# Interpreting reflexive theories in finitely many axioms

Fundamenta Mathematicae (1997)

- Volume: 152, Issue: 2, page 99-116
- ISSN: 0016-2736

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topShavrukov, V.. "Interpreting reflexive theories in finitely many axioms." Fundamenta Mathematicae 152.2 (1997): 99-116. <http://eudml.org/doc/212206>.

@article{Shavrukov1997,

abstract = {For finitely axiomatized sequential theories F and reflexive theories R, we give a characterization of the relation ’F interprets R’ in terms of provability of restricted consistency statements on cuts. This characterization is used in a proof that the set of $∏_1$ (as well as $∑_1$) sentences π such that GB interprets ZF+π is $Σ^0_3$-complete.},

author = {Shavrukov, V.},

journal = {Fundamenta Mathematicae},

keywords = {relative interpretability; finitely axiomatized sequential theories; reflexive theories; provability of restricted consistency statements on cuts},

language = {eng},

number = {2},

pages = {99-116},

title = {Interpreting reflexive theories in finitely many axioms},

url = {http://eudml.org/doc/212206},

volume = {152},

year = {1997},

}

TY - JOUR

AU - Shavrukov, V.

TI - Interpreting reflexive theories in finitely many axioms

JO - Fundamenta Mathematicae

PY - 1997

VL - 152

IS - 2

SP - 99

EP - 116

AB - For finitely axiomatized sequential theories F and reflexive theories R, we give a characterization of the relation ’F interprets R’ in terms of provability of restricted consistency statements on cuts. This characterization is used in a proof that the set of $∏_1$ (as well as $∑_1$) sentences π such that GB interprets ZF+π is $Σ^0_3$-complete.

LA - eng

KW - relative interpretability; finitely axiomatized sequential theories; reflexive theories; provability of restricted consistency statements on cuts

UR - http://eudml.org/doc/212206

ER -

## References

top- [1] A. Berarducci and P. D’Aquino, ${\Delta}_{0}$-complexity of the relation $y={\prod}_{i}\le nF\left(i\right)$, Ann. Pure Appl. Logic 75 (1995), 49-56.
- [2] A. Berarducci and R. Verbrugge, On the provability logic of bounded arithmetic, ibid. 61 (1993), 75-93. Zbl0803.03037
- [3] B S. R. Buss, Bounded Arithmetic, Bibliopolis, Napoli, 1986.
- [4] D P. D’Aquino, A sharpened version of McAloon’s theorem on initial segments of models of $I{\Delta}_{0}$, Ann. Pure Appl. Logic 61 (1993), 49-62.
- [5] P. Hájek and P. Pudlák, Metamathematics of First-Order Arithmetic, Springer, Berlin, 1993. Zbl0781.03047
- [6] J R. G. Jeroslow, Non-effectiveness in S. Orey's arithmetical compactness theorem, Z. Math. Logic Grundlangen Math. 17 (1971), 285-289. Zbl0234.02036
- [7] P. Lindström, Some results on interpretability, in: Proc. 5th Scandinavian Logic Sympos., F. V. Jensen, B. H. Mayoh and K. K. Møller (eds.), Aalborg Univ. Press, 1979, 329-361.
- [8] P. Lindström, On partially conservative sentences and interpretability, Proc. Amer. Math. Soc. 91 (1984), 436-443. Zbl0577.03028
- [9] J. Paris and A. Wilkie, Counting ${\Delta}_{0}$ sets, Fund. Math. 127 (1986), 67-76. Zbl0627.03018
- [10] J. B. Paris, A. J. Wilkie and A. R. Woods, Provability of the pigeonhole principle and the existence of infinitely many primes, J. Symbolic Logic 53 (1988), 1235-1244. Zbl0688.03042
- [11] P. Pudlák, Cuts, consistency statements and interpretations, J. Symbolic Logic 50 (1985), 423-441. Zbl0569.03024
- [12] P. Pudlák, On the length of proofs of finitistic consistency statements in first order theories, in: Logic Colloquium '84, J. B. Paris, A. J. Wilkie and G. M. Wilmers (eds.), North-Holland, Amsterdam, 1986, 165-196.
- [13] W. Sieg, Fragments of arithmetic, Ann. Pure Appl. Logic 28 (1985), 33-71. Zbl0558.03029
- [14] C. Smoryński, Nonstandard models and related developments, in: Harvey Friedman's Research on the Foundations of Mathematics, L. A. Harrington, M. D. Morley, A. Ščedrov and S. G. Simpson (eds.), North-Holland, Amsterdam, 1985, 179-229.
- [15] V. Švejdar, A sentence that is difficult to interpret, Comment. Math. Univ. Carolin 22 (1981), 661-666. Zbl0484.03032
- [16] A. Visser, Interpretability logic, in: Mathematical Logic, P. P. Petkov (ed.), Plenum Press, New York, 1990, 175-209. Zbl0793.03064
- [17] A. Visser, An inside view of EXP; or, the closed fragment of the provability logic of $I{\Delta}_{0}+{\Omega}_{1}$ with a propositional constant for EXP, J. Symbolic Logic 57 (1992), 131-165. Zbl0785.03008
- [18] A. Visser, The unprovability of small inconsistency, A study of local and global interpretability, Arch. Math. Logic 32 (1993), 275-298. Zbl0795.03080
- [19] W A. J. Wilkie, On sentences interpretable in systems of arithmetic, in: Logic Colloquium '84, J. B. Paris, A. J. Wilkie and G. M. Wilmers (eds.), North-Holland, Amsterdam, 1986, 329-342.
- [20] A. J. Wilkie and J. B. Paris, On the scheme of induction for bounded arithmetic formulas, Ann. Pure Appl. Logic 35 (1987), 261-302. Zbl0647.03046

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