule ( ⊥ ⊢ ). The right implication rule ( ⊢ ⇒ ) doesnot occur. In the left implication rule ( ⇒ ⊢ ) case, the new conditional must be an instanceof an axiom; concatenating the proofs for the two input sequents with this axiom, we obtaina proof for the output sequent.In cases where the rule application produces a Σ ( L ωω ) formula, our argument appeals todeep inference, following the pattern in our proof of completeness. This applies to the con-junction rules ( V ⊢ )( ⊢ V ), the disjunction rules ( W ⊢ )( ⊢ W ), the existential quantificationrules ( ∃ ⊢ )( ⊢ ∃ ), and the universal quantification rules ( ∀ ⊢ )( ⊢ ∀ ). We therefore considerthe cases in which an application of one of these rules does not produce a Σ ( L ωω ) formula;such an application cannot occur on the right.If the left conjunction rule ( V ⊢ ) produces a non- Σ ( L ωω ) formula, then that formula isa conjunction of universally closed axioms of τ , of content axioms, or of atomic axioms. Inthe first two cases, the Σ ( L ωω ) formulas in the output sequent are exactly the same as the Σ ( L ωω ) of the input sequent, so the proof for the input sequent works for the output sequent.In the third case, the loss of an atomic assumption ǫ is innocuous, because ⊤ ⇒ ǫ is atheorem of σ .If the left universal quantification rule ( ∀ ⊢ ) produces a non- Σ ( L ωω ) formula, then theuniversal quantifier is applied to a non- Σ ( L ωω ) input formula. Indeed, either the inputformula includes an implication symbol ⇒ , or the input formula is of the form W { t b } ∪ { t = a | a ∈ b } . In either case, the proof for the input sequent works for the outputsequent. The left existential quantification rule ( ∃ ⊢ ) only produces Σ ( L ωω ) formulas.If the left disjunction rule ( W ⊢ ) produces a non- Σ ( L ωω ) formula, then that formula isof the form W { t b } ∪ { t = a | a ∈ b } . Let x , . . . , x m be the free variables in the outputsequent; let φ ( x , . . . , x m ) , . . . , φ n ( x , . . . , x m ) be the Σ ( L ωω ) formulas on the left of theoutput sequent; and let ψ ( x , . . . , x m ) be the Σ ( L ωω ) formula on the right of the outputsequent, or the formula ⊥ if the there is no formula on the right of the output sequent. hus, there is a least ordinal α such that t ( x , . . . , x m ) b ∧ φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m ) ⇒ ψ ( x , . . . , x m )has a proof from ( τ + σ ) ( α ) , and for all a ∈ b , t ( x , . . . , x m ) = a ∧ φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m ) ⇒ ψ ( x , . . . , x m )has a proof from ( τ + σ ) ( α ) . It follows that ∀ a ∈ b : P ( α ) ( t ( x , . . . , x m ) = a ∧ φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m ) ⇒ ψ ( x , . . . , x m ))is a true Σ ( L ωω ) formula, where P ( α ) is the provability predicate for the theory ( τ + σ ) ( α ) .We now construct a proof of the desired conditional using the axioms of ( τ + σ ) ( α +1) : φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m ) ⇒ [ t ( x , . . . , x m ) b ∧ φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m )] ∨ [ t ( x , . . . , x m ) ∈ b ∧ φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m )] ⇒ ψ ( x , . . . , x m ) ∨ [ t ( x , . . . , x m ) ∈ b ∧ T ( φ ( x , . . . , x m )) ∧ . . . ∧ T ( φ n ( x , . . . , x m )) ∧ ∀ a ∈ b : P ( α ) ( t ( x , . . . , x m ) = a ∧ φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m ) ⇒ ψ ( x , . . . , x m )] ⇒ ψ ( x , . . . , x m ) ∨ [ T ( φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m )) ∧ P ( α ) ( t ( x , . . . , x m ) = t ( x , . . . , x m ) ∧ φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m ) ⇒ ψ ( x , . . . , x m )] ⇒ ψ ( x , . . . , x m ) ∨ [ T ( φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m )) ∧ T ( t ( x , . . . , x m ) = t ( x , . . . , x m )) ∧ P ( α ) ( t ( x , . . . , x m ) = t ( x , . . . , x m ) ∧ φ ( x , . . . , x m ) ∧ . . . ∧ φ n ( x , . . . , x m ) ⇒ ψ ( x , . . . , x m )] ⇒ ψ ( x , . . . , x m ) ∨ T ( ψ ( x , . . . , x m )) ⇒ ψ ( x , . . . , x m )In the second implication, we apply a theorem of σ ; in the fourth implication, we apply aproperty of the the truth predicate; in the fifth implication, we apply the validity of ( τ + σ ) ( α ) .In section 9, we defined the truth predicate just for sentences without function symbols;however, if we extend this definition to sentences with function symbols by including thecomputation of values into verifications, then the conditional ⊤ ⇒ T ( t ( x , . . . , x m ) = t ( x , . . . , x m ))is provable.Thus, we have a proof in ( τ + σ ) (Ω) from the conjunction of the Σ( L ωω ) formulas in theantecedent of the sequent V A, V E, φ , . . . , φ n ⊢ ψ , to ψ . If V E is not a Σ ( L ωω ) sentence,then we have a proof of φ ∧ . . . ∧ φ n ⇒ ψ , as desired. If V E is Σ ( L ωω ) formula, then weinstead have a proof of ( V E ) ∧ φ ∧ . . . φ n ⇒ ψ , which can be turned into a proof of thedesired conditional by applying the axiom ⊤ ⇒ V E of σ . (cid:3) Proposition 11.7 (base theory + every set is countable) . Let τ be a Σ ( L ωω ) theory extendingthe base theory. Let φ ⇒ ψ be a Σ ( L ωω ) conditional. Assume that φ ⇒ ψ is assertiblefrom τ . Conclude that φ ⇒ ψ is a theorem of ( τ + σ ) (Ω) .Proof. There is a set A of axioms of τ , and a set E of true Σ ( L ∞ ω ) formulas such that thesequent V A, V E ⊢ φ ⇒ ψ is derivable in LI ∞ ω . It follows that the sequent V A, V E, φ ⊢ ψ is also derivable. By the theorem, the theory ( τ + σ ) (Ω) proves φ ⇒ ψ . (cid:3) roposition 11.8 (base theory + every set is countable) . Let τ be a Σ ( L ωω ) theory extendingthe base theory. Let ψ be a Σ ( L ωω ) sentence. Assume that ψ is assertible from τ . Concludethat ⊤ ⇒ ψ is a theorem of ( τ + σ ) (Ω) .Proof. If φ is assertible, then ⊤ ⇒ φ is also assertible. Apply the above corollary. (cid:3) Definition 11.9 (base theory) . We extend the notion of a verification V to Σ ( L ∞ ω ) sen-tences as follows:(6) If W K is in V , then some element φ of K is in V .(7) If V K is in V , then each element φ of K is in V .The truth of a Σ ( L ∞ ω ) sentence is then defined, as before, as the existence of a verificationcontaining that sentence.This extended truth predicate evidently satisfies the expected axioms for set conjunctionand set disjunction:(6) T ( W K) ⇔ ∃ φ ∈ K : T ( φ )(7) T ( V K) ⇔ ∀ φ ∈ K : T ( φ )The extended truth predicate is again a Σ( L ωω ) formula, so we can reason about thetruth of Σ ( L ωω ) sentences using finitary, positivistic reasoning. In infinitary deductions itis convenient to establish the equivalence between T ( W K) and W φ ∈ K T ( φ ), and likewise forconjunction. Bridging the gap between between ∀ φ ∈ K : T ( φ ) and W φ ∈ K T ( φ ) requires aninfinitary assumption; the equivalence follows from a Σ ( L ωω ) sentence essentially listing theelements of K, for example a conjunction of the content axiom for K with the relevant atomicaxioms. Lemma 11.10 (base theory) . Let ψ ( x , . . . , x n ) be a Σ ( L ∞ ω ) formula. There is a set A ψ of axioms of the base theory and a set E ψ of true Σ ( L ∞ ω ) sentences such that thesequents V A ψ , V E ψ , ψ ( x , . . . , x n ) ⊢ T ( ψ ( x , . . . , x n )) and V A ψ , V E ψ , T ( ψ ( x , . . . , x n )) ⊢ ψ ( x , . . . , x n ) are provable in LI ∞ ω .Proof. The list of basic truth properties (1) - (7) is finite, so finitely many axioms of τ are sufficient to establish them. Let A ψ be the set of these axioms. Let E ψ consist of allatomic axioms and content axioms for elements in the transitive closure of ψ , includingcontent axioms for the conjunctions and disjunctions in ψ . As in the finitary case, wereduce T ( ψ ( x , . . . , x n )) to a Σ ( L ωω ) formula that differs from ψ ( x , . . . , x n ) only in that itexpresses infinitary conjunctions and infinitary disjunctions using bounded quantifiers; wethen use the content axioms and atomic axioms to show that this formula is equivalent to ψ ( x , . . . x n ). (cid:3) Theorem 11.11 (base theory + every set is countable + ( τ + σ ) (Ω) is valid) . Assume that τ is an Σ ( L ωω ) theory extending the base theory. The assertibility predicate for τ respectsthe semantics of positivistic connectives:(1) A τ ( φ ∧ ψ ) ⇔ A τ ( φ ) ∧ A τ ( ψ ) (2) A τ ( φ ∨ ψ ) ⇔ A τ ( φ ) ∨ A τ ( ψ ) (3) A τ ( V φ ∈ Φ φ ) ⇔ ∀ φ ∈ Φ : A τ ( φ ) (4) A τ ( W φ ∈ Φ φ ) ⇔ ∃ φ ∈ Φ : A τ ( φ ) (5) A τ ( ∃ v : ψ ) ⇔ ∃ a : A τ ( ψ va ) A τ ( ∀ v ∈ b : φ ) ⇔ ∀ a ∈ b : A τ ( ψ va ) The assertibility predicate satisfies modus ponens and the instantiation rule:(7) A τ ( φ ) ∧ A τ ( φ ⇒ ψ ) ⇒ A τ ( φ ) (8) A τ ( ∀ v : φ ) ⇒ A τ ( φ va ) Truths, tautologies, and the axioms of τ are assertible:(9) T ( φ ) ⇒ A τ ( φ ) , where the variable φ ranges over Σ ( L ωω ) sentences(10) D LI ( ∅ ⊢ φ ) ⇒ A τ ( φ ) , where the variable φ ranges over I ( L ωω ) sentences(11) τ ( φ ⇒ ψ ) ⇒ A τ ( φ ⇒ ψ ) , where the variables φ and ψ range over Σ ( L ωω ) formulas.Finally, assertibility is valid in the following sense:(12) A τ ( φ ) ⇔ T ( φ ) , where φ ranges over Σ ( L ωω ) sentences(13) T ( φ ) ∧ A τ ( φ ⇒ ψ ) ⇒ T ( ψ ) , where the variables φ and ψ range over Σ ( L ωω ) formulasProof. Property (1) is a simple consequence of rules for conjunction, as is property (3).Property (2) is a consequence of cut-elimination. Fix a cut-free derivation of V A , V E , ⊢ φ ∨ ψ , for some set A of axioms of τ , and some set E of true Σ ( L ∞ ω ) formulas. We willrework this derivation into a derivation of a sequent V A , V E ⊢ A τ ( φ ) ∨ A τ ( ψ ), for somelarger set A ⊇ A of axioms of τ , and some larger set E ⊇ E of true Σ ( L ∞ ω ) sentences.Thus, we will conclude A τ ( A τ ( φ ) ∨ A τ ( ψ )). By corollary, the conditional ⊤ ⇒ A τ ( φ ) ∨ A τ ( ψ )is provable from ( τ + σ ) (Ω) . By the validity of ( τ + σ ) (Ω) , we will conclude A τ ( φ ) ∨ A τ ( ψ ).We distinguish the “trunk” of the fixed cut-free derivation of V A , V E , ⊢ φ ∨ ψ as thefinal part of the proof tree that consists of sequents of the form ∆ ⊢ φ ∨ ψ ; we do not includesequents of this form that occur before sequents not of this form. We completely replaceeach subderivation of the fixed derivation that ends in some initial sequent of the trunk. Aninitial sequent of the trunk is of the form ∆ ⊢ φ ∨ ψ with each formula in ∆ a substitutioninstance of a subformula of V A or of V E . Thus, each formula in ∆ is a partially closedaxiom of τ or a conjunction of closed axioms of τ (we say that it is in ∆ A ), or it is a Σ ( L ∞ ω )formula (we say that it is in ∆ E ). In this exposition, we assume that ∆ does not contain aconjunction of closed axioms of τ ; in the case that ∆ does contain a conjunction of closedaxioms of τ the argument will need to be adjusted in a few inconsequential and messy ways.Fix an initial sequent ∆ ⊢ φ ∨ ψ of the trunk. Since none of the immediately precedingsequents in the fixed cut-free derivation have φ ∨ ψ in the succedent, it follows that thatthis initial sequent is produced by the right disjunction rule or the right weakening rule. Inthe former case, we have a derivation of ∆ ⊢ φ or of ∆ ⊢ ψ , and in the latter case we havea derivation of both. Without loss of generality, we suppose that we have a derivation of∆ ⊢ ψ . We continue this subderivation, closing the axioms of τ , and combining the closedaxioms of τ , and the Σ ( L ∞ ω ) formulas, into single formulas using the left conjunction rule,to obtain a derivation that looks quite like a witness to the assertibility of ψ , except that the Σ ( L ∞ ω ) conjunction may fail to be a sentence, and even if it is a sentence, it is not knownto be true: ^ ∆ A , ^ ∆ E ( x , . . . , x n ) ⊢ ψ We may formalize an argument that “if there is a derivation of the sequent V ∆ A , V ∆ E ( x , . . . , x n ) ⊢ ψ , and ∆ A is a set of closures of axioms of τ , and V ∆ E ( x , . . . , x n ), then the sentence ψ is ssertible”. The sentence “there is a derivation of the sequent V ∆ A , V ∆ E ( x , . . . , x n ) ⊢ ψ ”is a true Σ ( L ∞ ω ) sentence. The sentence “∆ A is a set of closures of axioms of τ ” is alsoa true Σ ( L ∞ ω ) sentence. Thus, there is a set of true Σ ( L ∞ ω ) sentences E ′ and a set A ′ ofaxioms of τ , and derivations of the following sequents ^ A ′ , ^ E ′ , ∆ E ( x , . . . , x n ) ⊢ A τ ( ψ ) ^ A ′ , ^ E ′ , ∆ A , ∆ E ( x , . . . , x n ) ⊢ A τ ( φ ) ∨ A τ ( ψ )Thus, for the given initial sequent ∆ A , ∆ E ( x , . . . , x n ) ⊢ φ ∨ ψ , we have obtained a derivationof the above sequent.We now rework the fixed cut-free derivation of V A, V E ⊢ φ ∨ ψ in the following way:we replace each derivation of an initial sequent of the trunk in the way described above;we replace each sequent ∆ ⊢ φ ∨ ψ of the trunk with the sequent ∆ ⊢ A τ ( φ ) ∨ A τ ( ψ ); wefinally propagate the new assumptions in the initial sequents of the trunk to the conclusion,enlarging the conjunctions as we do so. Every step of the derivation above an initial segmentof the trunk is according to LI ∞ ω because it is part of a derivation. Every other step whoseoutput sequent is not in the trunk is according to LI ∞ ω , because such a step has not beenaltered. Every step in the trunk is according to LI ∞ ω , because the rules applied here in thefixed cut-free derivation are all left rules, and we have not introduces any free variables. Theonly non-initial steps in the trunk that have an input sequent not in the trunk apply theleft implication rule; such implications are still valid because neither the new antecednentformulas nor the new succedent A τ ( φ ) ∨ A τ ( ψ ) play a role. Thus, we obtain a derivation of V A , V E ⊢ A τ ( φ ) ∨ A τ ( ψ ). The set A consists of the axioms in A , together with thenew axioms added at initial sequents of the trunk; likewise, E consists of the axioms of E , together with the new true Σ ( L ∞ ω ) formulas added at initial sequents of the trunk. Weconclude that A τ ( φ ) ∨ A τ ( ψ ) is assertible, and as previously argued, that A τ ( φ ) or A τ ( ψ ).The above argument generalizes more or less directly to property (4). The argumentfor property (5) involves more changes, particularly when the vocabulary includes functionsymbols. We fix a cut-free derivation of V A , V E ⊢ ∃ y : ψ ( y ), and, as expected, we definethe trunk to be the terminal part of this derivation that consists of sequents of the form∆ ⊢ ∃ y : ψ ( y ). Each initial sequent of the trunk is produced by the right weakening ruleor the right existential quantifier rule; in the later case, the preceding sequent is ∆ ⊢ ψ ( t )for some term t that may not be closed. Explicitly listing any free variables, we have aderivation of ^ ∆ A , ^ ∆ E ( x , . . . , x n ) ⊢ ψ ( t ( x , . . . , x n ))We may formalize the argument that “if y = t ( x , . . . , x n ), and ψ ( t ( x , . . . , x n )), then ψ ( y )”.Thus, there is a set of axioms A ′′ of τ so that we have a derivation of ^ ∆ A ∪ A ′′ , ^ ∆ E ( x , . . . , x n ) ∪ { y = t ( x , . . . x n ) } ⊢ ψ ( y ) . We may formalize an argument that “if there is a derivation of V ∆ A ∪ A ′′ , V ∆ E ( x , . . . , x n ) ∪{ y = t ( x , . . . x n ) } ⊢ ψ ( y ), and ∆ A is a set of closures of axioms of τ , and A ′′ is a set of axiomsof τ , and V ∆ E ( x , . . . , x n ), and y = t ( x , . . . , x n ), then the sentence ψ ( y ) is assertible”. Wemay also formalize an argument that “there is an object y such that y = t ( x , . . . , x n )”.Thus, there is a set of true Σ ( L ∞ ω ) sentences E ′ and a set A ′ of axioms of τ , and derivations f the following sequents ^ A ′ , ^ E ′ , ∆ E ( x , . . . , x n ) , y = t ( x , . . . , x n ) ⊢ A τ ( ψ ( y ))) ^ A ′ , ^ E ′ , ∆ E ( x , . . . , x n ) , ∃ y : y = t ( x , . . . , x n ) ⊢ ∃ y : A τ ( ψ ( y ))) ^ A ′ , ^ E ′ , ∆ E ( x , . . . , x n ) ⊢ ∃ y : A τ ( ψ ( y )))The rest of the argument follows the pattern of the disjunctive cases.For property (6), assume that ∀ v ∈ b : ψ is assertible from τ . If a is an element of b , then ψ va is assertible; we simply extend the derivation by adding a ∈ b to the antecedent, andinstantiating. Conversely, assume that for each element a in b , the formula ψ va is assertiblefrom τ . We combine the given derivations into a derivation whose consequent is V a ∈ b ψ va ; weconclude that ∀ v ∈ b : ψ is assertible, by adding to the antecedent the true Σ ( L ∞ ω ) sentenceslisting the elements of b .Properties (7) and (8) follow simply from the rules of LI ∞ ω . Properties (9), (10), and(11), follow from the definition of assertibility. For property (12), assume A τ ( φ ); it followsthat the conditional ⊤ ⇒ φ is a theorem of ( τ + σ ) (Ω) , so by the validity of ( τ + σ ) (Ω) ,we conclude that φ is true. Property (13) follows by combining property (7) and property(12). (cid:3) Philosophical remarks
The conservative response to objections that a formula with free variables implicitly ex-presses a universal proposition if it is meaningful at all is to interpret the free variablesas syntactic variables. If each free variable in a positivistic proof is replaced with a con-stant symbol, the proof remains valid at each step. The axioms of a theory are then alsointerpreted schematically; we must include all closed instances of deep inference.The drawback of this conservative interpretation is that it undermines the impact oftheorems that include free variables. Such a theorem may be interesting because its closedinstances are uniformly provable. Each instance of its proof is positivistically acceptable, butthis observation is not itself expressible as an inference under the conservative interpretation.Indeed, each instance of the observation itself is provable, but what is interesting is that theseinstances are uniformly provable; and we have come full circle.This difficulty is largely an artifact of our formalization of positivistic reasoning within theframework of first-order logic. Procedures, not formulas, are the elementary constituents ofa mathematical universe in the positivistic approach. We imagine a register machine whoseregister contents may change over the course of a machine procedure, or indeed, over thecourse of an argument. At each step of the argument, we predict that a certain procedure willhalt after the sequence of preceding procedures, on the basis that the sequence of precedingprocedures halts.We may analyze the operation of the register machine modally. Each machine procedureis a binary relation on the space of machine states. We interpret each positivistic formula φ ( x ) as the procedure that halts on its initial state if φ ( x ) can be verified for the registercontents of that state, and that otherwise does not halt. Ignoring deep inference, we mightthen interpret each conditional φ ( x ) ⇒ ψ ( x ) as the rule of inference the predicts ψ ( x ) after φ ( x ). However, our reasoning may also include a procedure that may halt on a state otherthan its initial state. ere the discussion splits into two threads. First, we discuss the extent to which universalquantification is implicit in this dynamic reasoning. Second, we formalize this dynamicreasoning in a way that incorporates deep inference and mathematical practice.Recall that existential quantification on x is defined in terms of a procedure that is pre-sumed to change the contents of register x arbitrarily. This presumed behavior is expressedby the logical axiom φ ( t ) ⇒ ∃ x : φ ( x ). If our argument assumes φ ( x ), and infers ψ ( x ),then x is essentially a schematic variable that denotes the object in register x , which re-mains unchanged over the course of the argument. In contrast, if our argument assumes theexistential quantifier operation for x followed by φ ( x ), and infers ψ ( x ), then x is essentiallya universally quantified variable, since we have inferred ψ ( x ) about an arbitrarily chosenobject that satisfies φ ( x ). That our inference carries a universal sense does not contradictthe positivistic rejection of universal quantification over potentialist totalities because it isan inference, not a proposition. We are simply inferring ψ from φ for an object that waschosen arbitrarily, i. e., produced by a specific procedure.We may summarize this point by saying that though the positivistic approach excludesthe possibility of a procedure that checks whether a given predicate holds for every object inthe universe, it requires a procedure that produces arbitrary objects. It is the procedure thataxiomatically produces arbitrary objects: to infer φ ( x ) for arbitrary x , is to infer φ ( x ) forthe object produced by this procedure, and to infer that this procedure produces arbitraryobjects is to infer that this procedure will produce an object equal to the object producedby this procedure. Our axioms for the existential quantifier express principles appropriateto this intuition.In particular, the correctness principle of an intentional theory is the inference that if anaxiom of that theory was produced arbitrarily, i. e., by the existential quantifier procedure,and the assumption of that axiom is true, then the conclusion of that axiom is true. Corollary9.8 shows that there are strong theories that prove this inference about themselves. Thesubstitution principle that we can replace an arbitrary object in an argument with a constantsymbol is a provable theorem; it also is formulated in terms of the existential quantifierprocedure: if an arbitrary free variable is replaced with an arbitrary constant symbol inan arbitrary proof, then the result is also a proof. Thus, we have a device for formulatinguniversal principles without admitting universal propositions.The major subtlety of dynamic reasoning is the treatment of state-preserving machineprocedures such as formulas. For example, an argument might assume φ , then infer χ , andthen infer ψ , leading us to claim that assuming φ , we can infer ψ . However, this summarypresumes that the procedure χ does not alter the machine state, a presumption that is notentirely trivial. After all, the verification of χ generally involves temporarily storing data invarious registers. We may address this difficulty by allowing hypothetical reasoning, whichreplaces a terminal part of the sequence of procedures in a deduction, rather than simplyadding to it. We may also simply adopt the convention that the occurrence of intermediateformulas is implicit in the statement of a theorem. In either case the direct formalization ofthe deductive system is somewhat alien to mathematical practice.Instead, we formalize dynamic reasoning in a way that synthesizes aspects of potentialism,of deep inference, and of mathematical practice. Any true Σ formula is true in some initialsegment of the universe, in which classical reasoning is positivistically valid. So, we takethe Σ reflection principle as a basic rule of inference, and we relativize the standard Hilbert ystem for classical deduction to transitive sets. We similarly relativize the axioms of thegiven positivistic theory to transitive sets. Finally, we formalize the dynamic aspect ofthe reasoning using the familiar phrase “let b be...”, to allow the introduction of constantsymbols after the deduction of an existential sentence.To simplify the ensuing discussion, we assume that the signature includes no functionsymbols of positive arity; we allow the negation connective ¬ to occur in Σ formulas, providedthat it does not occur above unbounded quantifiers; and we express the negated predicatesymbols and = using this connective. We recall that when φ ( x , . . . , x n ) is a classicalfirst-order formula, its relativization φ W ( x , . . . , x n ) to a set W is defined to be the result ofreplacing each existential quantifier ∃ v : by the bounded quantifier ∃ v ∈ W : , and likewisefor universal quantifiers. By convention, even bounded quantifiers are replaced in this way,so that ∀ y ∈ x : φ becomes ∀ y ∈ W : y x ∨ φ . Our notation for the relative universalclosure φ ( x , . . . , x n ) W may be read directly; it abbreviates the formula ∀ x ∈ W : · · · ∀ x n ∈ W : φ W ( x , . . . , x n ). Definition 12.1.
Let T be an extensional positivistic theory, i. e., a set of positivisticconditionals. The deductions of the system DT consist of Σ sentences. For the rules ofinference of DT listed below, φ , χ , and ψ denote arbitrary Σ formulas, and c , . . . , c n denotesa nonempty list of constant symbols that includes those of φ and χ . The notation Trans( X, c )abbreviates the formula ( c ∈ X ) ∧ ( c ∈ X ) ∧ · · · ∧ ( c n ∈ X ) ∧ ( ∀ w ∈ X : ∀ v ∈ w : v ∈ X ).(1) Nonlogical axioms . If φ ( x , . . . , x m ) ⇒ ψ ( x , . . . , x m ) is an axiom of T , fromTrans( C, c ), deduce ∀ x ∈ C : . . . ∀ x m ∈ C : ¬ φ C ( x , . . . , x m ) ∨ ψ ( x , . . . , x m ).(2) Logical axioms . If φ is an axiom of the standard Hilbert system for classical first-order logic with equality, from Trans( C, c ), deduce φ C .(3) Modus ponens . From Trans(
C, c ), and φ C , and ¬ φ ∨ χ C , deduce χ C .(4) Downward reflection.
From φ , deduce ∃ X : Trans( X, c ) ∧ φ X .(5) Upward reflection.
From ∃ X : Trans( X, c ) ∧ φ X , deduce φ .(6) Conjunction introduction.
From φ and χ , deduce φ ∧ χ .(7) Conjunction elimination.
From φ ∧ χ or χ ∧ φ , deduce φ .(8) Universal instantiation.
From Trans(
C, c ) and φ C , deduce φ ( x/c ) C .(9) Existential generalization.
From φ ( X/C ), deduce ∃ X : φ .(10) Existential instantiation.
From ∃ X : φ , deduce φ ( X/B ), where B is a new con-stant symbol, i. e., a constant symbol that is not in the signature and that does notoccur previously in the deduction.A Σ sentence ψ is deducible just in case it has a deduction from the Σ formula ⊤ . A Σconditional φ ( y , . . . , y n ) ⇒ ψ ( y , . . . , y n ) is deducible just in case for new constant symbols b , . . . , b n , the formula ψ ( b , . . . , b n ) can be deducible from φ ( b , . . . , b n ).Following standard mathematical practice, we gloss the introduction of a new constantsymbol using the word “let”. Mechanically, this introduction refers to an execution of theexistential quantifier procedure, whose output persists through the proof as the value of thenew constant symbol. Explicitly, in an application of the existential instantiation inferencerule, after we check ∃ X : φ , we infer that the procedure that looks for such an object andstores it in register B halts, and then we infer φ ( X/B ), so that we can later apply inferencesthat assume φ ( X/B ). We gloss an initial assumption φ ( b , b , . . . , b m ) in the deduction of a Σ onditional φ ( x ) ⇒ ψ ( x ) in just the same way; thus, such an initial assumption technicallycorresponds to a finite sequence of procedures, rather than a single procedure. Theorem 12.2 ( ZFC ) . Assume that χ ⇒ χ ′ is provable from a positivistic theory T . Then,this conditional is deducible in the system DT just defined.Proof. It’s sufficient to establish this theorem when χ ⇒ χ ′ is an instance of deep inferenceof some axiom of T , or some axiom of positivistic logic.Assume that χ ( y ) ⇒ χ ′ ( y ) is an instance of deep inference of an axiom of positivisticlogic, so it is a logical validity. It follows that ¬ χ ( y ) ∨ χ ′ ( y ) is derivable in the standardHilbert system. We now construct the desired deduction as follows: Assume χ ( b ). Let B be a transitive set that contains the constants of χ , χ ′ , and b , such that χ ( b ) B . Derive( ¬ χ ( b ) ∨ χ ′ ( b )) B . Apply modus ponens to derive χ ′ ( b ) B . Conclude χ ′ ( b ) by upward reflection.Assume that χ ( y ) ⇒ χ ′ ( y ) is an instance of deep inference of φ ( x ) ⇒ ψ ( x ), anaxiom of T . Write c for the constant symbols of χ and χ ′ . We now construct the desireddeduction as follows: Assume χ ( b ). Let B be a transitive set that contains b and c , suchthat χ ( b ) B . The constant symbols of φ must be among the constant symbols of χ , so derive ∀ x ∈ B : · · · ∀ x n ∈ B : ¬ φ B ( x , . . . , x m ) ∨ ψ ( x , . . . , x m ) by the nonlogical axioms rule.Applying conjunction introduction and downward reflection, obtain a transitive set B ′ thatcontains B , b , and c , such that (cid:0) χ ( b ) B ∧ Trans(
B, b, c ) ∧ ∀ x ∈ B : · · · ∀ x n ∈ B : ¬ φ B ( x ) ∨ ψ ( x ) (cid:1) B ′ . Since deep inference is logically valid, reasoning in the Hilbert system inside B ′ , derive χ ′ ( b , . . . , b n ) B ′ . Conclude χ ′ ( b , . . . , b n ) by upward reflection. (cid:3) Corollary 12.3 ( ZFC ) . Assume that χ ( y ) ⇒ χ ′ ( y ) is provable from a positivistic theory T . Let c be a sequence of constants symbols that includes those of χ . Then, the Σ formula ∀ y ∈ C : · · · ∀ y n ∈ C : ¬ χ C ( y , . . . , y n ) ∨ χ ′ ( y , . . . , y n ) is deducible from Trans(
C, c ) in the system DT , for any new constant symbol C .Proof. The classical first-order formula Trans(
Y, c ) implies ∀ y ∈ Y : · · · ∀ y n ∈ Y : ¬ χ Y ( y , . . . , y n ) ∨ χ ( y , . . . , y n ); this follows by induction on the complexity of χ because χ is a Σ formula. Theinduction hypothesis affirms the implication for all tuples of variables y . As an example,in the bounded universal quantifier case, we assume Trans( Y, c ) and let y , . . . , y n ∈ Y besuch that ( ∀ v ∈ t : χ ) Y , for some term t that is necessarily either a constant symbol or avariable among y . In either case, we have t ∈ Y . The formula ( ∀ v ∈ t : χ ) Y is equal to ∀ v ∈ Y : ¬ v ∈ t ∨ χ Y , and since Y is transitive, this formula implies ∀ v ∈ t : χ Y . By theinduction hypothesis, Trans( Y, c ) implies ∀ y ∈ Y : . . . ∀ y n ∈ Y : ∀ v ∈ Y : ¬ χ Y ∨ χ , so weconclude ∀ v ∈ t : χ .Thus, the theory T logically implies the conditionalTrans( Y, c ) ⇒ ∀ y ∈ Y : · · · ∀ y n ∈ Y : ∼ χ Y ( y , . . . , y n ) ∨ χ ′ ( y , . . . , y n ) , where the notation ∼ abbreviates taking the logical complement of a ∆ formula. By thecompleteness theorem, the theory T proves this conditional; and by theorem 12.2, the system DT derives ∀ y ∈ C : · · · ∀ y n ∈ C : ∼ χ C ( y , . . . , y n ) ∨ χ ′ ( y , . . . , y n ) from Trans( C, c ), forany new constant symbol C . We may replace the notation ∼ with the genuine negationconnective ¬ reasoning within the Hilbert system relativized to a large transitive set. (cid:3) ummary. The admissibility of free variables in positivistic proof derives from two as-pects of positivistic reasoning. First, a positivistic inference expresses that the consequentprocedure will halt after the antecedent procedure, if the antecedent procedure does halt, sodata can be passed from one procedure to the next. Second, procedures are innately non-deterministc: a priori any procedure may execute and halt in any number of ways. There isa procedure E that is presumed to produce arbitrary objects; the principle that φ is true ofeach object may be expressed by the inference “after E produces an object x , the procedure φ ( x ) will halt”.A theory that proves its own correctness principle, such as in corollary 9.8, is self-affirmingin the sense that it proves that if an object is produced by the procedure E , and that object isthe application of an axiom whose antecedent is true, then its consequent is also true. Such atheory may be of arbitrarily high consistency strength; there is no contradiction with G¨odel’ssecond incompleteness theorem because the full induction principle is not positivisticallyexpressible. References [1] J. Barwise,
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Every countable model of set theory embeds into its own constructible universe , Journal ofMathematical Logic (2013), no. 2, available at arXiv:1207.0963 .[7] J. Knight, A. Montalb´an, and N. Schweber, Computable structures in generic extensions (2014), availableat arXiv:1405.7456 .[8] P. Pudl´ak,
Improved bounds to the length of proofs of finitistic consistency statements , ContemporaryMathematics (1987).[9] M. Rathjen, A proof-theoretic characterization of the primitive recursive set functions , Journal of Sym-bolic Logic (1992), no. 3.[10] C. Rauszer and B. Sabalski, Notes on the Rasiowa-Sikorski Lemma , Studia Logica (1975), no. 3.[11] S. Simpson, Subsystems of Second Order Arithmetic , Cambridge University Press, 2009.[12] M. Takahashi,
A foundation of finite mathematics , Publications of the Research Institute for Mathe-matical Sciences, Kyoto University (1977).[13] N. Weaver, Truth and Assertibility , World Scientific, 2015.
Department of Mathematics, University of California, Davis, CA 95616
E-mail address : [email protected] -2) φ ⇒ φ (-1) φ ⇒ χ χ ⇒ ψφ ⇒ ψ (0) φ ⇒ ψφ at ⇒ ψ at (1) φ ⇒ ⊤ (2) ⊥ ⇒ ψ (3) φ ∧ ψ ⇒ φ (4) φ ∧ ψ ⇒ ψ (5) φ ⇒ χ φ ⇒ ψφ ⇒ χ ∧ ψ (6) φ ⇒ φ ∨ ψ (7) ψ ⇒ φ ∨ ψ (8) φ ⇒ ψ χ ⇒ ψφ ∨ χ ⇒ ψ (9) ( φ ∨ ψ ) ∧ χ ⇒ ( φ ∧ χ ) ∨ ( ψ ∧ χ )(10) φ xt ⇒ ∃ x : φ (11) φ ya ⇒ ψ ∃ y : φ ⇒ ψ for a not free in φ or ψ (12) χ ∧ ∃ y : φ ⇒ ∃ y : ( χ ∧ φ )(13) ∀ x ∈ s : φ ⇒ t s ∨ φ xa (14) ψ ⇒ a s ∨ φ ya ψ ⇒ ∀ y ∈ s : φ for a not free in φ or ψ (15) ∀ y ∈ s : ( χ ∨ φ ) ⇒ χ ∨ ( ∀ y ∈ s : φ )(16) ⊤ ⇒ a ∈ b ∨ a b (17) a ∈ b ∧ a b ⇒ ⊥ (18) ⊤ ⇒ a = b ∨ a = b (19) a = b ∧ a = b ⇒ ⊥ (20) ⊤ ⇒ a = a (21) a = b ⇒ b = a (22) a = b ∧ b = c ⇒ a = c (23) s = t ∧ φ at ⇒ φ as Figure 1. a complete calculus of conditionals(bound variables distinct from free variables) φ ⇒ ⊤ (2) ⊥ ⇒ ψ (3) φ ∧ ψ ⇒ φ (4) φ ∧ ψ ⇒ ψ (5) φ ⇒ φ ∧ φ (6) φ ⇒ φ ∨ ψ (7) ψ ⇒ φ ∨ ψ (8) ψ ∨ ψ ⇒ ψ (9) ( φ ∨ ψ ) ∧ χ ⇒ ( φ ∧ χ ) ∨ ( ψ ∧ χ )(10) φ xt ⇒ ∃ x : φ (11) ∃ y : ψ ⇒ ψ (12) χ ∧ ∃ y : φ ⇒ ∃ y : ( χ ∧ φ )(13) t ∈ s ∧ ∀ x ∈ s : φ ⇒ φ xt (14) ψ ⇒ ∀ y ∈ s : ( ψ ∧ y ∈ s )(15) ∀ y ∈ s : ( χ ∨ φ ) ⇒ χ ∨ ( ∀ y ∈ s : φ )(16) ⊤ ⇒ s ∈ t ∨ s y (17) s ∈ t ∧ s t ⇒ ⊥ (18) ⊤ ⇒ s = t ∨ s = t (19) s = t ∧ s = t ⇒ ⊥ (20) ⊤ ⇒ t = t (21) s = t ⇒ t = s (22) r = s ∧ s = t ⇒ r = t (23) s = t ∧ φ as ⇒ φ at Figure 2. a complete class of logical axioms for positivistic proof(bound variables distinct from free variables) φ ⇒ ⊤ (2) ⊥ ⇒ ψ (3) φ ∧ ψ ⇒ φ (4) φ ∧ ψ ⇒ ψ (5) φ ⇒ φ ∧ φ (6) φ ⇒ φ ∨ ψ (7) ψ ⇒ φ ∨ ψ (8) ψ ∨ ψ ⇒ ψ (9) ( φ ∨ ψ ) ∧ χ ⇒ ( φ ∧ χ ) ∨ ( ψ ∧ χ )(10) φ ⇒ ∃ x : φ (11) ∃ y : ψ ⇒ ψ for y not free in ψ (12) χ ∧ ∃ y : φ ⇒ ∃ y : ( χ ∧ φ ) for y not free in χ (13) x ∈ z ∧ ∀ x ∈ z : φ ⇒ φ (14) ψ ⇒ ∀ y ∈ z : ( ψ ∧ y ∈ z ) for y not free in ψ (15) ∀ y ∈ z : ( χ ∨ φ ) ⇒ χ ∨ ( ∀ y ∈ z : φ ) for y not free in χ (16) ⊤ ⇒ x ∈ y ∨ x y (17) x ∈ y ∧ x y ⇒ ⊥ (18) ⊤ ⇒ x = y ∨ x = y (19) x = y ∧ x = y ⇒ ⊥ (20) ⊤ ⇒ x = x (21) x = y ⇒ y = x (22) x = y ∧ y = z ⇒ x = z (23) x = y ∧ φ zx ⇒ φ zy for x and y substitutable for z in φ Figure 3. the logical axioms of positivistic proof ntroduction Rules ⊥ ⊢ ∅ ∅ ⊢ ⊤ Γ ⊢ ∆ , φ Γ , ¬ φ ⊢ ∆ Γ , φ ⊢ ∆Γ ⊢ ∆ , ¬ φ ( ∃ φ ∈ K) Γ , φ ⊢ ∆Γ , V K ⊢ ∆ ( ∀ φ ∈ K) Γ ⊢ ∆ , φ Γ ⊢ ∆ , V K( ∀ φ ∈ K) Γ , φ ⊢ ∆Γ , W K ⊢ ∆ ( ∃ φ ∈ K) Γ ⊢ ∆ , φ Γ ⊢ ∆ , W K( ∃ t ) Γ , φ vt ⊢ ∆Γ , ∀ v : φ ⊢ ∆ ( ∃ w Free(Γ , ∆)) Γ ⊢ ∆ , φ vw Γ ⊢ ∆ , ∀ v : φ ( ∃ w Free(Γ , ∆)) Γ , φ vw ⊢ ∆Γ , ∃ v : φ ⊢ ∆ ( ∃ t ) Γ ⊢ ∆ , φ vt Γ ⊢ ∆ , ∃ v : φ Γ ⊢ ∆Γ , φ ⊢ ∆ Γ ⊢ ∆Γ ⊢ ∆ , φ Structural RuleΓ ⊢ ∆Γ ′ ⊢ ∆ ′ (Γ ′ consists of the same formulas as Γ, and ∆ ′ consists of the same formulas as ∆)Cut RuleΓ ⊢ ∆ , φ Γ ′ , φ ⊢ ∆ ′ Γ , Γ ′ ⊢ ∆ , ∆ ′ Figure 4. the system LK ∞ ω (Γ, ∆ finite) ntroduction Rules ⊥ ⊢ ∅ ∅ ⊢ ⊤ Γ ⊢ φ Γ ′ , ψ ⊢ ∆Γ , Γ ′ , φ ⇒ ψ ⊢ ∆ Γ , φ ⊢ ψ Γ ⊢ φ ⇒ ψ ( ∃ φ ∈ K) Γ , φ ⊢ ∆Γ , V K ⊢ ∆ ( ∀ φ ∈ K) Γ ⊢ φ Γ ⊢ V K( ∀ φ ∈ K) Γ , φ ⊢ ∆Γ , W K ⊢ ∆ ( ∃ φ ∈ K) Γ ⊢ φ Γ ⊢ W K( ∃ t ) Γ , φ vt ⊢ ∆Γ , ∀ v : φ ⊢ ∆ ( ∃ w Free(Γ)) Γ ⊢ φ vw Γ ⊢ ∀ v : φ ( ∃ w Free(Γ , ∆)) Γ , φ vw ⊢ ∆Γ , ∃ v : φ ⊢ ∆ ( ∃ t ) Γ ⊢ φ vt Γ ⊢ ∃ v : φ Γ ⊢ ∆Γ , φ ⊢ ∆ Γ ⊢ ∅ Γ ⊢ φ Structural RuleΓ ⊢ ∆Γ ′ ⊢ ∆ ′ (Γ ′ consists of the same formulas as Γ, and ∆ ′ consists of the same formulas as ∆)Cut RuleΓ ⊢ φ Γ ′ , φ ⊢ ∆Γ , Γ ′ ⊢ ∆ Figure 5. the system LI ∞ ω (Γ finite, ∆ singleton or empty) ntroduction Rules ⊥ ⊢ ∅ ∅ ⊢ ⊤ Γ , φ ⊢ ∆Γ , φ ∧ ψ ⊢ ∆ Γ , ψ ⊢ ∆Γ , φ ∧ ψ ⊢ ∆ Γ ⊢ ∆ , φ Γ ⊢ ∆ , ψ Γ ⊢ ∆ , φ ∧ ψ Γ , φ ⊢ ∆ Γ , ψ ⊢ ∆Γ , φ ∨ ψ ⊢ ∆ Γ ⊢ ∆ , φ Γ ⊢ ∆ , φ ∨ ψ Γ ⊢ ∆ , ψ Γ ⊢ ∆ , φ ∨ ψ Γ , φ vw ⊢ ∆Γ , ∃ v : φ ⊢ ∆ Γ ⊢ ∆ , φ vt Γ ⊢ ∆ , ∃ v : φ Γ , t s ∨ φ vt ⊢ ∆Γ , ∀ v ∈ s : φ ⊢ ∆ Γ ⊢ ∆ , w s ∨ φ vw Γ ⊢ ∆ , ∀ v ∈ s : φ Γ ⊢ ∆Γ , φ ⊢ ∆ Γ ⊢ ∆Γ ⊢ ∆ , φ Structural RuleΓ ⊢ ∆Γ ′ ⊢ ∆ ′ Cut RuleΓ ⊢ ∆ , φ Γ ′ , ψ ⊢ ∆ ′ Γ , Γ ′ ⊢ ∆ , ∆ ′ Figure 6. the system L Σ ωω ( τ ) for a Σ ( L ωω ) theory τ For the cut rule, φ ⇒ ψ should be an axiom of τ . For the structural rule, Γ ′ shouldconsist of the same formulas as Γ, and ∆ ′ should consist of the same formulas as ∆. Forleft existential quantification rule, and for the right universal quantification rule, w shouldnot be free in Γ or ∆. n alternate axiomatization of PRS :(1) ( ∀ t ∈ a : t ∈ b ) ∧ ( ∀ t ∈ b : t ∈ a ) ⇒ a = b (2) ∃ t : t ∈ x ⇒ ∃ t ∈ x : ∀ s ∈ x : s t (3) z ∈ { x, y } ⇔ z = x ∨ z = y (4) z ∈ S x ⇔ ∃ y ∈ x : z ∈ y (5) y ∈ { t ∈ x | φ ( t, z ) } ⇔ y ∈ x ∧ φ ( y, z )(6) y ∈ { F ( t, z ) | t ∈ x } ⇔ ∃ t ∈ x : y = F ( t, z )(7) ⊤ ⇒ P ( x , . . . , x n ) = x i (8) ⊤ ⇒ K ( x , . . . , x m ) = F ( G ( x , . . . , x m ) , . . . , G n ( x , . . . , x m ))(9) ⊤ ⇒ F ( x, y ) = R ( { F ( t, y ) | t ∈ x } , x, y ) . The supplementary axioms:(1) ∀ s ∈ x : ∃ t : φ ( s, t, z ) ⇒ ∃ y : ∀ s ∈ x : ∃ t ∈ y : φ ( s, t, z )(2) ⊤ ⇒ ∃ f : Bij( f ) ∧ Ord(dom( f )) ∧ ran( f ) = x . Figure 7. the initial axioms of our base theorythe initial axioms of our base theory