``Gap 1'' two-cardinal principles and the omitting types theorem for \(\mathcal L(\mathcal Q)\) (Q1823234)

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.