×

zbMATH — the first resource for mathematics

Recursive logic frames. (English) Zbl 1093.03024
Summary: We define the concept of a logic frame, which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called complete (recursively compact, \(\aleph_0\)-compact), if every finite (respectively: recursive, countable) consistent theory has a model. We show that for logic frames built from the cardinality quantifiers “there exists at least \(\lambda\)” completeness always implies \(\aleph_0\)-compactness. On the other hand we show that a recursively compact logic frame need not be \(\aleph_0\)-compact.

MSC:
03C95 Abstract model theory
03C80 Logic with extra quantifiers and operators
03C55 Set-theoretic model theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abraham, Ann. Pure Applied Logic 59 pp 1– (1993)
[2] Barwise, Israel J. Math. 25 pp 47– (1976)
[3] Barwise, Ann. Math. Logic 13 pp 171– (1978)
[4] and (eds.), Model-theoretic logics (Springer-Verlag, 1985).
[5] Chang, Proc. Amer. Math. Soc. 16 pp 1148– (1965)
[6] Fuhrken, Fund. Math. 54 pp 291– (1964)
[7] Languages with added quantifier ”there exist at least {\(\alpha\)} ”. In: Theory of Models, Proceedings 1963 International Symposium Berkeley, pp. 121–131 (North-Holland, 1965).
[8] Hodges, J. Symbolic Logic 56 pp 300– (1991)
[9] Jensen, Ann. Math. Logic 4 pp 229– (1972)
[10] Languages with expressions of infinite length (North-Holland, 1964). · Zbl 0127.00901
[11] Models with orderings. In: Logic, Methodology and Philosophy of Science III, Proceedings Third International Congress, Amsterdam, 1967, pp. 35–62 (North-Holland, 1968).
[12] Keisler, Ann. Math. Logic 1 pp 1– (1970)
[13] Lindström, Theoria 32 pp 186– (1966)
[14] Lindström, Theoria 35 pp 1– (1969)
[15] Magidor, Ann. Math. Logic 11 pp 217– (1977)
[16] Mitchell, Ann. Math. Logic 5 pp 21– (1972/73)
[17] Partitions and models. In: Proceedings of the Summer School in Logic, Leeds, 1967, pp. 109–158 (Springer-Verlag, 1968).
[18] Mostowski, Fund. Math. 44 pp 12– (1957)
[19] Transfer theorems and their applications to logics. In: Model-theoretic logics, Perspectives in Mathematical Logic, pp. 177–209 (Springer-Verlag, 1985).
[20] Schmerl, J. Symbolic Logic 42 pp 174– (1977)
[21] Schmerl, J. Symbolic Logic 37 pp 531– (1972)
[22] Shelah, Israel J. Math. 9 pp 193– (1971)
[23] Shelah, J. Symbolic Logic 37 pp 247– (1972)
[24] Shelah, Trans. Amer. Math. Soc. 204 pp 342– (1975)
[25] Shelah, Ann. Math. Logic 14 pp 73– (1978)
[26] The pair (n , 0) may fail 0-compactness. In: Logic Colloquium 2001, Lecture Notes in Logic 20, pp. 402–433 (ASL, 2005).
[27] Vaught, Fund. Math. 54 pp 303– (1964)
[28] A Löwenheim-Skolem theorem for cardinals far apart. In: Theory of Models, Proceedings 1963 International Symposium Berkeley, pp. 390–401 (North-Holland, 1965).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.