On extensions of models of strong fragments of arithmetic
Author:
Roman Kossak
Journal:
Proc. Amer. Math. Soc. 108 (1990), 223232
MSC:
Primary 03F30; Secondary 03C62, 03H15
MathSciNet review:
984802
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Using a weak notion of recursive saturation (not always semiregularity) we prove that there are no finitely generated countable models of . We consider the problem of not almost semiregularity of models of . From a partial solution to this problem we deduce a generalization of the theorem of Smorynski and Stavi on cofinal extensions of recursively saturated models of arithmetic.
 [C1]
P.
Clote, Partition relations in arithmetic, Methods in
mathematical logic (Caracas, 1983) Lecture Notes in Math.,
vol. 1130, Springer, Berlin, 1985, pp. 32–68. MR 799036
(87f:03165), http://dx.doi.org/10.1007/BFb0075306
 [C2]
Peter
G. Clote, A note on the MacDowellSpecker theorem, Fund. Math.
127 (1987), no. 2, 163–170. MR 882624
(88d:03073)
 [GD]
Haim
Gaifman and Constantine
Dimitracopoulos, Fragments of Peano’s arithmetic and the MRDP
theorem, Logic and algorithmic (Zurich, 1980) Monograph. Enseign.
Math., vol. 30, Univ. Genève, Geneva, 1982,
pp. 187–206. MR 648303
(83j:03095)
 [KP]
L.
A. S. Kirby and J.
B. Paris, Initial segments of models of Peano’s axioms,
Set theory and hierarchy theory, V (Proc. Third Conf., Bierutowice, 1976),
Springer, Berlin, 1977, pp. 211–226. Lecture Notes in Math.,
Vol. 619. MR
0491157 (58 #10423)
 [K1]
Roman
Kossak, A certain class of models of Peano arithmetic, J.
Symbolic Logic 48 (1983), no. 2, 311–320. MR 704085
(84j:03076), http://dx.doi.org/10.2307/2273548
 [K2]
, Models with the property, to appear in Journal of Symbolic Logic.
 [Ku]
David
W. Kueker, Backandforth arguments and infinitary logics,
Infinitary logic: in memoriam Carol Karp, Springer, Berlin, 1975,
pp. 17–71. Lecture Notes in Math., Vol. 492. MR 0462940
(57 #2905)
 [L]
H. Lessan, Models of arithmetic, Ph.D. Thesis, Manchester 1978.
 [P]
J.
B. Paris, Some conservation results for fragments of
arithmetic, Model theory and arithmetic (Paris, 1979–1980)
Lecture Notes in Math., vol. 890, Springer, BerlinNew York, 1981,
pp. 251–262. MR 645006
(83f:03060)
 [PK]
J.
B. Paris and L.
A. S. Kirby, Σ_{𝑛}collection schemas in
arithmetic, Logic Colloquium ’77 (Proc. Conf., Wrocław,
1977) Stud. Logic Foundations Math., vol. 96, NorthHolland,
AmsterdamNew York, 1978, pp. 199–209. MR 519815
(81e:03056)
 [PW]
J. Paris and A. Wilkie, A note on the end extension problem, (to appear).
 [C1]
 P. Clote, Partition relations in arithmetic, in Lecture Notes in Math. vol. 1130 SpringerVerlag, Heidelberg, 1985, pp. 3268. MR 799036 (87f:03165)
 [C2]
 , A note on the MacDowellSpecker theorem, Fundamenta Mathematicae 127 (1986), pp. 163170. MR 882624 (88d:03073)
 [GD]
 H. Gaifman and C. Dimitracopoulos, Fragments of arithmetic and the MRDP theorem, Logic and Algorithmic, Monographie No. 30 de L'Enseignement Mathematique, pp. 187206. MR 648303 (83j:03095)
 [KP]
 L. Kirby and J. Paris, Initial segments of models of Peano's axioms, in Lecture Notes in Math. vol. 619, SpringerVerlag, Heidelberg, 1979, pp. 211226. MR 0491157 (58:10423)
 [K1]
 R. Kossak, A certain class of models of arithmetic, Journal of Symbolic Logic 48 (1983). pp. 311319. MR 704085 (84j:03076)
 [K2]
 , Models with the property, to appear in Journal of Symbolic Logic.
 [Ku]
 D. Kueker, Back and forth arguments in infinitary logics, in Lecture Notes in Math. vol. 492, SpringerVerlag, Heidelberg, 1975, pp. 1771. MR 0462940 (57:2905)
 [L]
 H. Lessan, Models of arithmetic, Ph.D. Thesis, Manchester 1978.
 [P]
 J. Paris, Some conservation results for fragments of arithmetic, in Lecture Notes in Math. vol. 890, SpringerVerlag, Heidelberg, 1980, pp. 251262. MR 645006 (83f:03060)
 [PK]
 J. Paris and L. Kirby, collection schemas in arithmetic, Logic Colloquium 77, North Holland, Amsterdam 1978. MR 519815 (81e:03056)
 [PW]
 J. Paris and A. Wilkie, A note on the end extension problem, (to appear).
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
03F30,
03C62,
03H15
Retrieve articles in all journals
with MSC:
03F30,
03C62,
03H15
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199009848022
PII:
S 00029939(1990)09848022
Keywords:
fragments of arithmetic,
recursive saturation,
end extensions,
cofinal extensions,
automorphisms
Article copyright:
© Copyright 1990
American Mathematical Society
