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Erschienen in:
Introduction: The Internal Logic of Arithmetic Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

7. The Internal Consistency of Arithmetic with Infinite Descent: A Syntactical Proof

verfasst von : Yvon Gauthier

Erschienen in: Towards an Arithmetical Logic

Verlag: Springer International Publishing

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Abstract

The question of the consistency or non-contradiction of arithmetic is a philosophical question, that is the certainty of a mathematical theory and it has become a logical problem requiring a mathematical solution. It is Hilbert who has put the original question and who has attempted a first answer. Having demonstrated in 1899 the consistency of elementary geometry on the basis of a consistent arithmetic of real numbers, he turned to the question of that arithmetic (the second problem of his 1900 list) which included besides the axiom of elementary arithmetic an axiom of continuity, i.e. the Archimedean axiom with syntactic completeness. Hilbert introduces functionals, that is second-order functions or functions of functions, in order the use induction over the ordinals. Transfinite induction proved successful in the hands of Gentzen, and thereafter Gödel’s second incompleteness results which aimed to show that Peano’s arithmetic cannot contain its own consistency.

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Vorheriges Kapitel The Internal Logic of Constructive Mathematics
Nächstes Kapitel Conclusion: Arithmetism Versus Logicism or Kronecker Contra Frege
Fußnoten
1
A first version of the proof was published in Modern Logic, VIII (1/2), 47–87 (2000) and contained some extraneous semantical and set-theoretical elements which have been removed here. See also Chapter 4 of my book Internal Logic. Foundations of Mathematics from Kronecker to Hilbert, Kluwer, Dordecht, 2002.
 
2
Kronecker had proven the unique factorization theorem in the following formulation: “Every integral form ( = polynomial) is representable as a product of irreducible (prime) forms in a unique way” (see Kronecker 1882, p. 352). Kronecker is interested in the theory of divisibility for prime forms (forms with no common divisors greater than 1), rather than prime polynomials in his work. The notion of integral domain and unique factorization domain are direct descendants to that theorem.
 
3
This can be seen as the precursor of the problem of quantification over empty domains. We know that we have
$$\displaystyle{\frac{A,A \supset B} {B} }$$
in an empty domain, provided that A and B have the same free variables. But Kronecker had a more general theory of inclusion or content of forms in mind and the transformation in question is a composition of contents, an internal constitution of polynomials (forms) where indeterminates are not the usual functional variables.
 
4
Mordell says of infinite descent in 1922 that you start with an arbitrary n—an arbitrary choice made once and for all—and descend finitely. Hasse’s principle of local solvability implying global solvability for quadratic forms relies on the same principle and is related to Legendre “positive” infinite descent (see Davenport 1968).
 
5
My emphasis is different from Edwards (1987a,b) who has chosen to look at divisor theory rather than the theory of forms which is, in my view, the encompassing theory.
 
6
See Kronecker (1889). Edwards (1987a,b) rightly says that Dedekind’s Prague theorem—a generalization of Gauss lemma to the algebraic case—is but a consequence of Kronecker’s result. See also Edwards et al. (1982).
 
7
One may notice in that context that Kohlenbach with collaborators has adopted the Dialectica interpretation for Kreisel’s proof mining project instead of Kreisel’s own no-counter-example interpretation. This turn in logic is somewhat ironical when one remembers that Kreisel downgraded the Dialectica Interpretation in his Gödel obituary in the Bibliographical Memoirs of the Royal Society 26, 1980.
 
8
The analogy is misleading though, since the algebrization of logic from Boole for the sentential calculus to Tarski (cylindrical algebra) and Halmos (polyadic algebra) for the predicate calculus does not have the generality or the extent of the polynomial calculus which is, as Kronecker has emphasized, a general arithmetic with purely arithmetical content. In contrast then to the algebra of logic, the polynomial calculus of the theory of forms pertains to the arithmetization of logic. Categorical logic as an algebraic logic does not fare better since it does only translate isomorphically logical calculi into categorical structures (from Lawvere’s elementary topos to closed cartesian categories or monoidal categories) only reflecting the surface features of deductive systems.
 
Literatur
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Y. Gauthier (2013a): Kronecker in Contempory Mathematics. General Arithmetic as a Foundational Programme. Reports on Mathematical Logic, 48:37–65.
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Metadaten
Titel
The Internal Consistency of Arithmetic with Infinite Descent: A Syntactical Proof
verfasst von
Yvon Gauthier
Copyright-Jahr
2015
Buch
Towards an Arithmetical Logic

Print ISBN: 978-3-319-22086-4

Electronic ISBN: 978-3-319-22087-1

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-319-22087-1_7