How the Law of Excluded Middle Pertains to the Second Incompleteness Theorem and its Boundary-Case Exceptions
aa r X i v : . [ m a t h . L O ] J un How the Law of Excluded Middle Pertains to theSecond Incompleteness Theorem and itsBoundary-Case Exceptions
Dan E.WillardState University of New York at Albany
Abstract
Our earlier publications showed semantic tableau admitspartial exceptions to the Second Incompleteness Theorem wherea formalism recognizes its self consistency and views multipli-cation as a 3-way relation (rather than as a total function).We now show these boundary-case evasions will collapse if theLaw of the Excluded Middle is treated by tableau as a schemaof logical axioms (instead of as derived theorems).
Keywords and Phrases:
Hilbert’s Second Open Question, Second Incom-pleteness Theorem, Semantic Tableau.
Mathematics Subject Classification:
Comment:
Short conference announcements of these results at ASL-2020’s VirtualN. American Meeting and at LFCS-2020.
Introduction
This article is intended to explore the “hidden significance” and unexplored implica-tions of G¨odel’s Second Incompleteness Theorem and its various generalizations. Inparticular, the existence of a deep chasm separating the goals of Hilbert’s consistencyprogram from the implications of the Second Incompleteness Theorem was evident,immediately, after G¨odel published [20]’s seminal announcement. We exhibited in[46, 47, 48, 50, 51, 52, 53, 54] a large number of articles about generalizations andboundary case exceptions to the Second Incompleteness Theorem, starting with our1993 article [46]. These papers, which included six papers published in the JSL andAPAL, showed every extension α of Peano Arithmetic can be mapped onto an axiomsystem α ∗ that can recognize its own consistency and prove analogs of all α ’s Π theorems (in a slightly different language, called L ∗ ).The term “Self-Justifying” arithmetic was employed in our articles. [47, 50, 51,52, 54]. These papers were able to verify their own consistency by containing a built-in self-referencing axiom that declared “I am consistent” (as will be explained later).In particular, our axiom systems α ∗ used the Fixed-Point Theorem to assure α ∗ ’sself-referencing analogs of the pronoun “I” would enable it to refer to itself in thecontext of its “I am consistent” axiomatic declaration.It turns out that such a self-referencing mechanism will produce unacceptableG¨odel-style diagonalizing contradictions, when either α ∗ or its particular deployeddefinition of consistency are too strong. This is because our methodologies only become contradiction-free when α ∗ uses sufficiently weak underlying structures.These weak structures obviously have significant disadvantages. Their virtue isthat their formalisms α ∗ can be arranged to prove more Π like theorems than PeanoArithmetic, while offering some type of partial knowledge about their own consistency.We will call such formalisms “Declarative Exceptions” to the Second Incomplete-ness Theorem.An alternative type of exception to the Second Incompleteness Theorem, whichwe shall call an “Infinite-Ranged Exception” , was recently developed by SergeiArtemov [4] (It is related to the works of Beklemishev [6] and Artemov-Beklemishev[5].) Artemov observed Peano Arithmetic can verify its own consistency, from aspecial infinite-ranging perspective. This means PA will generate an infinite set of1heorems T , T , T ... where each T i shows some subset S i of PA is unableto prove 0 = 1 and where PA equals the formal union of these special selected S i satisfying the inclusion property of S ⊂ S ⊂ S ⊂ ... .This perspective, which is certainly very useful, is also not a panacea. Thus,the abstract in [4] cautiously used the adjective of “somewhat” to describe how itsought to partially achieve the goals sought by Hilbert’s Consistency Program (withan infinite collection of theorems T , T , T ... replacing Hilbert’s intended goal offinding one unifying formal consistency theorem).Our “Declarative”’ exceptions to the Second Incompleteness Theorem and Arte-mov’s “Infinite Ranging” exceptions are two quite different rigorous results, whichare nicely compatible with each other. This is because each acknowledged that theSecond Incompleteness Theorem is a strong result, that will admit no full-scale ex-ceptions. Also, these results are of interest because G¨odel openly conjectured thatHilbert’s Consistency Program would ultimately, reach some levels of partial success (see next section). We will explain, herein, how G¨odel’s conjecture can be partiallyjustified, due to an unusual consequence of the Law of the Excluded Middle.More specifically, we shall focus on the semantic tableau deductive mechanismsof Fitting and Smullyan [15, 40] and their special properties from the perspective ofour JSL-2005 article [50]. Each instance of the Law of the Excluded Middle has beentreated by most tableau mechanisms as a provable theorem, rather than as a built-inlogical axiom. This may, at first, appear to be an insignificant distraction becausemost deductive methodologies do not have their consistency reversed when a theoremis promoted into becoming a logical axiom.Our self-justifying axiom systems are different, however, because their built-inself-referencing “I am consistent” axioms have their meanings change, fundamentally,when their self-referencing concept of “I” involves promoting a schema of theoremsverifying the Law of Excluded Middle into formal explicitly declared logical axioms.
This effect is counterintuitive because similar distinctions exist almost nowhereelse in Logic. Thus some confusion, that has surrounded our prior work, can beclarified when one realizes that an interaction between the self-referencing conceptof “I” with the Law of Excluded Middle causes the Second Incompleteness Theoremto become activated precisely when the Law of Excluded Middle is promoted into2ecoming a schema of logical axioms.The intuitive reason for this unusual effect is that the transforming of derivedtheorems into logical axioms can shorten proofs under the Fitting-Smullyan seman-tic tableau technology. In the particular context where § “I am consistent” axioms and views multiplication as a 3-way relation-ship, these conditions will be sufficient for enacting the full power of the SecondIncompleteness Theorem.The next chapter will explain how these issues are related to questions raised byG¨odel and Hilbert about feasible boundary-case exceptions to the Second Incomplete-ness Effect. Interestingly, neither G¨odel (unequivocally) nor Hilbert (after learning about G¨odel’swork) would dismiss the possibility of a compromise solution, whereby a fragment ofthe goals of Hilbert’s Consistency Program would remain intact. Thus, Hilbert neverwithdrew [26]’s statement ∗ for justifying his program: ∗ “ Let us admit that the situation in which we presently find ourselves withrespect to paradoxes is in the long run intolerable. Just think: in mathematics,this paragon of reliability and truth, the very notions and inferences, as every-one learns, teaches, and uses them, lead to absurdities. And where else wouldreliability and truth be found if even mathematical thinking fails?” G¨odel was, also, cautious (especially during the early 1930’s) not to speculatewhether all facets of Hilbert’s Consistency program would come to a termination. Hethus inserted the following cautious caveat into his famous 1931 paper [20]: ∗ ∗ “It must be expressly noted that Theorem XI” (e.g. the Second Incom-pleteness Theorem) “represents no contradiction of the formalistic standpointof Hilbert. For this standpoint presupposes only the existence of a consistencyproof by finite means, and there might conceivably be finite proofs which cannotbe stated in P or in ... ”
Several biographies of G¨odel [11, 22, 58] have noted that G¨odel’s intention (priorto 1930) was to establish Hilbert’s proposed objectives, before he formalized his fa-mous result that led in an opposite direction. Moreover, Yourgrau’s biography [58] ofG¨odel records how von Neumann found it necessary during the early 1930’s to “ar-gue against G¨odel himself ” about the definitive termination of Hilbert’s consistency3rogram, which “for several years” after [20]’s publication, G¨odel “was cautious notto prejudge” .It is known that G¨odel hinted the Second Incompleteness Theorem was more sig-nificant in a 1933 Vienna lecture [21]. Yet, G¨odel (who published only about 85 pagesduring his career) was frequently ambivalent about this point. Thus, a YouTube talkby Gerald Sacks [39] recalled G¨odel telling Sacks some type of revival of Hilbert’sConsistency Program was likely (see footnote for more details). Moreover, AnilNerode has told us [32] he recalled Stanley Tennenbaum having similar conversationswith G¨odel, where G¨odel again stated his suspicion that Hilbert’s Consistency Pro-gram would be partially revived. Many scholars have been caught by surprise byG¨odel’s private hesitation about the broader implications of the Second Incomplete-ness Effect. This is because G¨odel only published roughly 85 pages during his career,and he never publicly expanded upon [20]’s statement ** .The research that followed G¨odel’s seminal 1931 discovery has technically focusedon studying mostly generalizations of the Second Incompleteness Theorem (insteadof also examining its boundary-case exceptions). Many of these generalizations of theSecond Incompleteness Theorem [2, 3, 7, 8, 9, 10, 13, 16, 23, 24, 25, 29, 33, 34, 35,36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 51] are quite subtle.The author of this paper is especially impressed by a generalization of the SecondIncompleteness Effect, arrived at by the combined work of Pudl´ak and Solovay to-gether with added research by Nelson and Wilkie-Paris [31, 36, 42, 45]. These results,which have been further amplified in [10, 16, 23, 43, 47], show the Second Incomplete-ness Theorem does not require the presence of the Principle of Induction to apply tomost formalisms that use a Hilbert-Frege style of deduction.The next chapter’s Remark 3.5 will helpfully summarize such generalizations ofthe Second Incompleteness Effect. Some quotes from Sacks’s YouTube talk [39] are that G¨odel “did not think” the objectives ofHilbert’s Consistency Program “were erased” by the Incompleteness Theorem, and G¨odel believed(according to Sacks) it left Hilbert’s program “very much alive and even more interesting than itinitially was” . Main Notation and Background Literature
Let us call an ordered pair ( α, D ) a
Generalized Arithmetic Configuration (ab-breviated as a “GenAC” ) when its first and second components are defined asfollows:1. The
Axiom Basis “ α ” for a GenAC is defined as its set of proper axioms.2. The second component “ D ” of a GenAC, called its Deductive Apparatus ,is defined as the union of its logical axioms “ L D ” with its rules for obtaininginferences. Example 3.1
This notation allows us to separate the logical axioms L D , as-sociated with ( α, D ) , from its “basis axioms”, denoted as “ α ”. It also allows us tocompare different deductive apparatuses from the literature. Thus, the D E appara-tus, from Enderton’s textbook [12], uses only modus ponens as a rule of inference, butit deploys a complicated 4-part schema of logical axioms. This differs from the D M and D H apparatuses in the Mendelson [30] and H´ajek-Pudl´ak [25] textbooks. (Theyused a more reduced set of logical axioms but employed “generalization” as a secondrule of inference.) In contrast, the D F apparatus, from Fitting’s and Smullyan’stextbooks [15, 40], uses no logical axioms, but employs a broader “tableau style” ruleof inference. AN IMPORTANT POINT is that while proofs have different lengthsunder different apparatuses, all the common apparatuses produce the same set of finaltheorems from an initial common “axiom basis” of α (as footnote explains). Definition 3.2
Let α again denote an axiom basis, D designate a deductionapparatus, and ( α, D ) denote their GenAC. Henceforth, the configuration ( α, D ) willbe called Self-Justifying when i. one of ( α, D )’s theorems (or possibly one of α ’s axioms) states that the deduc-tion method D, applied to the basis system α, produces a consistent set oftheorems, and ii. the GenAC formalism ( α, D ) is actually, in fact, consistent. This is because all the common apparatuses satisfy the requirements of G¨odel’s CompletenessTheorem. xample 3.3 Using Definition 3.2’s notation, our prior research [46, 47, 50, 51,54] constructed GenAC pairs ( α, D ) that were “Self Justifying”. We also provedthat the Incompleteness Theorem implies specific limits beyond which self-justifyingformalisms simply cannot transgress. For any ( α, D ) , all our articles observed it waseasy to construct a system α D ⊇ α that satisfies the Part-i condition (in an isolatedcontext where the Part-ii condition is not also satisfied ). In essence, α D could consistof all of α ’s axioms plus the added “SelfRef ( α, D ) ” sentence, defined below: ⊕ There is no proof (using D ’s deduction method) of 0 = 1 from the union of the axiom system α with this sentence “SelfRef( α, D ) ” (lookingat itself).Kleene [28] was the first to notice how to encode analogs of SelfRef( α, D ) ’s abovestatement, which we often call an “I AM CONSISTENT” axiom. Each ofKleene, Rogers and Jeroslow [28, 38, 27] emphasized α D may be inconsistent (e.g.violate Part-ii of self-justification’s definition despite the assertion in SelfRef( α, D )’sparticular statement). This is because if the pair ( α, D ) is too strong then a quite con-ventional G¨odel-style diagonalization argument can be applied to the axiom basis of α D = α + SelfRef( α, D ) , where the added presence of the statement SelfRef( α, D )will cause this extended version of α , ironically, to become automatically inconsis-tent. Thus, an encoding for “SelfRef( α, D )” is relatively easy, via an application ofthe Fixed Point Theorem, but this sentence is potentially devastating. Definition 3.4
Let
Add ( x, y, z ) and M ult ( x, y, z ) denote two 3-way predicatesspecifying x + y = z and x ∗ y = z . (Obviously, arithmetic’s classic associative,commutative, identity and distributive axioms will have Π encodings when they areexpressed using these two predicates.) We will say that a formalized axiom basissystem of α recognizes successor, addition and multiplication as Total Functions iff it can prove all of (1) - (3) as theorems: ∀ x ∃ z Add ( x, , z ) (1) ∀ x ∀ y ∃ z Add ( x, y, z ) (2) ∀ x ∀ y ∃ z M ult ( x, y, z ) (3)
6e will call the GenAC system ( α, D ) a
Type-M formalism iff it proves (1) - (3) astheorems,
Type-A if it proves only (1) and (2), and it will be called
Type-S if itproves only (1) as a theorem. Also, ( α, D ) will be called
Type-NS iff it can provenone of (1) - (3).
Remark 3.5
The separation of GenAC systems into the categories of Type-NS,Type-S, Type-A and Type-M systems helps summarize the prior literature about gen-eralizations and boundary-case exceptions for the Second Incompleteness Theorem.This is because: i. The combined research of Pudl´ak, Solovay, Nelson and Wilkie-Paris [31, 36,42, 45], as formalized by Theorem ++ , implies that no natural Type − S system( α, D ) can recognize its own consistency when D represents one of Example 3.1’sthree Hilbert-Frege deductive methods of D E , D H and D M . It thus establishesthe following result: ++ (Solovay’s modification [42] of Pudl´ak [36]’s formalism usingsome of Nelson and Wilkie-Paris [31, 45]’s methods) : Let ( α, D )denote a Type-S GenAC system which assures the successor operationwill provably satisfy both x ′ = 0 and x ′ = y ′ ⇔ x = y . Then( α, D ) cannot verify its own consistency whenever simultaneously D is some type of a Hilbert-Frege deductive apparatus and α treatsaddition and multiplication as 3-way relations, satisfying their usualassociative, commutative, distributive and identity axioms.Essentially, Solovay [42] privately communicated to us in 1994 an analog oftheorem ++. Many authors have noted Solovay has been reluctant to publishhis nice privately communicated results on many occasions [10, 25, 31, 34, 36,45]. Thus, approximate analogs of ++ were explored subsequently by Buss-Ignjatovi´c, H´ajek and ˇSvejdar in [10, 23, 43], as well as in Appendix A of ourpaper [47] and in [49]. Also, Pudl´ak’s initial 1985 article [36] captured themajority of ++’s essence, chronologically before Solovay’s observations. Also,Friedman did some closely related work in [16]. ii. Part of what makes ++ interesting is that [47, 50, 51] presented two typesof self-justifying GenAC systems, whose natural hybrid is precluded by ++.Specifically, these results involve using Example 3.3’s self-referencing “I am onsistent” axiom (from statement ⊕ ). Thus, they establish that some (not all)Type-NS systems [47, 51] can verify their own consistency under a Hilbert-Fregestyle deductive apparatus , and some (not all) Type-A systems [46, 47, 50, 52]can, likewise, corroborate their consistency under a more restrictive semantictableau apparatus. Also, we observed in [48, 53] how one could refine ++with Adamowicz-Zbierski’s methods [2] to show most Type-M systems cannotrecognize their semantic tableau consistency. Remark 3.6 . Several of our papers, starting with our 1993 article [46], haveused Example 3.3’s “I am consistent” axiomatic declaration ⊕ for evading theSecond Incompleteness Effect. Other possible types of evasions rest on the cut-freemethods of Gentzen and Kreisel-Takeuti [19, 29], an interpretational approach (suchas what Adamowicz, Bigorajska, Friedman, Nelson, Pudl´ak and Visser had appliedin [1, 17, 31, 36, 44]), or Artemov’s Infinite-Range perspective [4] (where an infiniteschema of theorems replaces one single unified consistency theorem). We encouragethe reader to examine all these articles, each of which has their own separate merits.Our focus, in this paper, will be primarily on the next section’s Theorems 4.4 and 4.5.They show that some partial (and not full) evasions of the Second IncompletenessEffect can arise under a semantic tableau deductive apparatus. A function F is called Non-Growth when F ( a , . . . , a j ) ≤ M aximum ( a , . . . , a j )holds. Six examples of non-growth functions are: Integer Subtraction (where x − y is defined to equal zero when x ≤ y ) , Integer Division (where x ÷ y equals x when y = 0, and it equals ⌊ x/y ⌋ otherwise),3. M aximum ( x, y ) , Log ♠ ( x ) which is an abbreviation for ⌈ Log ( x + 1 ) ⌉ under the conventionalnotation. (The footnote explains the significance of this concept.) The Example 3.1 had provided three examples of Hilbert-Frege style deduction operators, called D E , D H and D M . It explained how these deductive operators differ from a tableau-style deductiveapparatus by containing a modus ponens rule. The H´ajek-Pudl´ak textbook [25] uses the notation “ | x | ” to designate what we shall call“ Log ♠ ( x ) ” Thus for x ≥ Log ♠ ( x ) denotes the number of symbols that will encode thenumber x , when it is written in a binary format. . Root ( x, y ) = ⌈ x /y ⌉ , and also6. Count ( x, j ) which designates the number of physical “1” bits that are stored among x ’s rightmost j bits. Our papers used the term
Grounding Function to refer to these six non-growthoperations. Also, the term
U-Grounding Function referred to a function thatcorresponds to either one of these six grounding primitives or the growth-oriented functional operations of Addition and
Double ( x ) = x + x .Our language L ∗ , defined in [50], was built out of the eight U-Grounding functionoperations plus the primitives of “0”, “1”, “ = ” and “ ≤ ”. This language differsfrom a conventional arithmetic by excluding a formal multiplication function symbol.(Instead, it treats multiplication as a 3-way relation, via the obvious employment ofits Division primitive.) This notation leads to a surprisingly strong and temptingevasion of the Second Incompleteness Effect. Definition 4.1
In a context where t is any term in our language L ∗ , the specialquantifiers used in the wffs ∀ v ≤ t Ψ( v ) and ∃ v ≤ t Ψ( v ) will be called boundedquantifiers . Also, any formula in our language L ∗ , all of whose quantifiers are sobounded, will be called a ∆ ∗ formula. The Π ∗ n and Σ ∗ n formulae are, thus, defined bythe usual rules, EXCEPT they
DO NOT contain multiplication function symbols.These rules are that:1. Every ∆ ∗ formula will also be a “ Π ∗ ” and “ Σ ∗ ” formula.2. A wff will be called Π ∗ n when it is encoded as ∀ v ... ∀ v k Φ with Φ being Σ ∗ n − .3. A wff will be called Σ ∗ n when it is encoded as ∃ v .. ∃ v k Φ , with Φ being Π ∗ n − . Remark 4.2 . A sentence Ψ will be called
Rank-1* when it can be encodedas either a Π ∗ or Σ ∗ sentence. Our definitions for Π ∗ or Σ ∗ formulae will differ fromArithmetic’s conventional counterparts by excluding multiplication function symbols.(This issue will turn out to be central to our evasions of the Second IncompletenessEffect.)There will be three variants of formal deductive apparatus methods, which we willcompare. The first is semantic tableau . It will receive an abbreviated name of “Tab” α, is a tree-structure that begins with the sentence ¬ Ψ stored inside the tree’sroot and whose every root − to − leaf path establishes a contradiction by containingsome pair of contradictory nodes that will “close” its path. The rules for generatinginternal nodes, along each root − to − leaf path, are that each node must be either aproper axiom of α or a deduction from an ancestor node via one of the Appendix’sstated “elimination” rules for the ∧ , ∨ , → , ¬ , ∀ , and ∃ symbols.Our second explored deductive apparatus is called Extended Tableau , and shall beabbreviated as “Xtab ”. Its definition is identical to
Tab- deduction, except that forany sentence φ in our language L ∗ , the sentence φ ∨ ¬ φ is allowed as an internal nodein an Xtab proof tree. (In other words, Xtab- deduction differs from
Tab- deductionby allowing all instances of the Law of Excluded Middle to appear as permittedlogical axioms. In contrast,
Tab- deduction will view these instances only as derivedtheorems.)Our third deductive apparatus was called
Tab-1 in [50]. It is, essentially, a com-promise between Tab and Xtab, where a “Tab-1” proof for Ψ from an axiom basis α corresponds to a set of ordered pairs ( p , φ ) , ( p , φ ) , ...., ( p k , φ k ) where1. φ k = Ψ2. Each p j is a Tab-proof of what we have called a Rank-1* sentence φ j fromthe union of α with the preceding Rank-1* sentences of φ , φ , ...., φ j − .The Rank-1* constraint ( defined by Remark 4.2 and utilized by the above Item2) is significant. This is because Tab-1 deduction is less efficient than Xtab when the former requires φ j be a Rank-1* sentence. (In contrast, Xtab does not impose asimilar Rank-1* requirement upon φ when its Law of the Excluded Middle allows φ ∨ ¬ φ to appear anywhere as a permissible logical axiom, for fully arbitrary φ. )Thus, Xtab is more desirable than Tab-1 when it can actually be feasibly (?) employed. Let us say an axiom system α owns a Level-1 appreciation of its own self-consistency (under a deductive apparatus D ) iff it can verify that D produces notwo simultaneous proofs for a Π ∗ sentence and its negation. Within this context,where β denotes any basis axiom system using L ∗ ’s U-Grounding language, IS D ( β )10as defined in [50] to be an axiomatic formalism capable of recognizing all of β ’s Π ∗ theorems and corroborating its own Level-1 consistency under D ’s deductive appara-tus. It consists of the following four groups of axioms: Group-Zero:
Two of the Group-zero axioms will define the constant-symbols, ¯ c and ¯ c , designating the integers of 0 and 1. The Group-zero axioms will alsodefine the growth functions of Addition and Double ( x ) = x + x. (They willenable our formalism to define any integer n ≥ · ⌈ Log n ⌉ logic symbols.) Group-1:
This axiom group will consist of a finite set of Π ∗ sentences, denoted as F , which can prove any ∆ ∗ sentence that holds true under the standard modelof the natural numbers. (Any finite set of Π ∗ sentences F , with this property,may be used to define Group-1, as [50] had noted.) Group-2:
Let p Φ q denote Φ’s G¨odel Number, and HilbPrf β ( p Φ q , p ) denote a ∆ ∗ formula indicating that p is a Hilbert-Frege styled proof of theorem Φ fromaxiom system β . For each Π ∗ sentence Φ, the Group-2 schema will contain thebelow axiom (4). (Thus IS D ( β ) can trivially prove all β ’s Π ∗ theorems.) ∀ p { HilbPrf β ( p Φ q , p ) ⇒ Φ } (4) Group-3:
The final part of IS D ( β ) will be a self-referencing Π ∗ axiom, that indicatesIS D ( β ) is “Level-1 consistent” under D ’s deductive apparatus. It thus amountsto the following declaration: No two proofs exist for a Π ∗ sentence and its negation, when D ’sdeductive apparatus is applied to an axiom system, consisting of theunion of Groups 0, 1 and 2 with this sentence (looking at itself ). One encoding for ∗ axiom, had appeared in [50]. Thus,Line (5) is a Π ∗ representation for a. Prf IS D ( β ) ( a, b ) is a ∆ ∗ formula indicating that b is a proof of a theorem a from the axiom basis IS D ( β ) under D ’s deductive apparatus, and b. Pair( x, y ) is a ∆ ∗ formula indicating that x is a Π ∗ sentence and y represents x ’s negation. 11 x ∀ y ∀ p ∀ q ¬ [ Pair( x, y ) ∧ Prf IS D ( β ) ( x, p ) ∧ Prf IS D ( β ) ( y, q ) ] (5)For the sake of brevity, we will not provide exact details about how Line (5) can beencoded under the Fixed Point Theorem. Adequate details are provided in [47, 50]. Definition 4.3
Let “ D ” denote any one of the Tab, Xtab or Tab-1 deductiveapparatus. Then we will say that the resulting mapping of IS D ( • ) is ConsistencyPreserving iff IS D ( β ) is automatically consistent whenever all the axioms of β holdtrue under the standard model of the natural numbers. The preceding definition raises questions about whether the mappings of IS
Tab ( • ),IS Tab − ( • ), and IS Xtab ( • ) are consistency preserving. It turns out that Theorem 4.4will show the first two of these mappings are consistency preserving, while Theorem4.5 explores how the Law of the Excluded Middle conflicts with IS Xtab ( • )’s Group-3axiom. Theorem 4.4
The IS
Tab − ( • ) and IS Tab ( • ) mappings are consistency pre-serving. (I.e. the axiom systems IS Tab − ( β ) and IS Tab ( β ) are automatically consistentwhenever all β ’s axioms hold true under the standard model of the Natural Numbers.) Theorem 4.5
In contrast, IS
Xtab ( • ) fails to be a consistency-preserving map-ping. (More specifically, IS Xtab ( β ) is automatically inconsistent whenever β provessome conventional Π ∗ theorems stating that addition and multiplication satisfy theirusual associative, commutative, distributive and identity properties.) The proofs of Theorems 4.4 and 4.5 would be quite lengthy, if they were derivedfrom first principles. Fortunately, it is unnecessary for us to do so here because wegave a detailed justification of Theorem 4.4’s result for IS
Tab − ( • ) in [50], and onecan incrementally modify the Remark 3.5’s special Invariant of ++ to justify Theorem4.5. Thus, it will be possible for the next two sections of this paper to adequatelysummarize the intuition behind Theorems 4.4 and 4.5, without delving into the fullformal details.Part of the reason Theorems 4.4 and 4.5 are of interest is because of their sur-prising contrast. Thus, some historians have wondered whether Hilbert and G¨odelwere entirely incorrect when their statements ∗ and ∗∗ suggested some form of the12onsistency Program would likely be viable. Moreover Gerald Sacks’s YouTube talk[39], as well as some added comments by Anil Nerode [32], have reinforced this point.This is because G¨odel repeated analogs of ∗∗ ’s statement on several occasions, duringthe later part of his career. Thus, the contrast between Theorems 4.4 and 4.5 providespossible evidence that a fractional portion of what Hilbert and G¨odel had advocated,might become feasible.This paper will not have the page space to go into the full details, but the nextseveral sections will summarize the gist behind the proofs for Theorems 4.4 and 4.5. Let us recall the acronym “Tab” stands for semantic tableau deduction. This wasdefined by Fitting [14, 15] to be a tree-like proof of a theorem Ψ from an axiombasis α , whose root consists of the temporary negated assumption of ¬ Ψ andwhose every root − to − leaf path establishes a contradiction by containing some pairof contradictory nodes that “close” its path. Each internal node along these pathsmust either be a proper axiom of α or be a deduction from an ancestor node via oneof the “elimination” rules associated with the logic symbols of ∧ , ∨ , → , ¬ , ∀ , or ∃ (that are illustrated in the Appendix.) Example 5.1
Let IS M Tab ( • ) denote a mapping transformation identical toTheorem 4.4’s formalism of IS Tab ( • ), except that IS M Tab shall contain a furthermultiplication function operation and, accordingly, have its Group-3 “I am consistent”axiom statements updated to recognize multiplication as a total function. It turnsout this change will cause IS M Tab ( • ) to stop satisfying the consistency-preservationproperty, which Theorem 4.4 attributed to IS Tab ( • ).The intuition behind this change can be roughly summarized if we let x , x , x , ... and y , y , y , ... denote the sequences defined by: x = 2 = y (6) x i = x i − + x i − (7) y i = y i − ∗ y i − (8)13or i > φ i and ψ i denote the sentences in (7) and (8) respectively. Also,let φ and ψ denote (6)’s sentence. Then φ , φ , ... φ n imply x n = 2 n +1 , and ψ , ψ , ... ψ n imply y n = 2 n . Thus, the latter sequence shall grow at an exponen-tially faster rate than the former. It turns out that this change in growth speed causesthe IS M Tab ( • ), and IS Tab ( • ) to have quite opposite self-justification properties.In particular, let the quantities Log( y n ) = 2 n and Log( x n ) = n + 1 representthe lengths for the binary codings for y n and x n . Thus, y n ’s coding will have alength 2 n , which is much larger than the n + 1 steps of ψ , ψ , ... ψ n (used to define y n ’s existence). In contrast, x n ’s binary encoding will have a sharply smaller lengthof size n + 1. These observations are significant because every proof establishing avariant of the Second Incompleteness Effect involves a G¨odel number z encoding acapacity to self-reference its own definition.The faster growing series y , y , , ... y n should, intuitively, have this self-referencingcapacity because y n ’s binary encoding has a 2 n +1 length that greatly exceeds thesize of the O ( n ) steps used to define its value. Leaving aside many of [48, 53]’s furtherdetails, this fast growth explains roughly why a Type-M logic, such as IS M Tab , satisfiesthe semantic tableau version of the Second Incompleteness Theorem, unlike IS
Tab .Our paradigm also explains why IS
Tab ’s Type-A formalism produces boundary-case exceptions for the semantic tableau version of the Second Incompleteness The-orem. This is because [50] showed that it was unable to construct numbers z thatcan self-reference their own definitions (when only the more slowly growing additionprimitive is available). In particular assuming only two bits are needed to encodeeach sentence in the sequence φ , φ , ...φ n , the length n + 1 for x n ’s binary encodingis insufficient for encoding this sequence.Leaving aside many of [50]’s details, this short length for x n explains the centralintuition behind [50]’s evasion of the Second Incompleteness Theorem under IS Tab .It arises essentially because of the sharp difference between the growth rates of thetwo sequences of x , x , x ... and y , y , y ... .There is obviously insufficient space for this extended abstract to provide moredetails, here. A fully detailed proof of Theorem 4.4 is available in [50]. It establishes(see ) that Peano Arithmetic can prove β ’s consistency implies both the consistency The exact meaning of this implication is subtle. This is because Peano Arithmetic (PA)
Tab − ( β ).Our more modest goal, within the present abbreviated paper, has been to merely summarize the intuition behind Theorem 4.4’s surprising evasion of the Second In-completeness Effect. It arises, intuitively, because of the striking difference in thegrowth rates between the two series of x , x , x ... and y , y , y ... . A formal proof of Theorem 4.5 is complex, but it can be nicely summarized. This isbecause this proposition’s proof is similar to the formal justification for Remark 3.5’sInvariant of ++ . (The latter’s insight has come from the combined work of Pudl´ak,Solovay, Nelson and Wilkie-Paris [36, 42, 31, 45]. It was, also, subsequently verifiedby several other authors [10, 16, 23, 43, 47] in slightly different forms.)The crucial aspect of the Hilbert-Frege deductive methodology is that its modusponens rule assures thar a proof of a theorem ψ from an axiom system α has a lengthno more than proportional to the sum of the proof-lengths used to derive φ and φ → ψ . This “Linear-Sum Effect” does not apply also to Tab- deduction (becausethe latter lacks a modus ponens rule).The
Xtab deductive methodology is , however, quite different from the
Tab formof deduction, in that only Xtab supports an analog of the prior paragraph’s “Linear-Sum Effect”. This is because any node of an Xtab proof-tree is allowed to store anysentence of the form φ ∨ ¬ φ (as a consequence of its allowed use of the Law ofExcluded Middle). This added feature will allow an Xtab proof for ψ to have a lengthproportional to the sum of the proof lengths for φ and φ → ψ . In particular, suchan Xtab proof for ψ will consist of the following four steps:1. The root of an Xtab proof for ψ consists of the usual temporary negated hy-pothesis of ¬ ψ (which the remainder of the proof tree will show is impossibleto hold). CANNOT KNOW whether β is consistent when β = P A . Thus, unlike the quite differentformalism of IS
Tab − ( P A ) , the system of PA shall linger in a state of self-doubt, about whetherboth PA and IS
Tab − ( P A ) are consistent.
The main point is, however, that we humans believe
PA is consistent, and we can use this fact to confirm that IS
Tab − ( P A ) is BOTH consistent andable to verify its self-consistency via its “I am consistent” axiom.
15. The child of this root node consists of an allowed invocation of the Law of theExcluded Middle of the particular form φ ∨ ¬ φ .3. The relevant Xtab proof tree will next employ the Appendix’s branching rulefor allowing the two sibling nodes of φ and ¬ φ to descend from Item 2’snode.4. Finally, our Xtab proof will insert below (3)’s left sibling node of φ a subtreethat is no longer than a proof for φ → ψ , and likewise insert a proof for φ below (3)’s right sibling of ¬ φ .The point is that the very last step of the above 4-part proof has a length nogreater than the sum of the two proof lengths for φ and φ → ψ . (This is analogousto the proof expansions resulting from a conventional modus ponens operation.) Itsfirst three steps will have entirely inconsequential effects that increase the overallproof length by no more than a tiny amount, that is proportional to the trivial sumof the lengths for the two individual sentences of “ φ ” and “ ψ ”.Hence, the preceeding “Linear-Sum Effect” allows us to construct an analog ofRemark 3.5 ’s earlier Theorem + + for Xtab deduction. It is formalized by thestatement J below: J Any axiom system A is automatically inconsistent whenever it satisfies the followingthree conditions: I. A can verify Successor is a total function (as Line (1) formalized). II. A can prove addition and multiplication (viewed as 3-way relations) satisfy theirusual associative, commutative, distributive and identity-operator properties. III. A proves an added theorem (which turns out to be false) affirming its ownconsistency when the Xtab deductive apparatus is used. It is not possible to provide a short proof for statement J because it willrest upon the very detailed “Definable Cut” machinery from pages 172-174 of theH´ajek-Pudl´ak textbook [25]. The intuition behind J is, however, quite simple.It is that statement J causes ++’s mechanism to generalize from Hilbert-Fregededuction to Xtab (because both satisfy the Linear-Sum Effect).16he nice aspect of J is that its machinery establishes Theorem 4.5. Thisis because if β satisfies Theorem 4.5’s hypothesis then IS Xtab ( β ) will satisfy theconditions I-III that cause IS Xtab ( β ) to become inconsistent. Our results in Theorems 4.4 and 4.5 demonstrate self-justifying methodologies applyto “Tab”’, but not also “Xtab” deduction. (This is because Xtab treats the the Lawof Excluded Middle as a formal schema of logical axioms, and the latter activates thepower of the Second Incompleteness Effect.)Our goal in this section will be to view this machinery in more meticulous de-tail. Thus, we will explore at what exact juncture the boundary is crossed betweengeneralizations of the Second Incompleteness Theorem and its permissible exceptions.
Definition 7.1
Let L ∗ again denote the base arithmetic language (that was de-fined in § Z denote an arbitrary set of sentences appearing in the language L ∗ (such as its set of Π ∗ sentences). Let us recall that the Appendix defined a semantictableau proof of a theorem Ψ from α ’s axiom system. Then a Z-Enriched modifica-tion for a semantic tableau proof of a theorem Ψ, from α ’s set of proper axioms, willbe defined as the particular refinement of the Appendix’s proof-tree formalism thatallows Line (9) as an added permissible logical axiom, for any Υ ∈ Z .Υ ∨ ¬ Υ (9)
Definition 7.2
It is also of interest to consider a slight modification of the pre-ceeding nomenclature, where Z is a set of formulae that are allowed to be free inthe single variable of x (instead of representing a sentence that contains no freevariables). In this case, Υ( x ) will designate a formula, within the subset of Z , andLine (10) will replace Line (9) as the added permissible logical axiom that can beallowed to appear inside a “Z-Base Variable Enriched” proof. ∀ x Υ( x ) ∨ ¬ Υ( x ) (10) Actually, IS
Xtab ( β ) will satisfy a requirement stronger than Item I because it recognizes additionas a total function.
17 fully detailed justification will not be provided here, but it turns out our re-sults from [47] can be expanded to show that their evasions of the semantic tableauversion of the Second Incompleteness Theorem can be extended to both the cases ofZ-Enriched and “Z-Base Variable Enriched” mechanisms, when Z represents the ∆ ∗ class of formulae. We can also extend our results from [49] to show that the com-parable evasions of the semantic tableau version of the Second Incompleteness Effectwill fail at and above the Π ∗ level.We conjecture the preceeding ∆ ∗ evasions of the Second Incompleteness Theoremwill continue at the Π ∗ level, but this fact has not yet been formally proven.A fascinating aspect about this subject is that semantic tableau deduction satis-fies its particular variant of G¨odel’s Completeness Theorem [15, 40]. Thus, the setof theorems proven by an axiom system α , via a conventional (unenriched) versionof semantic tableau deduction, is identical to those theorems proven by a Z-enricheddeductive mechanism. Yet despite this invariance, the proof-lengths change, quitesharply, under the Z-enriched formalisms of Defintions 7.1 and 7.2. This extremechange in proof-length causes the deployment of an “I am consistent” axiom to be-come fully infeasible when Υ in Line (9) is allowed to represent any arbitrary Π ∗ sentence (see footnote ). For the sake of simplicity, the previous sections had focused on the semantic tableaudeductive apparatus. However, it is known [15] that resolution shares numerouscharacteristics with tableau. Therefore, it turns out that Theorems 4.4 and 4.5 dogeneralize when resolution replaces semantic tableau.In particular, let us say a theorem T has a Res − proof from α ’s set of properaxioms when there is a resolution-based proof [15] of T from α . Also, the term Xres − proof of T refers to the obvious extension of a Res − proof that allows allinstances of the Law of Excluded Middle (from the base language of L ∗ ) to appearas formalized logical axioms. The point is that the sharp compression in proof lengths produces G¨odel-like Diagonalizationcompressions, similar to those particular Second Incompleteness Effects applicable to Π ∗ sentences,that are examined in [49].
18t turns out
Xres differs from
Res in the same manner
Xtab differed from
T ab .Thus, the obvious generalizations of Theorems 4.4 and 4.5 hold for
Res and
Xres .In particular, IS
Res ( • ) is a consistency preserving transformation, but IS Xres ( • )again is not.Some logicians may, also, wish to examine special speculations in [55]’s arXivarticle. It contemplated an alternative approach, where self-justifying arithmeticsemploy an unconventional “indeterminate” functional object, called the Θ primitive,to formalize the traditional properties of an endless sequence of integers.If a conjecture stated in [55] is correct (as we are almost certain it is), then such aself-justifying machine will be plausible for constructing the entire set of natural num-bers, without encountering the usual incompleteness difficulties that the Theorem ++(of Pudl´ak and Solovay) associated with Type-S formalisms (that recognize merelySuccessor as a total function). Interestingly, the θ function primitive of [55] shouldallow a substantial Type-NS arithmetic to exist that can simultaneously recognizeits own Hilbert-Frege consistency and possess a formalized ability to constructivelyenumerate the full infinite collection of integers 0 , , , , .... The initial 19-page draft of this article was accepted by the LFCS-2020 conference andwas published by Springer [57], shortly before the Covid crisis commenced. DuringJanuary 4-7, when LFCS met, there was little knowledge about the soon-to-appearepidemic. The nature of the Covid event did become apparent by March of 2020. Atthat time, the ASL changed its previously planned North American Annual Meetinginto a virtual conference (with a virtual presentation of our planned slides beingposted at the ASL’s web site).This ironic chronology is, perhaps, worth briefly recollecting because of the con-nection between Theorem 4.4 with the new world of computing that is now, currently,emerging.Thus, mankind will likely become increasingly dependent upon computers in thefuture. For instance, the spread of a serious epidemic can be more effectively con- This is because the Chinese authorities announced the presence of Covid only on December 31.Their announcement had not yet attracted any attention at the LFCS-2020 conference. only partially related to
Theorem 4.4’s self-justifyingIS
T ab mechanism. (For instance, future AI-machines will, certainly, need automatedskills that master the arts of visual learning, motion planning and several formsof decision-making.) Nevertheless, a quite fascinating point is that the early 20-thcentury predictions of Hilbert and G¨odel, in ∗ and ∗∗ , will gain some new positiveinterpretations, when they anticipated significant benefits from future generations ofthinking machines being aware about the consistency of, at least, their specializedrestricted forms of mathematical knowledge.We do not wish to pursue these points further, here, because there will, certainly,be many other types of unanticipated events, which also advance the need for moreelaborate forms of Artificial Intelligence in the future. These future events should beconsistent with, at least, the broad predictions that Hilbert and G¨odel made in theirfamous statements ∗ and ∗∗ .
10 Concluding Remarks
Our main results in this article are surprising because it is quite unusual for aninitially consistent formalism α to become inconsistent when its initial schema oftheorems (establishing the widespread validity of the Law of the Excluded Middle) istransformed into being a schema of logical axioms.20his unusual effect arose because the meaning of a Group-3 “I am consistent” axiom changes, quite substantially, when theorems are transformed into logical axioms(as illustrated by footnote ). Thus, unacceptable diagonalizing contradictions canoccur when an “I am consistent” axiom is able to reference itself in the context of aSUFFICIENTLY POWERFUL mathematical machine. The contrast between Theorems 4.4 and 4.5 (where only the former eschews diag-onalization effects) helpfully explains how Hilbert and G¨odel appreciated the SecondIncompleteness Effect, while they were simultaneously cautious about it.
Moreover,G¨odel’s particular remark ∗∗ should not be ignored when comments from GeraldSacks and Stanley Tennenbaum [32, 39] recalled how G¨odel reiterated the gist of his1931-published remark, many years after its printing. Indeed, it is noteworthy HarveyFriedman recorded a YouTube lecture [18], stating he was also tentatively open to thepossibility that the Second Incompleteness Theorems might allow partial exceptions.Thus, while there is no doubt that the Second Incompleteness Theorem will beremembered for its seminal impact, its part-way exceptions are also significant. Thisis because futuristic high-tech computers will better understand their self-capacities,if they own some partial awareness about their own consistency.There is no page space to delve into all details here. However, the distinctionbetween the initial “IS(A)” system, from our 1993 and 2001 papers [46, 47], withthe more sophisticated IS Tab − ( β ) formalism of our year-2005 article [50] should,also, be briefly mentioned. Our older “IS(A)” formalism was actually simpler, butit was substantially weaker because it only recognized the non-existence of a proofof 0 = 1 from itself. In contrast, IS Tab − ( β )’s Group-3 axiom can corroborate that no two simultaneous proofs exist for a Rank-1* sentence and its negation. This isan important distinction, because the First Incompleteness Theorem indicates nodecision procedure exists for separating all true from false Rank-1* sentences. (See[51, 52, 54, 55] for other particular refinements for our “IS(A)” formalism.)In summary, the main purpose of this article has been to explore the contrastbetween the opposing Theorems 4.4 and 4.5. The latter theorem, thus, provides another helpful reminder about the millennial importance of G¨odel’s seminal Second The point is that proofs are compressed when theorems are transformed into logical axioms, andsuch compressions can produce diagonalizing contradictions under some Type-A logics using “I amconsistent” axioms. partial exception s to G¨odel’s result do arise, as Hilbert and G¨odel predicted in theirstatements ∗ and ∗∗ .In essence, the 2-way contrast between Theorems 4.4 and 4.5 may be as significantas their individual actual results. This is because the Second Incompleteness Theoremis fundamental to Logic. Many historians have, thus, been perplexed by the partial reluctance that Hilbert and G¨odel had expressed about it in ∗ and ∗∗ . A partial reasonfor this reluctance is, perhaps, related to the contrast between these two opposingtheorems. ACKNOWLEDGMENTS:
I thank Seth Chaiken and James P. Torre, IV forseveral quite helpful comments about how to improve the presentation.22 ppendix providing a formal definition for a Se-mantic Tableau proof:
Our definition of a semantic tableau proof is similar to analogs from the textbooksby Fitting and Smullyan [15, 40]. A tableau proof of a theorem Ψ from a set ofproper axioms (denoted as α ) is therefore a tree structure, whose root containsthe temporary contradictory assumption of ¬ Ψ and whose every descending root-to-leaf branch affirms a contradiction by containing both some sentence φ and itsnegation ¬ φ . Each internal node in this tree will be either a proper axiom of α or adeduction from a higher ancestor in this tree via one of six elimination rules for thelogical connective symbols of ∧ , ∨ , → , ¬ , ∀ and ∃ . (These rules use anotation where “ A = ⇒ B ” is an abbreviation for a sentence B being an alloweddeduction from its ancestor of A .)1. Υ ∧ Γ = ⇒ Υ and Υ ∧ Γ = ⇒ Γ .2. ¬ ¬
Υ = ⇒ Υ . Other rules for the “ ¬ ” symbol are: ¬ (Υ ∨ Γ) = ⇒ ¬ Υ ∧ ¬ Γ, ¬ (Υ → Γ) = ⇒ Υ ∧ ¬ Γ , ¬ (Υ ∧ Γ) = ⇒ ¬ Υ ∨ ¬ Γ , ¬ ∃ v Υ( v ) = ⇒ ∀ v ¬ Υ( v )and ¬ ∀ v Υ( v ) = ⇒ ∃ v ¬ Υ( v )3. A pair of sibling nodes Υ and Γ is allowed when their ancestor is Υ ∨ Γ .
4. A pair of sibling nodes ¬ Υ and Γ is allowed when their ancestor is Υ → Γ.5. ∀ v Υ( v ) = ⇒ Υ( t ) where t may denote any term.6. ∃ v Υ( v ) = ⇒ Υ( p ) where p is a newly introduced parameter symbol.One minor difference in notation is we treat “ ∀ v ≤ s Φ( v ) ” as an abbreviationfor ∀ v { v ≤ s → Φ( v ) } and “ ∃ v ≤ s Φ( v ) ” as an abbreviation for ∃ v { v ≤ s ∧ Φ( v ) } . Therefore, Rules 5 and 6 imply the following hybrid rulesfor processing bounded universal and bounded existential quantifiers: a. ∀ v ≤ s Υ( v ) = ⇒ t ≤ s → Υ( t ) where t may be any arithmetic term. b. ∃ v ≤ s Υ( v ) = ⇒ p ≤ s ∧ Υ( p ) where p is a new parameter symbol. Added Comment:
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