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Syntactical truth predicates for second order arithmetic

Published online by Cambridge University Press:  12 March 2014

Loïc Colson  and
Serge Grigorieff
Loïc Colson
Affiliation:
L.L.A.I.C., UFR d'Informatique, Université Paris7, 2 Place Jussieu 75251 Paris, France, E-mail: [email protected]
Serge Grigorieff
Affiliation:
L.L.A.I.C. UFR d'Informatique, Université Paris 7, 2 Place Jussieu 75251 Paris, France, E-mail: [email protected]
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Abstract

We introduce a notion of syntactical truth predicate (s.t.p.) for the second order arithmetic PA2. An s.t.p. is a set T of closed formulas such that:

(i) T(t = u) if and only if the closed first order terms t and u are convertible, i.e., have the same value in the standard interpretation

(ii) T(A → B) if and only if (T(A) ⇒ T(B))

(iii) T(∀xA) if and only if (T(A[x ← t]) for any closed first order term t)

(iv) T(∀X A) if and only if (T(A[X ← ∆]) for any closed set definition ∆ = {x ∣ D(x)}).

S.t.p.'s can be seen as a counterpart to Tarski's notion of (model-theoretical) validity and have main model properties. In particular, their existence is equivalent to the existence of an ω-model of PA2, this fact being provable in PA2 with arithmetical comprehension only.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 1 , March 2001 , pp. 225 - 256
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Addison, J. W., Some consequences of the axiom of constructibility, Fundamenta Mathematicae, vol. 46 (1959), pp. 337–357.CrossRefGoogle Scholar
[2]Apt, K. R. and Marek, W., Second order arithmetic and related topics, Annals of Mathematical Logic, vol. 6 (1974), pp. 177–229.CrossRefGoogle Scholar
[3]Boolos, G. and Putnam, H., Degrees of unsolvability of constructible sets of integers, this Journal, vol. 33 (1968), no. 4, pp. 497–513.Google Scholar
[4]Boyd, R., Hensel, G., and Putnam, H., A recursion-theoretic characterization of the ramified analytical hierarchy, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 37–62.CrossRefGoogle Scholar
[5]Cohen, P. J., Minimal model for set theory, Bulletin of the American Mathematical Society, vol. 69 (1963), pp. 537–540.CrossRefGoogle Scholar
[6]Dragalin, A. G., Cut-elimination in the theory of definable sets of natural numbers, Abstracts of the iv-aja vsesojuznaja konferencija po mat. logike, kishinev, 1976, in Russian, p. 45.Google Scholar
[7]Dragalin, A. G., Cut-elimination in the theory of definable sets of natural numbers, Set theory and topology, 1st issue, Udmurt University Press, Izhevsk, 1977, in Russian, pp. 27–36.Google Scholar
[8]Dragalin, A. G., Cut-elimination in the theory of definable sets of natural numbers, Publicationes Mathematkae Debrecen, vol. 51 (1997), no. 1–2, pp. 153–164.CrossRefGoogle Scholar
[9]Feferman, S., Constructively provable well-orderings. Notices of the American Mathematical Society, vol. 8 (1961), p. 495.Google Scholar
[10]Feferman, S., Provable well-orderings and relations between predicative and ramified analysis, Notices of the American Mathematical Society, vol. 9 (1962), p. 323.Google Scholar
[11]Feferman, S., Systems of predicative analysis, this Journal, vol. 29 (1964), no. 1, pp. 1–30.Google Scholar
[12]Girard, J. Y., Proof theory and logical complexity, Studies in Proof Theory, vol. 1, Bibliopolis, Napoli, 1988.Google Scholar
[13]Gödel, K., The consistency of the axiom of choice and of the generalized continuum hypothesis, Proceedings of the National Academy of Sciences, USA, vol. 45 (1938), p. 93, reprinted in Collected works, vol. II (S. Feferman, editor), Oxford University Press, 1990.Google Scholar
[14]Hájek, P. and Pudlák, P., Metamathematics of first-order arithmetic, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, 1998.Google Scholar
[15]Jakubovic, A. M., On the consistency of the theory of types with the axiom of choice relative to the theory of types, Soviet Math. Doklady, vol. 24 (1981), pp. 621–624.Google Scholar
[16]Jensen, R. B., Definable subsets of minimal degree, Mathematical logic and foundations of set theory, Jerusalem, 1968 (Bar-Hillel, Y., editor), 1970, pp. 122–128.Google Scholar
[17]Jensen, R. B. and Johnsbraaten, H., A new construction of a non constructible Δ31 subset of ω, Fundamenta Mathematicae, vol. 81 (1974), pp. 279–290.CrossRefGoogle Scholar
[18]Jensen, R. B. and Solovay, R. M., Some applications of almost disjoint sets, Mathematical logic and foundations of set theory, Jerusalem, 1968 (Bar-Hillel, Y., editor), 1970, pp. 84–104.Google Scholar
[19]Kaye, R., Models of Peano arithmetic, Oxford Science Publ., 1991.CrossRefGoogle Scholar
[20]Kleene, S. C., Quantification of number-theoretic functions, Compositio Mathematicae, vol. 14, (1959), pp. 23–40.Google Scholar
[21]Kreisel, G., The axiom of choice and the class of hyperarithmetic functions, Dutch Academy A, vol. 65 (1962), pp. 307–319.Google Scholar
[22]Kreisel, G., A survey of proof theory, this Journal, vol. 33 (1998), no. 3, pp. 321–388.Google Scholar
[23]Leeds, S. and Putnam, H., Solution of a problem of Gandy's, Fundamenta Mathematicae, vol. 81 (1974), pp. 99–106.CrossRefGoogle Scholar
[24]Mostowski, A., A class of models for second order arithmetic, Bulletin de l'Academie Polonaise des Sciences, vol. 7 (1959), pp. 401–404.Google Scholar
[25]Mostowski, A., Models for second order arithmetic with definable Skolem functions, Fundamenta Mathematicae, vol. 75 (1972), pp. 223–234.CrossRefGoogle Scholar
[26]Schütte, K., Predicative well-orderings, North-Holland, 1965.CrossRefGoogle Scholar
[27]Simpson, S. G., Subsystems of second order arithmetic, Springer-Verlag, 1999.CrossRefGoogle Scholar
[28]Solovay, R. M., A non constructible Δ31 set of integers, Transactions of the American Mathematical Society, vol. 127 (1967), pp. 50–75.Google Scholar
[29]Takeuti, G., Proof theory, North-Holland, 1975.Google Scholar
[30]Zbierski, P., Models for higher order arithmetics, Bulletin de l'Académie Polonaise des Sciences, vol. 19 (1971), no. 7, pp. 557–562.Google Scholar