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“Gap 1” two-cardinal principles and the omitting types theorem for $$\mathcal L(\mathcal Q)$$. (English) Zbl 0681.03016
The author states two goals: (1) to give precise set-theoretic equivalents to two-cardinal transfer principles for strong limit singular cardinals, (2) to settle the relationship of two-cardinal transfer principles to omitting types theorems for $${\mathcal L}(Q)$$. The main result is the expansion of the following. For $$\lambda$$ a singular strong limit cardinal, and more generally if $$\lambda >2^{cof \lambda}$$, the following are equivalent:
(1) $$(\aleph_ 0,\aleph_ 1)\to (\lambda,\lambda^+),$$
(2) Completeness theorem for $${\mathcal L}(Q)$$ in the $$\lambda^+$$- interpretation,
(3) Omitting types for $${\mathcal L}(Q)$$ in the $$\lambda^+$$- interpretation,
(4) Various weak forms of $$\square_{\lambda}.$$
The latter part of the paper is concerned with how to carry out a Henkin construction for $${\mathcal L}(Q)$$ over a tree of approximations to a model of size $$\lambda^+$$, given a suitable $$\square$$-like principle.
Reviewer: J.M.Plotkin

MSC:
 03C55 Set-theoretic model theory 03E05 Other combinatorial set theory
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References:
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