45 research outputs found

    A Possible and Necessary Consistency Proof

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    After Gödel's incompleteness theorems and the collapse of Hilbert's programme Gerhard Gentzen continued the quest for consistency proofs of Peano arithmetic. He considered a finitistic or constructive proof still possible and necessary for the foundations of mathematics. For a proof to be meaningful, the principles relied on should be considered more reliable than the doubtful elements of the theory concerned. He worked out a total of four proofs between 1934 and 1939. This thesis examines the consistency proofs for arithmetic by Gentzen from different angles. The consistency of Heyting arithmetic is shown both in a sequent calculus notation and in natural deduction. The former proof includes a cut elimination theorem for the calculus and a syntactical study of the purely arithmetical part of the system. The latter consistency proof in standard natural deduction has been an open problem since the publication of Gentzen's proofs. The solution to this problem for an intuitionistic calculus is based on a normalization proof by Howard. The proof is performed in the manner of Gentzen, by giving a reduction procedure for derivations of falsity. In contrast to Gentzen's proof, the procedure contains a vector assignment. The reduction reduces the first component of the vector and this component can be interpreted as an ordinal less than epsilon_0, thus ordering the derivations by complexity and proving termination of the process.De begrÀnsningar av formella system som uppdagades av Gödels ofullstÀndighetsteorem Är 1931 innebÀr att Peanoaritmetikens konsistens endast kan bevisas med hjÀlp av fundamentala principer som inte kan formaliseras inom systemet. Trots att Hilberts finitistiska metoder inte kunde producera ett konsistensbevis, sÄ fortsatte sökandet efter ett bevis med konstruktiva metoder. För att ett bevis skall vara meningsfullt borde principerna som anvÀnds vara mera pÄlitliga Àn de element som betvivlas inom teorin. Avhandlingens titel hÀnvisar till ett citat av Gentzen dÄ han motiverar behovet av konsistensbevis för första ordningens aritmetik. Gentzen sjÀlv producerade fyra konsistensbevis och analyserade hur vÀl dessa stÀmde överens med Hilberts program. Gentzen anvÀnde konstruktiva metoder i sina bevis, men det debatteras huruvida dessa metoder kan anses vara finitistiska. Det tredje och mest kÀnda beviset presenterar en reduktion av hÀrledningar av kontradiktioner. Med hjÀlp av transfinit induktion visas att reduktionsprocessen terminerar i en enkel hÀrledning som konstateras vara omöjlig. DÀrför kan det inte finnas nÄgon hÀrledning av en kontradiktion. Avhandlingen undersöker och jÀmför Gentzens bevis frÄn olika aspekter. Konsistensen av intuitionistisk Heytingaritmetik bevisas bÄde i sekvenskalkyl och i naturlig deduktion. Det tidigare beviset Àr i Gentzens anda och innehÄller ett snittelimineringsbevis för kalkylen och en syntaktisk studie av den aritmetiska delen av systemet. Det senare beviset pÄminner om ett normaliseringsbevis och visar terminering med hjÀlp av en vektortilldelning.Gödelin vuonna 1931 jullkaisemista epÀtÀydellisyyslauseista seurausi rajoituksia formaalisille jÀrjestelmille: Niiden mukaan Peano-aritmetiikan ristiriidattomuus voidaan todistaa ainoastaan periaatteilla, jotka eivÀt ole formalisoitavissa jÀrjestelmÀn itsensÀ sisÀllÀ. Vaikka Hilbertin finitistisillÀ menetelmillÀ ei siksi pystytty tuottamaan konsistenssitodistusta, todistuksen etsiminen jatkui konstruktiivisillÀ menetelmillÀ. Jotta todistus olisi mielekÀs, siinÀ kÀytettyjen periaatteiden oli oltava luotettavampia kuin teorian itsensÀ sisÀltÀmÀt periaatteet. VÀitöskirjan otsikko viittaa Gentzenin kirjoitukseen, jossa hÀn perustelee ensimmÀisen kertaluvun aritmetiikan konsistenssitodistuksen tarvetta. Gentzen itse laati neljÀ sellaista konsistenssitodistusta ja analysoi, missÀ mÀÀrin ne olivat yhdenmukaisia Hilbertin ohjelman kanssa. Gentzen kÀytti konstruktiivisia menetelmiÀ todistuksissaan ja on paljon vÀitelty kysymys, voidaanko nÀitÀ menetelmiÀ pitÀÀ finitistisinÀ. Kolmannessa ja tunnetuimassa Gentzenin todistuksessa esitetÀÀn ristiriitaisuuksien pÀÀttelyn reduktiomenetelmÀ. TransfiniittistÀ induktiota kÀyttÀmÀllÀ osoitetaan, ettÀ reduktioprosessi pÀÀttyy yksinkertaiseen pÀÀttelyyn, jollainen on erikseen todettu mahdottomaksi. TÀmÀn vuoksi ristiriitaa ei voida pÀÀtellÀ. VÀitöskirjassa selvitetÀÀn ja vertaillaan Gentzenin todistuksia eri nÀkökulmista. Intuitionistisen Heyting-aritmetiikan ristiriidattomuus osoitetaan sekÀ sekvenssikalkyylissÀ ettÀ luonnollisessa pÀÀttelyssÀ. EnsimmÀinen todistus seuraa Gentzenin henkeÀ ja siinÀ sovelletaan ns. leikkaussÀÀnön eliminointitodistusta sekÀ syntaktista analyysia jÀrjestelmÀn aritmeettisesta osasta. JÀlkimmÀinen todistus muistuttaa luonnollisen pÀÀttelyn normalisointitodistusta ja nÀyttÀÀ reduktion pÀÀttymisen vektorimÀÀrityksen avulla

    Syntactical consistency proofs for term induction revisited

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    GIn [6] wird von Gerhard Gentzen die Widerspruchsfreiheit der Peano Arithmetik erster Stufe PA bewiesen. Die Methode geht dabei folgendermaßen vor: Man deniert einen simplen Teil SPA der Peano Arithmetik (SPA enthĂ€lt im speziellen keine Anwendung des Induktionsschemas) und zeigt zuerst die von SPA. Der Rest des Arguments verlĂ€uft indirekt. Man nimmt an, dass PA einen Widerspruch ableitet und zeigt das dessen Deduktion zu einer Deduktion in SPA transformiert werden kann, was der Widerspruchsfreiheit von SPA widerspricht. Diese Transformation verlĂ€uft wie folgt: Jeder Deduktion in PA wird eine Ordinalzahl (oder genauer, ein Ordinalzahlterm eines Ordinalzahlnotations Systems) zugeordnet, diese wird der Rang der Deduktion genannt. Dann wird gezeigt, dass es zu jeder Deduktion eines Widerspruches (die nicht in SPA verlĂ€uft) eine Deduktion (ebenfalls eines Widerspruches) gibt die einen kleineren Rang hat. Diese Methode benötigt daher die Wohlfundiertheit des verwendeten Ordinalzahlnotations Systems (in diesem Fall bis "0). Bei nĂ€herer Betrachtung von Gentzens Methode fĂ€llt auf, dass sie lediglich folgende Eigenschaften von PA verwendet: 1. Alle Axiome von PA sind AllsĂ€tze oder Instanzen des Induktionsschemas φ(0) ^ ∀ x[φ(x) -> φ(x + 1)] -> ∀ xφ(x): 2. Alle geschlossenen Terme sind beweisbar (in SPA) gleich zu einem Term der lediglich aus 0 und dem Symbol der Nachfolgerfunktion aufgebaut ist. Dies erlaubt eine Verallgemeinerung von Gentzens Methode. In dieser Diplomarbeit werden wir daher Theorien Tf = (Tf )0 [ (Tf )Ind betrachten die folgende Eigenschaften erfĂŒllen: 1. (Tf )0 besteht lediglich aus AllsĂ€tzen. 2. (Tf )Ind beinhaltet alle Instanzen des Induktionsschemas φ(c1)^:::^φ(cm)^∀ ~x[φ(x1)^:::^φ(xn) -> φ(f(x1; :::; xn))] -> ∀ xφ(x): 3. Der simple Teil von Tf beweist fĂŒr jeden geschlossenen Term t, dass t gleich einem Term _t ist der lediglich aus den symbolen c1; :::; cm und f aufgebaut ist. Die Widerspruchsfreiheit von Tf kann nun, wie in [6] fĂŒr PA, relativ zu ihrem simplen Teil (wo Induktion wie zuvor bei Gentzen nicht möglich ist) gezeigt werden. Eine Konsequenz dieses Resultates ist das folgende Korollar. Korollar. Sei φ(a) quantorenfrei und Tf widerspruchsfrei. Wenn Tf j= 9xφ(x), dann (Tf )0 j= 9xφ(x). Insbesondere ist Tf _0 1 -konservativ ĂŒber (Tf )0. Es scheint mir als wĂ€re die Methode, die von Kurt SchĂŒtte in seinem Widerspruchsfreiheitsbeweis von PA verwendet wird, eine gĂ€nzlich andere. SchĂŒtte, Tait und Andere verwenden KalkĂŒle mit unendlichen Deduktionsregeln um, in einem gewissen Sinne, die Beweistheoretische Ordinalzahl einer Theorie zu berechnen. Dies erfolgt ĂŒber eine Transformation der endlichenDeduktionen der Theorie (in der Logik erster Stufe) in Deduktionen ineinem unendlichen KalkĂŒl, das Schnittelimination erlaubt. Im Gegensatz zu Gentzens Methode hat die von SchĂŒtte eine enge Beziehung zu den beweistheoretischenOrdinalzahlen. Auf die Unterschiede der beiden Methoden wird nicht weiter eingegangen werden. Anstatt dieses Vergleiches wird lediglich eine Variante von Taits Methode dazu verwendet die _1 1-Ordinalzahl, wie von Wolfram Pohlers in [∀ ] beschrieben, von Theorien TAf (aufgefasst als TaitkalkĂŒhl) zu messen. Es wird angenommen das TAf folgende Eigenschaften erfĂŒllt: 1. TAf enthĂ€lt fĂŒr jede primitiv rekursive Funktion die denierenden Formeln als Axiome. 2. Weiters enthĂ€lt TAf alle Instanzen des Schemas φ(c1)^:::^φ(cm)^∀ ~x[φ(x1)^:::^φ(xn) -> φ(f(x1; :::; xn))] -> ∀ xφ(x): Hierbei ist f ein m-stelliges Symbol einer gleichstelligen primitiv rekursiven Funktion und c1; :::; cl Individuenkonstanten. 3. Es wird außerdem angenommen das jedes n 2 N gleich einer Komposition aus fN und den natĂŒrlichen Zahlen cN1;:::; cN l ist.Gerhard Gentzen proves the consistency of first-order Peano arithmetic \textbf{PA}. His method works as follows: Define a simple part \textbf{SPA} of peano arithmetic (\textbf{SPA} does in particular not contain induction) and first show the consistency of \textbf{SPA}. Now assume towards a contradiction that \textbf{PA} deducts an contradiction. Show that this deduction can be transformed into a deduction in \textbf{SPA}, this contradicts the consistency of \textbf{SPA}. How to get a deduction in \textbf{SPA}: We assign an ordinal (more exact an ordinal term of an ordinal notation system) to each deduction in \textbf{PA}, called the rank of the deduction. Next show that for each deduction which deducts a contradiction (and is not in \textbf{SPA}) there is a deduction (also deducting and contradiction) with smaller rank. This method requires that the ordinal notation system (which goes up to Δ0\varepsilon_0) is well-founded. It turns out that Gentzen's method requires only to the following properties of \textbf{PA}: \begin{enumerate} \item All axioms of \textbf{PA} are universal sentences or instances of the induction schema φ(0)∧∀x[φ(x)→φ(x+1)]→∀xφ(x). \varphi(0) \wedge \forall x [\varphi(x) \rightarrow \varphi(x+1)] \rightarrow \forall x \varphi(x). \item All closed terms are provable equal to a term build up just from 00 and the symbol of the successor function. \end{enumerate} This allows a slight generalisation of Gentzen's method. In this Diploma Thesis we consider theories Tf=(Tf)0âˆȘ(Tf)IndT_f = (T_f)_0 \cup (T_f)_{\text{Ind}} with the following properties: \begin{enumerate} \item (Tf)0(T_f)_0 contains only universal sentences. \item (Tf)Ind(T_f)_{\text{Ind}} contains all instances of the general induction schema φ(c1)∧...∧φ(cm)∧∀x⃗[φ(x1)∧...∧φ(xn)→φ(f(x1,...,xn))]→∀xφ(x).\varphi(c_1)\wedge ... \wedge \varphi(c_m) \wedge \forall \vec{x} [\varphi(x_1)\wedge...\wedge \varphi(x_n) \rightarrow \varphi(f(x_1,...,x_n))] \rightarrow \forall x \varphi(x). \item The simple part of TfT_f proves for every closed term tt the equality of tt to a term tˉ\bar{t} build up just from c1,...,cmc_1,...,c_m and ff. \end{enumerate} As Gentzen did for \textbf{PA}, the consistency of TfT_f can be shown with respect to their simple part which corresponds to the simple part of Gentzen (also without induction). As a consequence, one gets the following result for all such theories. \begin{cor2} Assume φ(a)\varphi(a) is quantifier free and TfT_f consistent.\\ If Tf⊹∃xφ(x)T_f \models \exists x \varphi(x), then (Tf)0⊹∃xφ(x)(T_f)_0 \models \exists x \varphi(x).\\ I.e., TfT_f is ÎŁ10\Sigma_1^0-conservative over (Tf)0(T_f)_0. \end{cor2} It seems that this method is different in an essential way to the method Kurt Sch\"{u}tte uses in his consistency proof of \textbf{PA}. Sch\"{u}tte, Tait and others uses calculi with infinite deduction rules. These methods compute, in some sense, the proof theoretical ordinal of the considered theory by embedding the deductions of the theory (in ordinary first-order logic) in an infinite system which allows cut-elimination. In contrast to Gentzen's method Sch\"{u}tte's and Tait's methods are closely related to the proof theoretical ordinals.\\ We do not provide an analysis of the disparities of both methods. Instead we present the point of view Wolfram Pohlers propose, to measure the Π11\Pi_1^1-ordinal of theories TAf\mathsf{TA}_f (presented as a Tait-calculus) satisfying the following conditions: \begin{enumerate} \item TAf\mathsf{TA}_f includes all defining axioms for primitive recursive functions. \item All instances of the schema φ(c1)∧...∧φ(cm)∧∀x⃗[φ(x1)∧...∧φ(xn)→φ(f(x1,...,xn))]→∀xφ(x)\varphi(c_1)\wedge ... \wedge \varphi(c_m) \wedge \forall \vec{x} [\varphi(x_1)\wedge...\wedge \varphi(x_n) \rightarrow \varphi(f(x_1,...,x_n))] \rightarrow \forall x \varphi(x) are included. Here ff is an mm-array primitive recursive function constant and c1,...,clc_1,...,c_l are individual constants. \item Every n∈Nn \in \mathbb{N} is equal to a composition of fNf^\mathbb{N} and the elements c1N,...,clNc_1^\mathbb{N},...,c_l^\mathbb{N}. \end{enumerate

    A Complete Axiomatization of the Three valued Completion of Logic Programs

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    We prove the completeness of extended SLDNF-resolution for the new class of e-programs with respect to the three-valued completion of a logic program. Not only the class of allowed programs but also the class of definite programs are contained in the class of Δ-programs. To understand better the three-valued completion of a logic program we introduce a formal system for three-valued logic in which one can derive exactly the three-valued consequences of the completion of a logic program. The system is proof theoretically interesting, since it is a fragment of Gentzen's sequent calculus L

    Aspects of the constructive omega rule within automated deduction

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    In general, cut elimination holds for arithmetical systems with the w -rule, but not for systems with ordinary induction. Hence in the latter, there is the problem of generalisation, since arbitrary formulae can be cut in. This makes automatic theorem -proving very difficult. An important technique for investigating derivability in formal systems of arithmetic has been to embed such systems into semi- formal systems with the w -rule. This thesis describes the implementation of such a system. Moreover, an important application is presented in the form of a new method of generalisation by means of "guiding proofs" in the stronger system, which sometimes succeeds in producing proofs in the original system when other methods fail

    Specifying Theorem Provers in a Higher-Order Logic Programming Language

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    Since logic programming systems directly implement search and unification and since these operations are essential for the implementation of most theorem provers, logic programming languages should make ideal implementation languages for theorem provers. We shall argue that this is indeed the case if the logic programming language is extended in several ways. We present an extended logic programming language where first-order terms are replaced with simply-typed λ-terms, higher-order unification replaces firstorder unification, and implication and universal quantification are allowed in queries and the bodies of clauses. This language naturally specifies inference rules for various proof systems. The primitive search operations required to search for proofs generally have very simple implementations using the logical connectives of this extended logic programming language. Higher-order unification, which provides sophisticated pattern matching on formulas and proofs, can be used to determine when and at what instance an inference rule can be employed in the search for a proof. Tactics and tacticals, which provide a framework for high-level control over search, can also be directly implemented in this extended language. The theorem provers presented in this paper have been implemented in the higher-order logic programming language λProlog
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