Generalized quantifiers and compact logic

Shelah, Saharon

doi:10.1090/S0002-9947-1975-0376334-6

Generalized quantifiers and compact logic
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Trans. Amer. Math. Soc. 204 (1975), 342-364 Request permission

Abstract:

We solve a problem of Friedman by showing the existence of a logic stronger than first-order logic even for countable models, but still satisfying the general compactness theorem, assuming e.g. the existence of a weakly compact cardinal. We also discuss several kinds of generalized quantifiers.

References

K. Jon Barwise, Absolute logics and $L_{\infty \omega }$, Ann. Math. Logic 4 (1972), 309–340. MR 337483, DOI 10.1016/0003-4843(72)90002-2

Two refinements of Morley’s method on omitting types of elements

11

C. C. Chang, A note on the two cardinal problem, Proc. Amer. Math. Soc. 16 (1965), 1148–1155. MR 193016, DOI 10.1090/S0002-9939-1965-0193016-3

Model theory

A. Ehrenfeucht and A. Mostowski, Models of axiomatic theories admitting automorphisms, Fund. Math. 43 (1956), 50–68. MR 84456, DOI 10.4064/fm-43-1-50-68

Why first order logic

Harvey Friedman, Beth’s theorem in cardinality logics, Israel J. Math. 14 (1973), 205–212. MR 317923, DOI 10.1007/BF02762675
S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57–103. MR 108455, DOI 10.4064/fm-47-1-57-103
Gebhard Fuhrken, Languages with added quantifier “there exist at least ${\bf \aleph }_{\alpha }$”, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 121–131. MR 0241282
W. Hanf, Incompactness in languages with infinitely long expressions, Fund. Math. 53 (1963/64), 309–324. MR 160732, DOI 10.4064/fm-53-3-309-324

Hanf numbers for some generalizations of first-order languages

11

L. Henkin, Some remarks on infinitely long formulas, Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959) Pergamon, Oxford; Państwowe Wydawnictwo Naukowe, Warsaw, 1961, pp. 167–183. MR 0143691
R. Björn Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR 309729, DOI 10.1016/0003-4843(72)90001-0

Some combinatorial properties of $L$ and $V$

H. Jerome Keisler, Some model theoretic results for $\omega$-logic, Israel J. Math. 4 (1966), 249–261. MR 221922, DOI 10.1007/BF02771640
H. Jerome Keisler, Logic with the quantifier “there exist uncountably many”, Ann. Math. Logic 1 (1970), 1–93. MR 263616, DOI 10.1016/S0003-4843(70)80005-5
H. Jerome Keisler, Models with tree structures, Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1974, pp. 331–348. MR 0357108
H. Jerome Keisler and Michael Morley, Elementary extensions of models of set theory, Israel J. Math. 6 (1968), 49–65. MR 237321, DOI 10.1007/BF02771605

Constructible sets and weakly compact cardinals

12

David W. Kueker, Löwenheim-Skolem and interpolation theorems in infinitary languages, Bull. Amer. Math. Soc. 78 (1972), 211–215. MR 290942, DOI 10.1090/S0002-9904-1972-12921-5
Azriel Lévy, Axiom schemata of strong infinity in axiomatic set theory, Pacific J. Math. 10 (1960), 223–238. MR 124205, DOI 10.2140/pjm.1960.10.223
Per Lindström, First order predicate logic with generalized quantifiers, Theoria 32 (1966), 186–195. MR 244012, DOI 10.1111/j.1755-2567.1966.tb00600.x
Per Lindström, On extensions of elementary logic, Theoria 35 (1969), 1–11. MR 244013, DOI 10.1111/j.1755-2567.1969.tb00356.x

Compact extensions of

20

On continuous quantifiers

20

Michael Morley, Omitting classes of elements, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 265–273. MR 0201305
A. Mostowski, On a generalization of quantifiers, Fund. Math. 44 (1957), 12–36. MR 89816, DOI 10.4064/fm-44-1-12-36
Michael Morley and Robert Vaught, Homogeneous universal models, Math. Scand. 11 (1962), 37–57. MR 150032, DOI 10.7146/math.scand.a-10648
Jack H. Silver, Some applications of model theory in set theory, Ann. Math. Logic 3 (1971), no. 1, 45–110. MR 409188, DOI 10.1016/0003-4843(71)90010-6
James H. Schmerl, An elementary sentence which has ordered models, J. Symbolic Logic 37 (1972), 521–530. MR 427060, DOI 10.2307/2272738
Saharon Shelah, Two cardinal compactness, Israel J. Math. 9 (1971), 193–198. MR 302437, DOI 10.1007/BF02771584
Saharon Shelah, On models with power-like orderings, J. Symbolic Logic 37 (1972), 247–267. MR 446955, DOI 10.2307/2272971
James H. Schmerl and Saharon Shelah, On power-like models for hyperinaccessible cardinals, J. Symbolic Logic 37 (1972), 531–537. MR 317925, DOI 10.2307/2272739

The uniqueness of elementary logic

20

R. L. Vaught, The Löwenheim-Skolem theorem, Logic, Methodology and Philos. Sci. (Proc. 1964 Internat. Congr.), North-Holland, Amsterdam, 1965, pp. 81–89. MR 0210574