Journal of Symbolic Logic

Models with Second Order Properties in Successors of Singulars

Rami Grossberg

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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.

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

First available in Project Euclid: 6 July 2007

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Grossberg, Rami. Models with Second Order Properties in Successors of Singulars. J. Symbolic Logic 54 (1989), no. 1, 122--137.

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