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Theories very close to PA where Kreisel's Conjecture is false

Published online by Cambridge University Press:  12 March 2014

Pavel Hrubeš
Pavel Hrubeš*
Affiliation:
Mathematical Institute, Academy of Sciences of the Czech Republic, Prague, Czech Republic. E-mail: [email protected]
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Abstract

We give four examples of theories in which Kreisel's Conjecture is false: (1) the theory PA(-) obtained by adding a function symbol minus, ‘—’, to the language of PA, and the axiom ∀x∀y∀z (x − y = z) ≡ (x = y + z ∨ (x < y ∧ z = 0)); (2) the theory L of integers; (3) the theory PA(q) obtained by adding a function symbol q (of arity ≥ 1) to PA, assuming nothing about q; (4) the theory PA(N) containing a unary predicate N(x) meaning ‘x is a natural number’. In Section 6 we suggest a counterexample to the so called Sharpened Kreisel's Conjecture.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 1 , March 2007 , pp. 123 - 137
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

Baaz, M. and Pudlák, P. [1993], Kreisel's conjecture for L∃1, Arithmetic, proof theory, and computation complexity, Papers from the Conference Held in Prague, 07 2–5, 1991, Oxford University Press, New York, pp. 30–60.Google Scholar
Friedman, H. [1975], One hundred and two problems in mathematical logic, this Journal, vol. 40, pp. 113–129.Google Scholar
Krajiček, J. and Pudlák, P. [1988], The number of proof lines and the size of proofs in first order logic, Archive for Mathematical Logic, vol. 27, pp. 69–84.CrossRefGoogle Scholar
Miyatake, T. [1980], On the lengths of proofs in formal systems, Tsukuba Journal of Mathematics, vol. 4, pp. 115–125.CrossRefGoogle Scholar
Parikh, R. [1973], Some results on the length of proofs, Transactions of the American Mathematical Society, vol. 177, pp. 29–36.CrossRefGoogle Scholar
Yukami, T. [1978], A note on a formalized arithmetic with function symbols and +, Tsukuba Journal of Mathematics, vol. 7, pp. 69–73.Google Scholar
Yukami, T. [1984], Some results on speed-up, Annals of the Japan Association for Philosophy and Science, vol. 6, pp. 195–205.Google Scholar