Journal of Symbolic Logic

Models with Second Order Properties in Successors of Singulars

Rami Grossberg

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Abstract

Let $L(\mathbf{Q})$ be first order logic with Keisler's quantifier, in the $\lambda^+$ interpretation (= the satisfaction is defined as follows: $M \models (\mathbf{Q}x)\varphi(x)$ means there are $\lambda^+$ many elements in $M$ satisfying the formula $\varphi(x))$. Theorem 1. Let $\lambda$ be a singular cardinal; assume $\square_\lambda$ and GCH. If $T$ is a complete theory in $L(\mathbf{Q})$ of cardinality at most $\lambda$, and $p$ is an $L(\mathbf{Q})$ 1-type so that $T$ strongly omits $p ( = p$ has no support, to be defined in $\S1$), then $T$ has a model of cardinality $\lambda^+$ in the $\lambda^+$ interpretation which omits $p$. Theorem 2. Let $\lambda$ be a singular cardinal, and let $T$ be a complete first order theory of cardinality $\lambda$ at most. Assume $\square_\lambda$ and GCH. If $\Gamma$ is a smallness notion then $T$ has a model of cardinality $\lambda^+$ such that a formula $\varphi(x)$ is realized by $\lambda^+$ elements of $M$ iff $\varphi(x)$ is not $\Gamma$-small. The theorem is proved also when $\lambda$ is regular assuming $\lambda = \lambda^{< \lambda}$. It is new when $\lambda$ is singular or when $|T| = \lambda$ is regular. Theorem 3. Let $\lambda$ be singular. If $\operatorname{Con}(ZFC + GCH + (\exists\kappa)$ [$\kappa$ is a strongly compact cardinal]), then the following in consistent: ZFC + GCH + the conclusions of all above theorems are false.

Article information

Source
J. Symbolic Logic, Volume 54, Issue 1 (1989), 122-137.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183742856

Mathematical Reviews number (MathSciNet)
MR987327

Zentralblatt MATH identifier
0673.03023

JSTOR
links.jstor.org

Citation

Grossberg, Rami. Models with Second Order Properties in Successors of Singulars. J. Symbolic Logic 54 (1989), no. 1, 122--137. https://projecteuclid.org/euclid.jsl/1183742856


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