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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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