“Gap 1” two-cardinal principles and the omitting types theorem for \(\mathcal L(\mathcal Q)\).

*(English)*Zbl 0681.03016The 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.

(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

##### Keywords:

set-theoretic equivalents to two-cardinal transfer principles; strong limit singular cardinals; omitting types theorems; Henkin construction
Full Text:
DOI

**OpenURL**

##### References:

[1] | S. Ben-David and S. Sheiah,The two-cardinal transfer property and resurrection of supercompactness, to appear. #248. |

[2] | R. Grossberg,Models with second order properties in successors of strong limits, J. Symb. Logic, to appear. · Zbl 0673.03023 |

[3] | H. J. Keisler,Logic with the quantifier ’there exist uncountably many’, Ann. Math. Logic1 (1970), 1–93. · Zbl 0206.27302 |

[4] | S. Shelah,Two cardinal compactness, Isr. J. Math.9 (1971), 193–198. #8. · Zbl 0212.02001 |

[5] | S. Shelah,Models with second order properties I. Boolean algebras with no undefinable automorphisms, Ann. Math. Logic14 (1978), 57–72. #72. · Zbl 0383.03018 |

[6] | S. Shelah,Models with second order properties II. Trees with no undefined branches. Appendix: the Vaught two-cardinal theorem revisited, Ann. Math. Logic14 (1978), 223–226. #74. · Zbl 0383.03020 |

[7] | S. Shelah,Models with second order properties III. Omitting types theorems for L(Q) in {\(\lambda\)} +, Arch. Math. Logik21 (1981), 1–12. #82. · Zbl 0502.03016 |

[8] | B. Hart, C. Laflamme and S. Shelah,Models with second order properties V. Ann. Pure Appl. Logic, to appear. #162 |

[9] | S. Shelah,Uncountable constructions for Boolean algebras, existentially complete groups, and Banach spaces, Isr. J. Math.51 (1985), 273–297. #128. · Zbl 0589.03012 |

[10] | S. Shelah,Can you take Solovay’s inaccessible away?, Isr. J. Math.48 (1984), 1–47. #176. · Zbl 0596.03055 |

[11] | S. Shelah,Models with second order properties II. Trees with no undefined branches, Ann. Math. Logic14 (1978), 73–87. #73. · Zbl 0383.03019 |

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.