Partially conservative extensions of arithmetic
Author:
D. Guaspari
Journal:
Trans. Amer. Math. Soc. 254 (1979), 4768
MSC:
Primary 03F30; Secondary 03F25, 03F40, 03H15
MathSciNet review:
539907
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let T be a consistent r.e. extension of Peano arithmetic; , the usual quantifierblock 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, SpringerVerlag,
BerlinNew York, 1975. An approach to definability theory; Perspectives in
Mathematical Logic. MR 0424560
(54 #12519)
 [Barwise, 2]
Jon
Barwise, Infinitary methods in the model theory of set theory,
Logic Colloquium ’69 (Proc. Summer School and Colloq., Manchester,
1969), NorthHolland, Amsterdam, 1971, pp. 53–66. MR 0277370
(43 #3103)
 [HajkovaHajek]
Marie
Hájková and Petr
Hájek, On interpretability in theories containing
arithmetic, Fund. Math. 76 (1972), no. 2,
131–137. MR 0307897
(46 #7012)
 [Hajek, 1]
Petr
Hájek, On interpretability in set theories, Comment.
Math. Univ. Carolinae 12 (1971), 73–79. MR 0311470
(47 #32)
 [Hajek, 2]
Petr
Hájek, On interpretability in set theories. II,
Comment. Math. Univ. Carolinae 13 (1972), 445–455.
MR
0323566 (48 #1922)
 [Feferman]
S.
Feferman, Arithmetization of metamathematics in a general
setting, Fund. Math. 49 (1960/1961), 35–92. MR 0147397
(26 #4913)
 [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
(50 #102)
 [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
(28 #5000)
 [KrivineMcAloon]
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 (50 #9580)
 [Levy]
Azriel
Lévy, A hierarchy of formulas in set theory, Mem. Amer.
Math. Soc. No. 57 (1965), 76. MR 0189983
(32 #7399)
 [Löb]
M.
H. Löb, Solution of a problem of Leon Henkin, J. Symb.
Logic 20 (1955), 115–118. MR 0070596
(17,5b)
 [MacintyreSimmons]
A.
Macintyre and H.
Simmons, Gödel’s diagonalization technique and related
properties of theories, Colloq. Math. 28 (1973),
165–180. MR 0332465
(48 #10792)
 [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, NorthHolland, Amsterdam, 1971,
pp. 171–177. MR 0354340
(50 #6820)
 [Montague]
Semantical closure and nonfinite axiomatizability, in Infinitistic Methods, Pergamon Press, New York, 1959, pp. 4569.
 [Shoenfield]
Joseph
R. Shoenfield, Mathematical logic, AddisonWesley Publishing
Co., Reading, Mass.LondonDon Mills, Ont., 1967. MR 0225631
(37 #1224)
 [Solovay]
On interpretability in set theories (to appear).
 [Barwise, 1]
 Admissible sets and structures, SpringerVerlag, 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. 5366. MR 0277370 (43:3103)
 [HajkovaHajek]
 On interpretability in theories containing arithmetic, Fund. Math. 76 (1972), 131137. MR 0307897 (46:7012)
 [Hajek, 1]
 On interpretability in set theories, Comment. Math. Univ. Carolinae 12 (1971), 7379. MR 0311470 (47:32)
 [Hajek, 2]
 On interpretability in set theories. II, Comment. Math. Univ. Carolinae 13 (1972), 445455. MR 0323566 (48:1922)
 [Feferman]
 Arithmetization of metamathematics in a general setting, Fund. Math. 49 (1960), 3592. 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. 539573. 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), 139158. MR 0161796 (28:5000)
 [KrivineMcAloon]
 Some true unprovable formulas for set theory, Bertrand Russell Memorial Logic Conference, Leeds, 1973, pp. 332341. 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), 115118. MR 0070596 (17:5b)
 [MacintyreSimmons]
 Gödel's diagonalization technique and related properties of theories, Colloq. Math. 28 (1973), 165180. 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. 4569.
 [Shoenfield]
 Mathematical logic, AddisonWesley, Reading, Mass., 1967. MR 0225631 (37:1224)
 [Solovay]
 On interpretability in set theories (to appear).
Similar Articles
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:
http://dx.doi.org/10.1090/S00029947197905399077
PII:
S 00029947(1979)05399077
Article copyright:
© Copyright 1979
American Mathematical Society
