Partially conservative extensions of arithmetic

Author:
D. Guaspari

Journal:
Trans. Amer. Math. Soc. **254** (1979), 47-68

MSC:
Primary 03F30; Secondary 03F25, 03F40, 03H15

DOI:
https://doi.org/10.1090/S0002-9947-1979-0539907-7

MathSciNet review:
539907

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let *T* be a consistent r.e. extension of Peano arithmetic; , the usual quantifier-block classification of formulas of the language of arithmetic (bounded quantifiers counting ``for free"); and , variables through the set of all classes , . The principal concern of this paper is the question: When can we find an independent sentence which is -conservative in the following sense: Any sentence in which is provable from is already provable from *T*? (Additional embellishments: Ensure that is not provably equivalent to a sentence in any class ``simpler'' than ; that is not conservative for classes ``more complicated'' than .) The answer, roughly, is that one can find such a , embellishments and all, unless and are so related that such a *obviously* cannot exist. This theorem has applications to the theory of interpretations, since `` is -conservative'' is closely related to the property `` is interpretable in *T*"-or to variants of it, depending on . Finally, we provide simple model theoretic characterizations of -conservativeness. Most results extend straightforwardly if extra symbols are added to the language of arithmetic, and most have analogs in the Levy hierarchy of set theoretic formulas (*T* then being an extension of *ZF*).

**[Barwise, 1]***Admissible sets and structures*, Springer-Verlag, Berlin, 1975. MR**0424560 (54:12519)****[Barwise, 2]***Infinitary methods in the model theory of set theory*, Logic Colloquium, 1969 (R. O. Gandy and M. Yates, Editors), North Holland, Amsterdam, 1971, pp. 53-66. MR**0277370 (43:3103)****[Hajkova-Hajek]***On interpretability in theories containing arithmetic*, Fund. Math.**76**(1972), 131-137. MR**0307897 (46:7012)****[Hajek, 1]***On interpretability in set theories*, Comment. Math. Univ. Carolinae**12**(1971), 73-79. MR**0311470 (47:32)****[Hajek, 2]***On interpretability in set theories*. II, Comment. Math. Univ. Carolinae**13**(1972), 445-455. MR**0323566 (48:1922)****[Feferman]***Arithmetization of metamathematics in a general setting*, Fund. Math.**49**(1960), 35-92. MR**0147397 (26:4913)****[Friedman]***Countable models of set theories*, Proc. Cambridge Summer School in Math Logic, 1971 (A. R. D. Mathis and H. Rogers, Editors), Lectures Notes in Math., vol. 337, Springer, Berlin and New York, 1973, pp. 539-573. MR**0347599 (50:102)****[Keisler]***Model theory for infinitary logic*, North Holland, Amsterdam, 1971.**[Kreisel]***On weak completeness of intuitionistic predicate logic*, J. Symbolic Logic**27**(1962), 139-158. MR**0161796 (28:5000)****[Krivine-McAloon]***Some true unprovable formulas for set theory*, Bertrand Russell Memorial Logic Conference, Leeds, 1973, pp. 332-341. MR**0357112 (50:9580)****[Levy]***A hierarchy of formulas in set theory*, Mem. Amer. Math. Soc. No. 57 (1965). MR**0189983 (32:7399)****[Löb]***Solution of a problem of Leon Henkin*, J. Symbolic Logic**20**(1955), 115-118. MR**0070596 (17:5b)****[Macintyre-Simmons]***Gödel's diagonalization technique and related properties of theories*, Colloq. Math.**28**(1973), 165-180. MR**0332465 (48:10792)****[Matijasevic]***Diophantine representation of recursively enumerable predicates*, Proc. Second Scandinavian Logic Symposium (J. E. Fenstad, Editor), North Holland, Amsterdam, 1971. MR**0354340 (50:6820)****[Montague]***Semantical closure and nonfinite axiomatizability*, in Infinitistic Methods, Pergamon Press, New York, 1959, pp. 45-69.**[Shoenfield]***Mathematical logic*, Addison-Wesley, Reading, Mass., 1967. MR**0225631 (37:1224)****[Solovay]***On interpretability in set theories*(to appear).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
03F30,
03F25,
03F40,
03H15

Retrieve articles in all journals with MSC: 03F30, 03F25, 03F40, 03H15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0539907-7

Article copyright:
© Copyright 1979
American Mathematical Society