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 Free Access

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]**Jon Barwise,*Admissible sets and structures*, Springer-Verlag, Berlin-New York, 1975. An approach to definability theory; Perspectives in Mathematical Logic. MR**0424560****[Barwise, 2]**Jon Barwise,*Infinitary methods in the model theory of set theory*, Logic Colloquium ’69 (Proc. Summer School and Colloq., Manchester, 1969), North-Holland, Amsterdam, 1971, pp. 53–66. MR**0277370****[Hajkova-Hajek]**Marie Hájková and Petr Hájek,*On interpretability in theories containing arithmetic*, Fund. Math.**76**(1972), no. 2, 131–137. MR**0307897****[Hajek, 1]**Petr Hájek,*On interpretability in set theories*, Comment. Math. Univ. Carolinae**12**(1971), 73–79. MR**0311470****[Hajek, 2]**Petr Hájek,*On interpretability in set theories. II*, Comment. Math. Univ. Carolinae**13**(1972), 445–455. MR**0323566****[Feferman]**S. Feferman,*Arithmetization of metamathematics in a general setting*, Fund. Math.**49**(1960/1961), 35–92. MR**0147397****[Friedman]**Harvey Friedman,*Countable models of set theories*, Cambridge Summer School in Mathematical Logic (Cambridge, 1971) Springer, Berlin, 1973, pp. 539–573. Lecture Notes in Math., Vol. 337. MR**0347599****[Keisler]***Model theory for infinitary logic*, North Holland, Amsterdam, 1971.**[Kreisel]**G. Kreisel,*On weak completeness of intuitionistic predicate logic*, J. Symbolic Logic**27**(1962), 139–158. MR**0161796**, https://doi.org/10.2307/2964110**[Krivine-McAloon]**J. L. Krivine and K. McAloon,*Some true unprovable formulas for set theory*, The Proceedings of the Bertrand Russell Memorial Conference (Uldum, 1971), Bertrand Russell Memorial Logic Conf., Leeds, 1973, pp. 332–341. MR**0357112****[Levy]**Azriel Lévy,*A hierarchy of formulas in set theory*, Mem. Amer. Math. Soc. No.**57**(1965), 76. MR**0189983****[Löb]**M. H. Löb,*Solution of a problem of Leon Henkin*, J. Symb. Logic**20**(1955), 115–118. MR**0070596****[Macintyre-Simmons]**A. Macintyre and H. Simmons,*Gödel’s diagonalization technique and related properties of theories*, Colloq. Math.**28**(1973), 165–180. MR**0332465****[Matijasevic]**Yu. V. Matijasevič,*Diophantine representation of recursively enumerable predicates*, Proceedings of the Second Scandinavian Logic Symposium (Univ. Oslo, Oslo, 1970) Studies in Logic and the Foundations of Mathematics, Vol. 63, North-Holland, Amsterdam, 1971, pp. 171–177. MR**0354340****[Montague]***Semantical closure and nonfinite axiomatizability*, in Infinitistic Methods, Pergamon Press, New York, 1959, pp. 45-69.**[Shoenfield]**Joseph R. Shoenfield,*Mathematical logic*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. MR**0225631****[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