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Gödel sentences of bounded arithmetic

Published online by Cambridge University Press:  12 March 2014

Gaisi Takeuti
Gaisi Takeuti*
Affiliation:
1420 Locust Street 35R, Philadelphia, PA 19102, USA E-mail: takeuti@saul.cis.upenn.edu
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Extract

In [1], S. Buss introduced the systems of Bounded Arithmetic for (i = 0,1,2,…) which has a close relationship to classes in polynomial hierarchy.

In [4], we defined a very special kind of proof-predicate Prfi for which gives detailed information on bounds of free variables used in the proof. There we also introduced infinitely many Gödel sentences for Prfi (k = 0, 1, 2, …) and showed that the properties of Prfi and are closely related to the P ≠ NP problem. Then we presented many conjectures on Prfi and which imply P ≠ NP.

Now in [2], Feferman emphasized that the arithmetization of metamathematics must be carried out intensionally. Bounded Arithmetic is a very interesting case in this sense.

In this paper, we also introduce the usual proof-predicate PRFi for and infinitely many Gödel sentences for PRFi(k= 0, 1, 2, …). Then we show that (Prfi, )and (PRFi, ) form a good contrast, this contrast is also closely related to the P ≠ NP problem, and present more conjectures which imply P ≠ NP.

As in [4] we define to be the following extension of Buss' original .

(1) We add finitely many function symbols which express polynomial time computable functions to Buss' original language of .

(2) All basic axioms on function symbols and ≤ can be expressed by initial sequents without logical symbols.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 3 , September 2000 , pp. 1338 - 1346
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Buss, S., Bounded arithmetic, Bibliopolis, Napoli, 1986.Google Scholar
[2]Feferman, S., Arithmetization of metamathematics in a general setting, Fundamenta Mathematicae, vol. 49 (1960), pp. 35–92.CrossRefGoogle Scholar
[3]Takeuti, G., Bounded arithmetic and truth definition, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 75–104.CrossRefGoogle Scholar
[4]Takeuti, G., Incompleteness theorems and versus , Logic Colloquium '96 (Larrazabal, J. M., Lascar, D., and Mints, G., editors), LN in Logic, no. 12, Springer, 1998, pp. 247–261.Google Scholar
[5]Wilkie, A. J. and Paris, J. B., On the scheme of induction for bounded arithmetic formulas, Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261–302.CrossRefGoogle Scholar