Lorenzen's proof of consistency for elementary number theory [with an edition and translation of "Ein halbordnungstheoretischer Widerspruchsfreiheitsbeweis'']
aa r X i v : . [ m a t h . HO ] J un Lorenzen’s proof of consistency for elementarynumber theory
Thierry Coquand
Computer science and engineering department, University of Gothenburg, Sweden, [email protected] . Stefan Neuwirth
Laboratoire de mathématiques de Besançon, Université Bourgogne Franche-Comté, France, [email protected] . Abstract
We present a manuscript of Paul Lorenzen that provides a proof of con-sistency for elementary number theory as an application of the construc-tion of the free countably complete pseudocomplemented semilattice overa preordered set. This manuscript rests in the Oskar-Becker-Nachlass atthe Philosophisches Archiv of Universität Konstanz, file OB 5-3b-5. Ithas probably been written between March and May 1944. We also com-pare this proof to Gentzen’s and Novikov’s, and provide a translation ofthe manuscript.Keywords: Paul Lorenzen, consistency of elementary number the-ory, free countably complete pseudocomplemented semilattice, inductivedefinition, ω -rule. We present a manuscript of Paul Lorenzen that arguably dates back to 1944and provide an edition and a translation, with the kind permission of Lorenzen’sdaughter, Jutta Reinhardt.It provides a constructive proof of consistency for elementary number theoryby showing that it is a part of a trivially consistent cut-free calculus. The proofresorts only to the inductive definition of formulas and theorems.More precisely, Lorenzen proves the admissibility of cut by double induction,on the complexity of the cut formula and of the derivations, without using anyordinal assignment, contrary to the presentation of cut elimination in moststandard texts on proof theory.Prior to that, he proposes to define a countably complete pseudocomple-mented semilattice as a deductive calculus, and shows how to present it for1onstructing the free countably complete pseudocomplemented semilattice overa given preordered set.He arrives at the understanding that the existence of this free kind of latticecaptures the formal content of the consistency of elementary number theory,the more so as he has come to understand that the existence of another freekind of lattice captures the formal content of ideal theory. In this way, latticetheory provides a bridge between algebra and logic: by the concept of preorder,the divisibility of elements in a ring becomes commensurate with the materialimplication of numerical propositions; the lattice operations give rise to theideal elements in algebra and to the compound propositions in logic.The manuscript has remained unpublished, being superseded by Lorenzen’s‘Algebraische und logistische Untersuchungen über freie Verbände’ that ap-peared in 1951 in
The Journal of Symbolic Logic . These ‘Algebraic and logisticinvestigations on free lattices’ have immediately been recognised as a landmarkin the history of infinitary proof theory, but their approach and method of proofhave not been incorporated into the corpus of proof theory.
In 1938, Paul Lorenzen defends his Ph.D. thesis under the supervision of HelmutHasse at Göttingen, an ‘Abstract foundation of the multiplicative ideal theory’,i.e. a foundation of divisibility theory upon the theory of cancellative monoids.He is in a process of becoming more and more aware that lattice theory isthe right framework for his research. Lorenzen ( , footnote on p. 536)thinks of understanding a system of ideals as a lattice, with a reference to
Köthe 1937 ; in the definition of a semilattice-ordered monoid on p. 544, hecredits Dedekind’s two seminal articles of 1897 and 1900 for developing theconcept of lattice. On 6 July 1938 he reports to Hasse: ‘Momentarily, I am atmaking a lattice-theoretic excerpt for Köthe’. He also reviews several articleson this subject for the
Zentralblatt , e.g.
Klein 1939 and
George 1939 whichboth introduce semilattices,
Whitman 1941 which studies free lattices. He alsoknows about the representation theorem for boolean algebras in
Stone 1936 and he discusses the axioms for the arithmetic of real numbers in
Tarski 1937 with Heinrich Scholz. Helmut-Hasse-Nachlass, Niedersächsische Staats- und Universitätsbibliothek Göttingen,Cod. Ms. H. Hasse 1:1022, edited in
Neuwirth 2019 , § 4. See the collection of documents grouped together by Scholz under the title ‘
Paul Lorenzen :Gruppentheoretische Charakterisierung der reellen Zahlen [Group theoretic characterisationof the real numbers]’ and deposited at the Bibliothek des Fachbereichs Mathematik undInformatik of the Westfälische Wilhelms-Universität Münster, as well as several letters filed
2n 1939, he becomes assistant to Wolfgang Krull at Bonn. During WorldWar II, he serves first as a soldier and then, from 1942 on, as a teacher atthe naval college Wesermünde. He devotes his ‘off-duty evenings all alone on[his] own’ to mathematics with the goal of habilitating. On 25 April 1944, hewrites to his advisor that ‘[. . . ] it became clear to me—about 4 years ago—thata system of ideals is nothing but a semilattice’. He will later recall a talk by Gerhard Gentzen on the consistency of ele-mentary number theory in 1937 or 1938 as a trigger for his discovery that thereformulation of ideal theory in lattice-theoretic terms reveals that his ‘algeb-raic works [. . . ] were concerned with a problem that had formally the samestructure as the problem of freedom from contradiction of the classical calculusof logic’; compare also his letter to Eckart Menzler-Trott (see Menzler-Trott2001 , p. 260).In his letter dated 13 March 1944, he announces: ‘Subsequently to an al-gebraic investigation of orthocomplemented semilattices, I am now trying toget out the connection of these questions with the freedom from contradictionof classical logic. [. . . ] actually I am much more interested into the algebraicside of proof theory than into the purely logical’. The concept of ‘orthocomple-mentation’ (see p. 19 for its definition) must have been motivated by logicalnegation from the beginning. On the one hand, such lattices correspond to thecalculus of sequents considered by Gentzen ( , section IV), who shows thata given derivation can be transformed into a derivation ‘in which the connect-ives ∨ , ∃ and ⊃ no longer occur’ and provides a proof of consistency for thiscalculus (see section 3 below). On the other hand, note that Lorenzen reviews Ogasawara 1939 for the
Zentralblatt . The result of this investigation can be found in the manuscript ‘Ein halb-ordnungstheoretischer Widerspruchsfreiheitsbeweis’. in the Heinrich-Scholz-Archiv at Universitäts- und Landesbibliothek Münster, the earliestdated 7 April 1944. Carbon copy of a letter to Krull, 13 March 1944, Paul-Lorenzen-Nachlass, PhilosophischesArchiv, Universität Konstanz, PL 1-1-131, edited in
Neuwirth 2019 , § 6. Carbon copy of a letter to Krull, PL 1-1-132, edited in
Neuwirth 2019 , § 6. Letter to Carl Friedrich Gethmann, see
Gethmann 1991 , p. 76. PL 1-1-131, edited in
Neuwirth 2019 , § 6. The terminology might be adapted from
Stone 1936 , where it has a Hilbert space back-ground; today one says ‘pseudocomplementation’. ‘A proof of freedom from contradiction within the theory of partial order’, Oskar-Becker-Nachlass, Philosophisches Archiv, Universität Konstanz, OB 5-3b-5, https://archive.org/
3e believe that it is the one that he assertedly sends to Wilhelm Ackermann,Gentzen, Hans Hermes and Heinrich Scholz between March and May 1944, andfor which he gets a dissuasive answer from Gentzen, dated 12 September 1944:‘I have looked through your attempt at a consistency proof, not in detail, forwhich I lack the time. However I say this much: the consistency of numbertheory cannot be proven so simply’. Our identification of the manuscript is made on the basis of the followingdating: Lorenzen mentions such a manuscript and its recipients in his lettersto Scholz dated 13 May 1944 and 2 June 1944, and in a postcard to Hassedated 25 July 1945; a letter by Ackermann dated 11 November 1946 statesthat he lost a manuscript by Lorenzen ‘at the partial destruction of his flat bybombs’. Our identification is also consistent with the content of Lorenzen’sletter to Menzler-Trott mentioned above. On the other hand, we have notfound any hint at another manuscript by Lorenzen for which it could havebeen mistaken. The generalisation of his proof of consistency to ramifiedtype theory is first mentioned in a letter from Scholz to Bernays dated 11December 1945: it corresponds to the manuscript ‘Die Widerspruchsfreiheitder klassischen Logik mit verzweigter Typentheorie’ and is the future part IIof his 1951 article.This manuscript renews the relationship between logic and lattice theory:whereas boolean algebras were originally conceived for modeling the classicalcalculus of propositions, and Heyting algebras for modeling the intuitionisticone, here logic comes at the rescue of lattice theory for studying countablycomplete pseudocomplemented semilattices.We have found only three contemporaneous occurrences of the notion ofcountably complete lattice other than σ -fields of subsets of a given set used inmeasure theory: Birkhoff ( , p. 795; article reviewed by Lorenzen for the details/lorenzen-ein_halbordnungstheoretischer_widerspruchsfreiheitsbeweis . Thefile OB 5-3b consists of documents related to Lorenzen, the oldest being the 1944 manu-script and the youngest a letter from 1951. Lorenzen and Becker are both at Bonn from 1945to 1956 and have been in close contact since at least 1947: see Lorenzen’s letter to Gethmann(in Gethmann 1991 , p. 77). The letter is reproduced in
Menzler-Trott 2001 , p. 372, and translated in
Menzler-Trott2007 . Heinrich-Scholz-Archiv and PL 1-1-138. Cod. Ms. H. Hasse 1:1022, edited in
Neuwirth 2019 , § 7. ‘So ist auch ein Manuskript, das Sie mir seiner Zeit zuschickten, bei der teilweisen Zer-störung meiner Wohnung durch Bomben verschwunden’ (PL 1-1-125). The ‘unpublished’ manuscript ‘Ein finiter Logikkalkül’ mentioned by Lorenzen ( ,p. 20) may be dated to 1947 even if we have not spotted a copy of it: the review given thereshows that it corresponds to a thread of research described in a letter to Bernays dated 21February 1947 (ETH-Bibliothek, Hochschularchiv, Hs 975:2950). Hs 975:4111. entralblatt ) speaks of ‘ σ -lattice, by analogy with the usual notions of σ -ringsand σ -fields of sets’; in the appendix The somen as elements of partially orderedsets of the posthumously published book
Carathéodory 1956 ; the ‘ ℵ -lattice’ in von Neumann 1937 .Lorenzen describes a countably complete pseudocomplemented semilatticeas a deductive calculus on its own, without any reference to a larger formalframework: this conception dates back to the ‘system of sentences’ of Hertz( , ). The rules of the calculus construct the free countably completepseudocomplemented semilattice over a given preordered set by taking as ax-ioms the inequalities in the set, by defining inductively formal meets and formalnegations, and by introducing inequalities between the formal elements. Theintroduction rule for formal countable meets, stating thatif c a , c a , . . . , then c V M , where M = ( a , a , . . . )(rule c on p. 21), stands out: it has an infinity of premisses, so that it is an‘ ω -rule’ in today’s terminology. Lorenzen’s boldness is most probably due to histraining in algebra, where such a rule is very natural, so that when he arrivesat a clear constructive understanding of ideal theory, he has also got a clearconstructive understanding of the ω -rule.In ideal theory, Lorenzen ( , § 4) defines a system of ideals for a pre-ordered set as the free semilattice generated by it: it consists in the formalmeets a ∧ · · · ∧ a m of finitely many elements a , . . . , a m ; this formal element isintroduced with the following rules: if c a , . . . , c a m , then c a ∧· · ·∧ a m ; a ∧ · · · ∧ a m a , . . . , a ∧ · · · ∧ a m a m . The ω -rule is the infinitary coun-terpart of the first rule, and the infinitary counterpart of the second rule is theadmissible rule ε on p. 23.Lorenzen’s presentation of elementary number theory can be compared tothat of Gödel 1933 as follows.• Lorenzen starts with ‘prime formulas’, i.e. the numerical propositions ase.g. 1 = 1 ′′ or 1 + 1 = 1 ′ . These are preordered by material implicationand may be combined into compound formulas. Lorenzen works in aconstructive metatheory, in which infinitely many propositions may besupervised if given by a construction, e.g. the propositions C → A (1) , C → A (2) , . . . , and rule c on p. 31 is the rule of introduction of the universalquantifier that one may infer from these C → ( x ) A ( x ).• Gödel starts with ‘elementary formulas’, which may also contain variables.He works in a finitary metatheory in which only finitely many propositions In contradistinction to the ‘consequence relation’ of
Tarski 1930 which presupposes settheory.
Gödel 1931 apredicate A ( x ) such that each of the propositions C → A (1) , C → A (2) , . . . holds, but C → ( x ) A ( x ) does not.In elementary number theory, the rule of complete induction plays a centralrôle. The statement of this rule is complex from a logical point of view becauseof the presence of a free variable, of a universal quantifier, or of an implication.The ω -rule appears as an analysis of this complexity: the rule of completeinduction is the derivation of A (1) → ( x ) A ( x ) from A ( a ) → A ( a + 1) with afree variable a ; in the latter, replacement of a by 1 , , . . . and the cut ruleyield A (1) → A (2) , A (1) → A (3) , . . . ; therefore this rule is a combination of theadmissible cut rule k on p. 31 with the ω -rule that derives A (1) → ( x ) A ( x ) from A (1) → A (1) , A (1) → A (2) , A (1) → A (3) , . . . . Conversely, the only expecteduses of the ω -rule correspond to the rule of complete induction and to therule of introduction of the universal quantifier. The ω -rule has a very simplestructure: its premisses are stated without further need of free variables andquantifiers; however, there are infinitely many. Its main feature is that it allowsfor derivations without detour.Sundholm ( ) and Feferman ( ) provide a historical account of suchrules. Hilbert ( , b ) states an ω -rule with the motivation of, respectively,proving the completeness of arithmetic and the law of excluded middle. Hedeclares that it is a ‘finitary deduction rule’, that it has a ‘rigorously finitarycharacter’. Lorenzen makes no reference to these articles, but, in the 1945 manu-script ‘Die Widerspruchsfreiheit der klassischen Logik mit verzweigter Typen-theorie’, he expands on the finitary character of its usage: ‘One has to persuadeoneself at each appearance of this rule that its application occurs to the effect ofa “finitary deduction”, because the proof of freedom from contradiction wouldotherwise become meaningless’. E.g. in the derivation of the rule of completeinduction on p. 33, the infinitely many premisses A (1) → A (1) , A (1) → A (2) , . . . Hilbert ( ) states a restricted ω -rule, in the sense that its premisses must be decidable(i.e. numerical); he states the axiom of complete induction separately. This is noted in theletter that Bernays addresses to Gödel on 18 January 1931 (Feferman, Dawson, Goldfarb,Parsons et al. 2003 , pp. 80–91), where he formulates its unrestricted counterpart. See alsoGödel’s answer dated 2 April 1931. The ω -rule in Hilbert 1931b is not restricted. Compare
Ewald, Sieg, Hallett, Majer, and Schlimm 2013 , pp. 788–805, 967–973, 983–984. ‘Man hat sich bei jedem Vorkommen dieser Regeln zu überzeugen, daß ihre Anwendungim Sinne des “finiten Schließens” geschieht, weil sonst der Wf-Beweis sinnlos würde’ (‘Thefreedom from contradiction of classical logic with ramified type theory’; a version of thismanuscript can be found in Niedersächsische Staats- und Universitätsbibliothek Göttingen,Cod. Ms. G. Köthe M 10). m follows therefrom at once A (1) → A ( m ) by m -fold application of the rule of inference k ’. Lorenzen shares this intuitionisticframework with Gentzen ( , p. 526): ‘After all, we need not associate theidea of a closed infinite number of individual propositions with this [( x ) A ( x ),where A shall not yet contain an universal or existential quantifier], but can,rather, interpret its sense “finitistically” as follows: “If, starting with 1, wesubstitute for x successive natural numbers then, however far we may progressin the formation of numbers, a true proposition results in each case”’.In his letter to Bernays dated 2 April 1931, Gödel points out that such arule presupposes a framework in which this infinity of premisses may be as-serted: ‘the very complicated and problematical concept “finitary proof” isassumed [. . . ] without having been made mathematically precise’ (see Fefer-man, Dawson, Goldfarb, Parsons et al. , p. 97). This framework is thus aninformal one; and, as the proof of consistency rests on its reliability, this frame-work is to be the intuitionistic one, as Herbrand ( , ‘groupe D’, p. 5) andNovikoff ( , p. 231) state, i.e. the constructive one (
Lorenzen 1951 , p. 82).In this sense, a calculus including the ω -rule is of a different nature than amechanical calculus, where we can check by a finitary process the correctnessof a given derivation. In fact, neither the Hilbert program nor Lorenzen’s proofof consistency take place in a mechanical formal system, i.e. in a system whoseobjects are finitary and whose derivations are finitary and decidable.The proof that the calculus thus defined is a countably complete pseudocom-plemented semilattice illustrates, as Lorenzen realises a posteriori, that thestrategy of Gentzen’s dissertation ( , IV, § 3) for proving the consistencyof elementary number theory without complete induction may be maintainedfor proving the consistency of all of elementary number theory: the introduc-tion rules (rules a to f on p. 21) introduce inequalities for formal elementsof increasing complexity, i.e. no inequality can result from a detour; then thecorresponding elimination rules (rules γ to ε on p. 23) are shown to hold by aninduction on the complexity of the introduced inequality (in Lorenzen’s laterterminology, one would say that these rules are shown to be ‘admissible’ andcan be considered as resulting from an ‘inversion principle’); at last transitivityof the preorder, i.e. the cut rule (rule β on p. 23: if a b and b c , then a c ), is established by proving a stronger rule through an induction on thecomplexity of the cut element b nested with inductions on the complexity ofthe derivation of the rule’s premisses.The inductions used here are the ones accurately described by Jacques This is how we interpret the beginning of the second paragraph on p. 17: ‘Withoutknowledge of [. . . ]’. , pp. 4–5) after having been emphasised by David Hilbert( , p. 76): the first proceeds along the construction of formulas startingfrom prime formulas through rules, and has no special name (it will be called‘formula induction’ in
Lorenzen 1951 ); the second proceeds along the construc-tion of theorems starting from prime theorems through deduction rules, and iscalled ‘premiss induction’. In other words, Lorenzen starts with a preordered set P, constructs the freecountably complete pseudocomplemented semilattice K over P and emphasisesconservativity, i.e. that no more inequalities come to hold among elements of Pviewed as a subset of K than the ones that have been holding before: onesays that P is embedded into K and that the preorder of P is embedded intothe countably complete preorder of K.Then the consistency of elementary number theory with complete inductionis established in § 3 by constructing the free countably complete pseudocomple-mented semilattice over its ‘prime formulas’, i.e. the numerical formulas, viewedas a set preordered by material implication.Note the presence of rule g on pp. 21 and 31, a contraction rule. This shouldbe put in relation• with the rôle of contraction, especially for steps 13. 5 1–13. 5 3, in Gentzen’sproofs of consistency ( , );• with the calculus of P. S. Novikoff ( , lemma 6), in which contractionmay be proved. There are similarities and differences with respect to the strategy developed byGentzen for proving the consistency of elementary number theory with completeinduction. In his first proof, submitted in August 1935, withdrawn and finallypublished posthumously by Bernays in (after its translation by
Szabo1969 ), Gentzen defines a concept of reduction procedure for a sequent andshows that such a procedure may be specified for every derivable sequent butnot for the contradictory sequent → See
Lorenzen 1939b for his interest in the foundation of inductive definitions. This is exactly the approach of Skolem ( , § 2) for constructing the free lattice overa preordered set, in the course of studying the decision problem for lattices. ∀ x F ( x ), the following stepof the reduction procedure consists in replacing it by F ( n ), where n is anumber to be chosen freely.• A reduction procedure is defined as the specification of a sequence ofsteps for all possible free choices, with the requirement that the reductionterminates for every such choice.In his letter to Bernays dated 4 November 1935, Gentzen visualises a reduc-tion procedure as a tree whose every branch terminates.The proof that a reduction procedure may be specified for every derivablesequent is by theorem induction. For this, a lemma is needed, claiming that ifreduction procedures are known for two sequents Γ → D and D, ∆ → C , thena reduction procedure may be specified for their cut sequent Γ , ∆ → C . Theproof goes by induction on the construction of the cut formula D and traces theclaim back to the same claim with the same cut formula, but with the sequent D, ∆ → C replaced by a sequent D, ∆ ∗ → C ∗ resulting from it after one or morereduction steps and the cut sequent replaced by Γ , ∆ ∗ → C ∗ . By definition ofthe reduction procedure, this tracing back must terminate eventually.This last kind of argument may be considered as an infinite descent inthe reduction procedure. In his letter to Bernays, Gentzen seems to indicatethat this infinite descent justifies an induction on the reduction procedure; asanalysed by William W. Tait ( ), this would be an instance of the Bartheorem. But in his following letter, dated 11 December 1935, he writes that‘[his] proof is not satisfactory’ and announces another proof, to be submittedin February 1936: in it, he defines the concept of reduction procedure for aderivation (and not for a sequent), associates inductively an ordinal to everyderivation, and shows that a reduction procedure may be specified for everyderivation by an induction on the ordinal.Let us compare this strategy with Lorenzen’s.• The free choice is subsumed in a deduction rule, an ω -rule as describedabove (rules c and j on p. 31). • Elementary number theory is constructed as the cut-free derivations start-ing from the numerical formulas, so that it is trivially consistent, and thecut rule (rule k on p. 31) is shown to be admissible: if derivations areknown for two sequents A → B and B → C , then a derivation may be Hs 975:1652, translated by von Plato ( , pp. 241–244). Hs 975:1653, translated by von Plato ( , p. 244). Compare Bernays’ suggestion in his letter to Gentzen dated 9 May 1938, Hs 975:1661,translated by von Plato ( , pp. 254–255). A → C by a formula induction on the cutformula B nested with several instances of a theorem induction.In this way, Lorenzen’s strategy may be used to realise the endeavour expressedby Tait ( ): ‘the gap in Gentzen’s argument is filled, not by the Bar The-orem, but by taking as the basic notion that of a [cut-free] deduction tree inthe first place rather than that of a reduction tree’. His 1944 proof can thusbe seen as a formal improvement on Gentzen’s 1935 argument, which is all themore remarkable given Gentzen’s reaction to Lorenzen’s proof. Novikoff ( ) introduces an intuitionistic calculus that contains an ω -rule(rule 6 on p. 233). He defines in § 4 the concept of ‘regular formula’ thatexpresses that the formula has a cut-free proof, and shows in § 8 that it isan explanation of classical truth. In fact, he proves essentially that cut (‘therule of inference’) is admissible. This proof does not use any induction on thecut formula, contrary to Gentzen’s and Lorenzen’s proofs (see Mints 1991 and
Tupailo 1992 ). In his introduction, Novikov writes: ‘As a basis, the consistencyof which is assumed, the intuitionistic mathematics is taken. From such a pointof view it appears to be possible to prove the consistency of [elementary numbertheory]’.
On p. 23, the premiss induction that establishes rule γ is given the form of areductio ad absurdum, but the reasoning may easily be unraveled into a directform.The calculus N presented on p. 31 is in fact common to intuitionistic andclassical arithmetic: recall that ‘the connectives ∨ , ∃ and ⊃ no longer occur’.It may be criticised for its sloppy way of treating variables.Furthermore, the introduction of free variables seems dispensable in thepresence of an ω -rule. Rule j and the corresponding elimination rule p may beomitted from the calculus at the affordable price of giving complete inductionthe less elegant form A (1) & ( x ) A ( x ) & A ( x ′ ) → ( x ) A ( x ) as in Lorenzen 1962 .10
Conclusion
Proof theory continues to focus on measures of complexity by ordinal numbers.The fact that Lorenzen does not resort to ordinals in his proof of consistencyshould be considered as a feature of his approach.Lorenzen’s article is remarkable for its metamathematical standpoint. Amathematical object is presented as a construction described by rules. A claimon the object is established by an induction that expresses the very meaning ofthe construction.The relations between these objects, of the form of an inequality or of animplication, also admit such a presentation: it has the feature that the con-struction of a relation proceeds as accumulatively (‘without detour’, i.e. cut) asthe construction of the formulas appearing in the relation. It is only in a secondplace that the corresponding elimination rules and the cut rule are shown tobe admissible.In elementary number theory and for the free countably complete pseudo-complemented semilattice, the construction of a relation uses an ω -rule thatis stronger than the rule of complete induction but requires infinitely manypremisses, so that a relation corresponds to a well-founded tree.Lorenzen’s standpoint holds equally well for a logical calculus and for a lat-tice: ‘logical calculuses are semilattices or lattices’ ( Lorenzen 1951 , p. 89).The consistency of the logical calculus of elementary number theory is recog-nised as a consequence of the following fact: a preordered set embeds into thefree countably complete pseudocomplemented semilattice generated by it in aconservative way.Other reflections on the philosophical significance of Lorenzen’s approachto logic are addressed by Matthias Wille ( , ). Acknowledgments
We thank Brigitte Parakenings for having provided ideal working conditionsand her expertise at Philosophisches Archiv of Universität Konstanz, and HenriLombardi and Jan von Plato for helpful discussions. This research has been sup-ported through the program ‘Research in pairs’ of Mathematisches Forschungsin-stitut Oberwolfach in 2016 and through the hospitality of the university ofGothenburg. We prefer this plural with Curry ( ). eferences Birkhoff, G. 1938. ‘Lattices and their applications’.
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Abh. Math. Semin. Univ. Hambg. ,6, 65–85. doi:10.1007/BF02940602. Translation by S. Bauer-Mengelberg and D. Følles-dal: ‘The foundations of mathematics’, in van Heijenoort 1967 , 464–479.Hilbert, D. 1931a. ‘Die Grundlegung der elementaren Zahlenlehre’.
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Naturforschungund Medizin in Deutschland 1939–1946, 1 [Fiat Rev. German Sci.]: Reine Mathem-atik, I . Wiesbaden: Dieterich’sche Verlagsbuchhandlung, 11–22.Lorenzen, P. 1950. ‘Über halbgeordnete Gruppen’.
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Monatsh. Math. Phys. , 37(1), 361–404. doi:10.1007/BF01696782. Trans-lation by J. H. Woodger: ‘Fundamental concepts of the methodology of the deductivesciences’, in J. H. Woodger (ed.),
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Ann. of Math. (2) , 42, 325–330.doi:10.2307/1969001. Review by Lorenzen: Zbl. 0024.24501.Wille, M. 2013. ‘Zwischen Algebra und Erlanger Schule: Paul Lorenzens Beiträge zurBeweistheorie’.
Siegener Beiträge zur Geschichte und Philosophie der Mathematik , 1,79–108. http://nbn-resolving.de/urn:nbn:de:hbz:467-8245 .Wille, M. 2016. ‘Verzweigte Typentheorie, relative Konsistenz und Fitch-Beweis: wieLorenzen (nach eigener Auskunft) Hilberts Forderungen für die Analysis erfüllte’. InJ. Mittelstraß (ed.),
Paul Lorenzen und die konstruktive Philosophie . Münster: mentis,163–174.
Ein halbordnungstheoretischer Widerspruchsfreiheitsbeweis.
Die Dissertation von G. Gentzen enthält einen Wf-beweis der reinen Zah-lentheorie ohne vollständige Induktion, der auf dem folgenden Grundgedankenberuht: jede herleitbare Sequenz muß sich auch ohne Umwege herleiten lassen,sodaß während der Herleitung nur die Verknüpfungen eingeführt werden, dieunbedingt notwendig sind, nämlich diejenigen, die in der Sequenz selbst en-thalten sind. In dem Wf-beweis der Zahlentheorie mit vollständiger Induktiontritt dieser Grundgedanke gegenüber anderen zurück. Ich möchte jedoch imfolgenden zeigen, daß er allein genügt, auch diese Wf. zu erhalten.Ohne Kenntnis der Dissertation von Gentzen bin ich auf diese Möglichkeitauf Grund einer halbordnungstheoretischen Frage gekommen. Diese lautete:wie läßt sich eine halbgeordnete Menge in einen orthokomplementären voll-ständigen Halbverband einbetten? Im allgemeinen sind mehrere solche Ein-bettungen möglich – unter den möglichen Einbettungen ist aber eine ausgezeich-net, nämlich die, welche sich in jede andere homomorph abbilden läßt. DieExistenz dieser ausgezeichneten Einbettung wird in § 2 bewiesen.Um hieraus in § 3 den gesuchten Wf-beweis zu erhalten, ist nur noch eineÜbersetzung des halbordnungstheoretischen Beweises in die logistische Sprachenotwendig. Denn der Kalkül, den wir betrachten und auf den sich die üb-lichen Kalküle zurückführen lassen, ist in der ausgezeichneten Einbettung derhalbgeordneten Menge der zahlentheoretischen Primformeln enthalten. | § 1. Eine Menge M heißt ha lbg eo r dnet, wenn in M eine zweistelligeRelation definiert ist, sodaß für die Elemente a, b, . . . von M gilt: a aa b, b c ⇒ a c .Gilt a b und b a , so schreiben wir a ≡ b .Gilt a x für jedes x ∈ M, so schreiben wir a . Ebenso schreiben wir a ,wenn x a für jedes x gilt. ( bedeutet also, daß x y für jedes x, y ∈ Mgilt.)Eine halbgeordnete Menge M heißt Ha lbver ba nd, wenn es zu jedem a, b ∈ M ein c ∈ M gibt, sodaß für jedes x ∈ M gilt x a, x b ⇐⇒ x c . c heißt die Konjunktion von a und b : c ≡ a ∧ b .16P. LORENZEN] A proof of freedom from contradiction within the theory of partialorder.
The dissertation of G. Gentzen contains a proof of freedom from contradic-tion of elementary number theory without complete induction that relies onthe following basic thought: every derivable sequent must also be derivablewithout detour, so that during the derivation only those connectives are beingintroduced that are absolutely necessary, i.e. those that are contained in thesequent itself. In the proof of freedom from contradiction of number theorywith complete induction, this basic thought steps back with regard to others. Iwish however to show in the following that it alone suffices to obtain also thisfreedom from contradiction.Without knowledge of the dissertation of Gentzen, I have arrived at this pos-sibility on the basis of a semilattice-theoretic question. This question is: howmay a partially ordered set be embedded into an orthocomplemented completesemilattice? In general, several such embeddings are possible – but among thepossible embeddings one is distinguished, i.e. the one which may be mapped ho-momorphically into every other. The existence of this distinguished embeddingwill be proved in § 2.In order to obtain from this in § 3 the sought-after proof of freedom fromcontradiction, now just a translation of the semilattice-theoretic proof into thelogistic language is necessary. For the calculus that we consider, and to whichthe usual calculuses may be reduced, is contained in the distinguished embed-ding of the partially ordered set of the number-theoretic prime formulas. | § 1. A set M is called partially ordered if a binary relation is definedin M so that for the elements a, b, . . . of M holds: a aa b, b c ⇒ a c .If a b and b a holds, then we write a ≡ b .If a x holds for every x ∈ M, then we write a . We write as well a if x a holds for every x . ( means thus that x y holds for every x, y ∈ M.)A partially ordered set M is called semilattice if to every a, b ∈ M there isa c ∈ M so that for every x ∈ M holds x a, x b ⇐⇒ x c .17in Halbverband M heißt o r tho ko mplementä r, wenn es zu jedem a ∈ Mein b ∈ M gibt, so daß für jedes x ∈ M gilt a ∧ x ⇐⇒ x b . b heißt das Orthokomplement von a : b ≡ ¯ a .Ein Halbverband M heißt ω -vo llstä ndig, wenn es zu jeder abzählbarenFolge M = a , a , . . . in M ein c ∈ M gibt, so daß für jedes x ∈ M gilt: ( für jedes n : x a n ) ⇐⇒ x c . c heißt die Konjunktion der Elemente von M : c ≡ V n a n ≡ V M .Sind M und M ′ halbgeordnete Mengen, so heißt M ein Teil von M ′ , wenn M Untermenge von M ′ ist und für jedes a, b ∈ M genau dann a b in M ′ gilt,wenn a b in M gilt.Sind M und M ′ halbgeordnete Mengen, so verstehen wir unter einer Ab-bildung von M in M ′ eine Zuordnung, die jedem a ∈ M ein a ′ ∈ M ′ zuordnet,so daß gilt a ≡ b ⇒ a ′ ≡ b ′ . | Sind M und M ′ orthokomplementäre ω -vollständige Halbverbände, so ver-stehen wir unter einem Ho mo mo r phismus von M in M ′ eine Abbildung → von M in M ′ , so daß für jedes a, b ∈ M und a ′ , b ′ ∈ M ′ mit a → a ′ und b → b ′ gilt: a ∧ b → a ′ ∧ b ′ ¯ a → a ′ .Ferner soll für jede Folge M = a , a , . . . in M und M ′ = a ′ , a ′ , . . . in M ′ mit a n → a ′ n gelten: V M → V M ′ .Wir wollen jetzt beweisen, daß es zu jeder halbgeordneten Menge P einenorthokomplementären ω -vollständigen Halbverband K gibt, so daß1) P ein Teil von K ist,2) K in jeden orthokomplementären ω -vollständigen Halbverband, der P alsTeil enthält, homomorph abbildbar ist.Wäre K ′ ein weiterer orthokomplementärer ω -vollständiger Halbverband, derdie Bedingungen 1) und 2) erfüllt, so gäbe es eine Zuordnung, durch die K in K ′ und K ′ in K homomorph abgebildet würde, d. h. K und K ′ wären iso mo r ph.18 is called the conjunction of a and b : c ≡ a ∧ b .A semilattice M is called orthocomplemented if to every a ∈ M there isa b ∈ M so that for every x ∈ M holds a ∧ x ⇐⇒ x b . b is called the orthocomplement of a : b ≡ ¯ a .A semilattice M is called ω -complete if to every countable sequence M = a , a , . . . in M there is a c ∈ M so that for every x ∈ M holds: ( for every n : x a n ) ⇐⇒ x c .If M and M ′ are partially ordered sets, then M is called a part of M ′ if M isa subset of M ′ and for every a, b ∈ M a b holds in M ′ exactly if a b holdsin M .If M and M ′ are partially ordered sets, we understand by a mapping of M into M ′ an assignment that to every a ∈ M assigns an a ′ ∈ M ′ so that a ≡ b ⇒ a ′ ≡ b ′ . | If M and M ′ are orthocomplemented ω -complete semilattices, we understandby a homomorphism of M into M ′ a mapping → of M into M ′ , so that forevery a, b ∈ M and a ′ , b ′ ∈ M ′ with a → a ′ and b → b ′ holds: a ∧ b → a ′ ∧ b ′ ¯ a → a ′ .Moreover, for every sequence M = a , a , . . . in M and M ′ = a ′ , a ′ , . . . in M ′ with a n → a ′ n is to hold: V M → V M ′ .We want to prove now that to every partially ordered set P there is anorthocomplemented ω -complete semilattice K so that1) P is a part of K ,2) K may be mapped homomorphically into every orthocomplemented ω -complete semilattice that contains P as part.If K ′ were a further orthocomplemented ω -complete semilattice that fulfils con-ditions 1) and 2), then there would be an assignment by which K would bemapped homomorphically into K ′ and K ′ into K , i.e. K and K ′ would be iso-morphic . K is thus determined uniquely up to isomorphism by conditions19 ist also durch die Bedingungen 1) und 2) bis auf Isomorphie eindeutigbestimmt. Wir nennen K den a usg ezeichneten o r tho ko mplemen tä r en ω -vo llstä ndig en Ha lbver ba nd üb er P . § 2. Satz : Über jeder halbgeordneten Menge gibt es den ausgezeichnetenorthokomplementären ω -vollständigen Halbverband. Wir konstruieren zu der halbgeordneten Menge P eine Menge K auf folgendeWeise:1) K enthalte die Elemente von P . (Diese nennen wir die P r imelementevon K .) | K enthalte mit endlich vielen Elementen a , a , . . . , a n auch die hierausgebildete Ko mbina tio n als Element. (Diese bezeichnen wir durch a ∧ a ∧ · · · ∧ a n .)3) K enthalte mit jedem Element a auch ein Element ¯ a .4) K enthalte mit jeder abzählbaren Folge M auch ein Element V M .Jedes Element von K läßt sich also eindeutig als Kombination a ∧ a ∧· · ·∧ a n von Primelementen und Elementen der Form ¯ a oder V M schreiben.Wir definieren eine Relation in K auf folgende Weise:1) Für Primelemente p, q gelte p q in K , wenn p q in P gilt. (DieseRelationen nennen wir die Grundrelationen.)2) Es soll jede Relation in K gelten, die sich aus den Grundrelationen mitHilfe der folgenden Regeln herleiten läßt: c a c ba ) c a ∧ ba ∧ c b ) c ¯ ac a , . . . , c a n , . . .c ) c V M a cd ) a ∧ b ca be ) a ∧ ¯ b ca n ∧ b cf ) V M ∧ b c ( M = a , a , . . . ) a ∧ a ∧ b cg ) | a ∧ b c
20) and 2). We call K the distinguished orthocomplemented ω -complete semi-lattice over P . § 2. Theorem : There is over every partially ordered set the distinguishedorthocomplemented ω -complete semilattice. We construct for the partially ordered set P a set K in the following way:1) Let K contain the elements of P . (These we call the prime elements of K .) |
2) Let K contain with finitely many elements a , a , . . . , a n also the combin-ation formed out of these as element. (These we designate by a ∧ a ∧· · · ∧ a n .)3) Let K contain with every element a also an element ¯ a .4) Let K contain with every countable sequence M also an element V M .Every element of K may thus be written uniquely as combination a ∧ a ∧· · · ∧ a n of prime elements and elements of the form ¯ a or V M .We define a relation in K in the following way:1) For prime elements p, q let p q hold in K if p q holds in P . (Theserelations we call the basic relations.)2) Every relation that may be derived from the basic relations by the aidof the following rules is to hold in K : c a c ba ) c a ∧ ba ∧ c b ) c ¯ ac a · · · c a n · · · c ) c V M a cd ) a ∧ b ca be ) a ∧ ¯ b ca n ∧ b cf ) V M ∧ b c ( M = a , a , . . . ) a ∧ a ∧ b cg ) | a ∧ b c We call the relations above the line the premisses of the relation below theline. 21ir nennen die Relationen über dem Strich die P r ä missen der Relation unterdem Strich.Wir haben jetzt zunächst zu zeigen, daß K ein orthokomplementärer ω -voll-ständiger Halbverband bezügl. der Relation ist. Dazu müssen wir beweisen α ) a aβ ) a b, b c ⇒ a cγ ) c a ∧ b ⇒ c aδ ) c ¯ a ⇒ a ∧ c ε ) c V M ⇒ c a n ( M = a , a , . . . ) Diese Eigenschaften zusammen mit a ) , b ) und c ) drücken nämlich aus, daß K ein orthokomplementärer ω -vollständiger Halbverband ist. α ) gilt für Primelemente. Gilt α ) für a und b , so auch für a ∧ b wegen a aa ∧ b a b ba ∧ b ba ∧ b a ∧ b Gilt α ) für jedes a n ∈ M , so auch für V M wegen a a V M a · · · a n a n V M a n · · · V M V M Gilt α ) für a , so auch für ¯ a , wegen a aa ∧ ¯ a ¯ a ¯ a Dadurch ist α ) allgemein bewiesen. | Da β ) am schwierigsten zu beweisen ist, nehmen wir zunächst γ ) .Um γ ) zu beweisen, haben wir zu zeigen, daß, wenn c a ∧ b herleitbar ist,dann auch stets c a herleitbar sein muß.Wir führen den Beweis indirekt durch eine transfinite Induktion . Es sei c a ∧ b herleitbar, aber nicht c a . Der letzte Schritt der Herleitung von c a ∧ b
22e have now to show first that K is an orthocomplemented ω -completesemilattice w.r.t. the relation . For this we must prove α ) a aβ ) a b, b c ⇒ a cγ ) c a ∧ b ⇒ c aδ ) c ¯ a ⇒ a ∧ c ε ) c V M ⇒ c a n ( M = a , a , . . . ) These properties together with a ) , b ) , and c ) express in fact that K is anorthocomplemented ω -complete semilattice. α ) holds for prime elements. If α ) holds for a and b , then also for a ∧ b because of a aa ∧ b a b ba ∧ b ba ∧ b a ∧ b If α ) holds for every a n ∈ M , then also for V M because of a a V M a · · · a n a n V M a n · · · V M V M If α ) holds for a , then also for ¯ a , because of a aa ∧ ¯ a ¯ a ¯ a Hereby α ) is proved in general. | As β ) is the most difficult to prove, we take first γ ) .In order to prove γ ) , we have to show that if c a ∧ b is derivable, thenalso c a must always be derivable.We lead the proof indirectly by a transfinite induction . Let c a ∧ b bederivable, but not c a . Then the last step of the derivation of c a ∧ b cannot be c a c bc a ∧ b , likewise not c c c ∧ c a ∧ b ( c = c ∧ c ), as then c c c ∧ c a would be derivable at once.23ann dann nicht sein c a c bc a ∧ b ebenfalls nicht c c c ∧ c a ∧ b ( c = c ∧ c ) da dann sofort c c c ∧ c a herleitbar wäre.Für den letzten Schritt bleiben nur die Möglichkeiten c a ∧ bc ∧ c a ∧ b c ∧ c ∧ c a ∧ b ( c = c ∧ c ) c ∧ c a ∧ bc ∧ c ′ a ∧ b · · · c n ∧ c ′ a ∧ b · · · M = a , a , . . .c = V M ∧ c ′ !V M ∧ c ′ a ∧ b Hier muß jetzt c a bezw. c ∧ c ∧ c a bezw. für mindestens ein nc n ∧ c ′ a nicht herleitbar sein, da sonst sofort c a herleitbar wäre. In derHerleitung von c a ∧ b wäre also schon für eine Prämisse die Behauptung γ ) falsch. Gehe ich in der Herleitung von einer Relation zu einer Prämisse über,von dieser wieder zu einer Prämisse usw., so bin ich nach endlich vielen Schrittenbei einer Grundrelation. Wir erhielten also eine Grundrelation, für die dieBehauptung γ ) falsch wäre. Da dieses aber unmöglich ist, ist damit γ ) bewiesen.Wir nennen die Induktion, die wir hier durchgeführt haben, eine P r ä mis-seninduktio n.Mit Hilfe von Prämisseninduktionen verläuft der Beweis für δ ) und ε ) ebensoeinfach wie für γ ) , so daß ich hierauf nicht weiter eingehe. | Es bleibt nur noch β ) zu zeigen. Statt dessen beweisen wir die stärkereBehauptung ζ ) a b, b ∧ b ∧ · · · ∧ b ∧ c d ⇒ a ∧ c d um hierauf Prämisseninduktionen anwenden zu können.Es seien zunächst b , c und d Primelemente. Dann gilt ζ ) für jede Grun-drelation a b . Wir nehmen als Induktionsvoraussetzung an, daß ζ ) für jedePrämisse von a b gelte.Da b ein Primelement ist, kann der letzte Schritt der Herleitung von a b nur sein: a ba ∧ a b a ∧ a ∧ a b ( a = a ∧ a ) a ∧ a ba a ( a = a ∧ a ) a ∧ a b a n ∧ a ′ b M = a , a , . . .a = V M ∧ a ′ !V M ∧ a ′ b Nach der Induktionsvoraussetzung ist dann a ∧ c d bezw. a ∧ a ∧ a ∧ c c a ∧ bc ∧ c a ∧ b c ∧ c ∧ c a ∧ b ( c = c ∧ c ) c ∧ c a ∧ bc ∧ c ′ a ∧ b · · · c n ∧ c ′ a ∧ b · · · M = a , a , . . .c = V M ∧ c ′ !V M ∧ c ′ a ∧ b Here must now c a resp. c ∧ c ∧ c a resp. for at least one nc n ∧ c ′ a not be derivable, as otherwise at once c a would be derivable.In the derivation of c a ∧ b the claim γ ) would thus already be false for apremiss. If in the derivation of a relation I go over to a premiss, of this againto a premiss, etc., then I am after finitely many steps at a basic relation. Wewould thus obtain a basic relation for which the claim γ ) would be false. Butas this is impossible, γ ) is thereby proved.We call the induction that we have undertaken here a premiss induction .By the aid of premiss inductions, the proof for δ ) and ε ) proceeds just assimply as for γ ) , so that I am not going into this any further. | It remains only to show in addition β ) . Instead of this we prove the strongerclaim ζ ) a b, b ∧ b ∧ · · · ∧ b ∧ c d ⇒ a ∧ c d in order to be able to apply premiss inductions hereupon.Let first b , c and d be prime elements. Then ζ ) holds for every basic rela-tion a b . We assume as induction hypothesis that ζ ) holds for every premissof a b .As b is a prime element, the last step of the derivation of a b can only be: a ba ∧ a b a ∧ a ∧ a b ( a = a ∧ a ) a ∧ a ba a ( a = a ∧ a ) a ∧ a b a n ∧ a ′ b M = a , a , . . .a = V M ∧ a ′ !V M ∧ a ′ b According to the induction hypothesis, then a ∧ c d resp. a ∧ a ∧ a ∧ c d resp. a n ∧ a ′ ∧ c d is derivable. In every case a ∧ c d is at once derivable,as well from a a because of a a a ∧ a da ∧ c d bezw. a n ∧ a ′ ∧ c d herleitbar. In jedem Falle ist sofort a ∧ c d herleitbar,ebenso aus a a wegen a a a ∧ a da ∧ c d Damit ist ζ ) bewiesen für Primelemente b , c und d .Jetzt sei nur noch b ein Primelement. Dann gilt also ζ ) für beliebiges a und Primelemente c , d . Eine Prämisseninduktion ergibt jetzt, daß ζ ) für jedeRelation b ∧ b ∧ · · · ∧ b ∧ c d gilt. Jede Prämisse von b ∧ b ∧ · · · ∧ b ∧ c d hat nämlich wieder die Form b ∧ · · · ∧ b ∧ c d . Damit ist ζ ) allgemein für Primelemente b bewiesen.Gilt ζ ) für Elemente b und b , so auch ersichtlich für b ∧ b . Gilt ζ ) für jedes b n ∈ M , so auch für b = V M . (Beweis durch Prämisseninduktion: V M ∧ V M ∧ · · · ∧ V M ∧ c d kann folgende Prämisse haben: b n ∧ V M ∧ · · · ∧ V M ∧ c d .Nach Induktionsvoraus | setzung gilt dann a V M , b n ∧ V M ∧ · · · ∧ V M ∧ c d ⇒ b n ∧ a ∧ c d . Da ζ ) aber auch für b = b n vorausgesetzt ist, und wegen a V M ⇒ a b n gilt auch a V M , b n ∧ a ∧ c d ⇒ a ∧ a ∧ c d . Aus a ∧ a ∧ c d ist aber a ∧ c d herleitbar. Jede andere Prämisse von V M ∧ V M ∧ · · · ∧ V M ∧ c d isttrivial.)Gilt ζ ) für b , so auch für ¯ b . (Beweis durch Prämisseninduktion: ¯ b ∧ ¯ b ∧ · · · ∧ ¯ b ∧ c d kann die folgende Prämisse haben: ¯ b ∧ · · · ∧ ¯ b ∧ c b . Dann gilt nachInduktionsvoraussetzung a ¯ b, ¯ b ∧ · · · ∧ ¯ b ∧ c b ⇒ a ∧ c b .Da ζ ) auch für b vorausgesetzt ist, gilt auch a ∧ c b, a ∧ b d ⇒ a ∧ a ∧ c d .Also gilt auch a ¯ b, ¯ b ∧ · · · ∧ ¯ b ∧ c b ⇒ a ∧ c d wegen a ¯ b ⇒ a ∧ b d . Jede andere Prämisse ist wieder trivial.)Also ist ζ ) allgemein gültig. Damit ist bewiesen, daß K ein orthokomple-mentärer ω -vollständiger Halbverband ist.26hereby ζ ) is proved for prime elements b , c and d .Now let only b still be a prime element. Then ζ ) holds thus for arbitrary a and prime elements c , d . A premiss induction results now in ζ ) holding forevery relation b ∧ b ∧ · · · ∧ b ∧ c d . Every premiss of b ∧ b ∧ · · · ∧ b ∧ c d hasin fact again the form b ∧ · · · ∧ b ∧ c d . Thereby ζ ) is proved in general for prime elements b .If ζ ) holds for elements b and b , then obviously also for b ∧ b . If ζ ) holdsfor every b n ∈ M , then also for b = V M . (Proof by premiss induction: V M ∧ V M ∧· · ·∧ V M ∧ c d can have the following premiss: b n ∧ V M ∧ · · · ∧ V M ∧ c d . According toinduction hypo | thesis holds then a V M , b n ∧ V M ∧· · ·∧ V M ∧ c d ⇒ b n ∧ a ∧ c d .But as ζ ) is also assumed for b = b n , and because of a V M ⇒ a b n ,also a V M , b n ∧ a ∧ c d ⇒ a ∧ a ∧ c d holds. But from a ∧ a ∧ c d maybe derived a ∧ c d . Every other premiss of V M ∧ V M ∧ · · · ∧ V M ∧ c d is trivial.)If ζ ) holds for b , then also for ¯ b . (Proof by premiss induction: ¯ b ∧ ¯ b ∧· · ·∧ ¯ b ∧ c d can have the following premiss: ¯ b ∧ · · · ∧ ¯ b ∧ c b . Then holds according toinduction hypothesis a ¯ b, ¯ b ∧ · · · ∧ ¯ b ∧ c b ⇒ a ∧ c b .As ζ ) is also assumed for b , also holds a ∧ c b, a ∧ b d ⇒ a ∧ a ∧ c d .Thus holds also a ¯ b, ¯ b ∧ · · · ∧ ¯ b ∧ c b ⇒ a ∧ c d because of a ¯ b ⇒ a ∧ b d . Every other premiss is again trivial.)Thus ζ ) is valid in general. This proves that K is an orthocomplemented ω -complete semilattice. P is a part of K , as p q in P ⇐⇒ p q in K holds. We have for this to convince ourselves that no relation p q is derivablein K that is not already holding in P . But this goes without saying, as none ofthe rules except g ) actually yields relations p q below the line. A derivation27 ist ein Teil von K , da p q in P ⇐⇒ p q in K gilt. Wir haben uns dazu zu überzeugen, daß keine Relation p q in K her-leitbar ist, die nicht schon in P gilt. Das ist aber selbstverständlich, da keineder Regeln außer g ) überhaupt Relationen p q unter dem Strich liefert. EineHerleitung einer Relation p q kann also nur die Regeln d ) und g ) benutzen.Mit diesen sind aber nur die Grundrelationen herleitbar.Zum Beweis unseres Satzes bleibt jetzt noch zu zeigen, daß sich K in jedenanderen orthokomplementären ω -vollständigen Halbverband K ′ , der P als Teilenthält, homomorph abbilden | läßt. Diese Abbildung definieren wir durch1) für Primelemente p gilt p → p ,2) ferner soll gelten a → a ′ , b → b ′ ⇒ a ∧ b → a ′ ∧ b ′ a → a ′ ⇒ ¯ a → a ′ a n → a ′ n ⇒ V M → V M ′ M = a , a , . . .M ′ = a ′ , a ′ , . . . ! Dadurch wird ersichtlich ein Homomorphismus definiert, denn es gilt für a → a ′ und b → b ′ stets a b ⇒ a ′ b ′ .Jede Herleitung von a b beweist nämlich sofort auch a ′ b ′ , da dieHerleitungsschritte a ) - g ) in jedem orthokomplementären ω -vollständigen Hal-bverband stets richtig sind. § 3. Um aus dem im § 2 bewiesenen Satz die Widerspruchsfreiheit derreinen Zahlentheorie mit vollständiger Induktion beweisen zu können, benutzenwir die folgende Formalisierung. Als Primformeln nehmen wir die Zeichen fürzahlentheoretische Prädikate A( . . . ) , B( . . . ) , . . . mit den Zahlen , ′ , ′′ , . . . als Argumenten, z. B. ′′ , ′ .Diese Primformeln P , Q , . . . bilden eine halbgeordnete Menge, wenn wir P → Q setzen, falls das Prädikat P das Prädikat Q impliziert. Zu den Grun-drelationen P → Q nehmen wir auch noch die Relationen der Form → P , P → , → hinzu, soweit sie inhaltlich richtig sind.Über dieser halbgeordneten Menge P der Primformeln konstruieren wir jetztwie in § 2 den ausgezeichneten orthokomplementären ω -vollständigen Halbverb-and. Wir benutzen dazu die logistischen Zeichen, also → statt , & statt ∧ .28f a relation p q can thus use only the rules d ) and g ) . But with these onlythe basic relations are derivable.For the proof of our theorem, it remains now in addition to show that K may be mapped homomorphically into every other orthocomplemented ω -complete semilattice K ′ that contains P as part | . This mapping we defineby1) for prime elements p holds p → p ,2) moreover is to hold a → a ′ , b → b ′ ⇒ a ∧ b → a ′ ∧ b ′ a → a ′ ⇒ ¯ a → a ′ a n → a ′ n ⇒ V M → V M ′ M = a , a , . . .M ′ = a ′ , a ′ , . . . ! Hereby obviously a homomorphism is being defined, for with a → a ′ and b → b ′ always holds a b ⇒ a ′ b ′ .Every derivation of a b proves in fact at once also a ′ b ′ , as the deriv-ation steps a ) – g ) are always correct in every orthocomplemented ω -completesemilattice. § 3. In order to be able to prove the freedom from contradiction of element-ary number theory with complete induction from the theorem proved in § 2,we use the following formalisation. We take as prime formulas the signs fornumber-theoretic predicates A( . . . ) , B( . . . ) , . . . with the numbers , ′ , ′′ , . . . as arguments, e.g. ′′ , ′ .These prime formulas P , Q , . . . form a partially ordered set if we set P → Q in case the predicate P implies the predicate Q . To the basic relations P → Q we are also adding the relations of the form → P , P → , → , as far as theyare correct in terms of content.Over this partially ordered set P of the prime formulas, we construct nowas in § 2 the distinguished orthocomplemented ω -complete semilattice. We usefor this the logistic signs, thus → instead of , & instead of ∧ .To the formulas belong thus the prime formulas, with A and B also A & B , with A also A . We restrict the conjunction of countable | sequences tothe sequences of the form A (1) , A (1 ′ ) , . . . . We designate this conjunction by ( x ) A ( x ) . 29u den Formeln gehören also die Primformeln, mit A und B auch A & B ,mit A auch A . Die Konjunktion abzählbarer | Folgen beschränken wir aufdie Folgen der Form A (1) , A (1 ′ ) , . . . Diese Konjunktion bezeichnen wir durch ( x ) A ( x ) .Ferner führen wir noch freie Variable a = a , b , . . . ein durch folgendeSchlußregel: sind A (1) , A (1 ′ ) , . . . herleitbare Relationen, sosoll auch A ( a ) herleitbar sein.Hierdurch werden die Beweise von § 2 nur unwesentlich modifiziert. Wirerhalten insgesamt einen Kalkül N mit den folgenden Schlußregeln C → A C → B a ) C → A & BA & C → b ) C → AC → A (1) · · · C → A ( n ) · · · c ) C → ( x ) A ( x ) A → C d ) A & B → CA → B e ) A & B → CA ( n ) & B → C f ) ( x ) A ( x ) & B → CA & A & B → C g ) A & B → CA & B → C h ) B & A → C A & ( B & C ) → D i ) ( A & B ) & C → D A (1) · · · A ( n ) · · · j ) | A ( a ) Die Schlußregeln h ) und i ) waren in § 2 überflüssig, da wir dort a ∧ b ∧ c . . . sofort als Zeichen für die Kombination von a, b, c, . . . eingeführt haben.Der Beweis in § 2 liefert jetzt das folgende Ergebnis: Der Kalkül N istwiderspruchsfrei, z. B. ist die leere Relation → nicht herleitbar, da nur dieinhaltlich richtigen Relationen in P gelten und P ein Teil von N ist. Zu demKalkül N können die folgenden Schlußregeln hinzugenommen werden, ohne daßdie Menge der herleitbaren Relationen vergrößert wird: A → B B → C k ) A → CC → A & B l ) C → A C → A & B m ) C → BC → A n ) A & C → C → ( x ) A ( x ) o ) C → A ( n ) a = a , b , . . . by the fol-lowing rule of inference:if A (1) , A (1 ′ ) , . . . are derivable relations, then A ( a ) is also to be derivable.By this the proofs of § 2 are only modified unessentially. We obtain overalla calculus N with the following rules of inference C → A C → B a ) C → A & BA & C → b ) C → AC → A (1) · · · C → A ( n ) · · · c ) C → ( x ) A ( x ) A → C d ) A & B → CA → B e ) A & B → CA ( n ) & B → C f ) ( x ) A ( x ) & B → CA & A & B → C g ) A & B → CA & B → C h ) B & A → C A & ( B & C ) → D i ) ( A & B ) & C → D A (1) · · · A ( n ) · · · j ) | A ( a ) The rules of inference h ) and i ) were dispensable in § 2, as we have intro-duced there a ∧ b ∧ c . . . at once as sign for the combination of a, b, c, . . . .The proof in § 2 yields now the following result: the calculus N is consistent,e.g. the empty relation → is not derivable, as only the relations correct in termsof content hold in P and P is a part of N . To the calculus N the following rulesof inference can be added without increasing the set the derivable relations: A → B B → C k ) A → CC → A & B l ) C → A C → A & B m ) C → BC → A n ) A & C → C → ( x ) A ( x ) o ) C → A ( n ) To the basic relations can be added A → A .This result from § 2 we can now complete:31u den Grundrelationen kann A → A hinzugenommen werden.Dieses Ergebnis aus § 2 können wir jetzt ergänzen:1) es kann auch die Schlußregel A ( a ) p ) A ( n ) hinzugenommen werden.Der Beweis wird wieder durch eine transfinite Prämisseninduktion geführt.Ist A ( a ) herleitbar in N und ist die letzte Schlußregel dieser Herleitung nicht A (1) · · · A ( n ) · · · A ( a ) so hat die Prämisse die Form A ′ ( a ) . Nehmen wir als Induktionsvoraussetzungan, daß für jede Prämisse A ′ ( a ) auch A ′ ( n ) herleitbar ist, so folgt sofort A ( n ) . |
2) Zu den Grundrelationen darf A → A hinzugenommen werden.Für jede Primformel P gilt nämlich stets → P oder P → . Wegen → PP → PP →→ PP → P ist also für jede Primformel stets P → P herleitbar. Hieraus folgtallgemein die Herleitbarkeit von A → A (vergl. etwa Hilbert-Bernays, Grundla-gen der Mathematik II ).3) Es kann auch die vollständige Induktion A ( a ) → A ( a ′ ) q ) A (1) → A ( b ) zu den Schlußregeln hinzugenommen werden ohne die Menge der herleit-baren Relationen zu vergrößern.Ist nämlich A ( a ) → A ( a ′ ) herleitbar, so auch die Relation A ( n ) → A ( n ′ ) fürjede Zahl n .Für jede Zahl m folgt daraus durch m -malige Anwendung der Schlußregel k ) sofort A (1) → A ( m ) .Wegen A (1) → A (1) · · · A (1) → A ( m ) · · · A (1) → A ( b ) ist also auch A (1) → A ( b ) herleitbar.Damit ist die Wf. der reinen Zahlentheorie bewiesen, da die insgesamtzulässigen Schlußregeln einen Kalkül definieren, der den klassischen Prädikaten-kalkül ersichtlich enthält. 32) The rule of inference A ( a ) p ) A ( n ) can also be added.The proof is again being led by a transfinite premiss induction . If A ( a ) isderivable in N and if the last rule of inference of this derivation is not A (1) · · · A ( n ) · · · A ( a ) then the premiss has the form A ′ ( a ) . If we assume as induction hypothesis thatfor every premiss A ′ ( a ) also A ′ ( n ) is derivable, then A ( n ) follows at once. |
2) To the basic relations may be added A → A .For every prime formula P holds in fact always → P or P → . Because of → PP → P P →→ PP → P , P → P is thus always derivable for every prime formula.From this follows in general the derivability of A → A (cf. e.g. Hilbert-Bernays, Grundlagen der Mathematik II ).3) The complete induction A ( a ) → A ( a ′ ) q ) A (1) → A ( b ) can also be added to the rules of inference without increasing the set thederivable relations.In fact, if A ( a ) → A ( a ′ ) is derivable, then also the relation A ( n ) → A ( n ′ ) for every number n .For every number m follows therefrom at once A (1) → A ( m ) by m -foldapplication of the rule of inference k ) .Because of A (1) → A (1) · · · A (1) → A ( m ) · · · A (1) → A ( b ) also A (1) → A ( b ))