Journal of Symbolic Logic

On Models with Power-Like Ordering

Saharon Shelah

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We prove here theorems of the form: if $T$ has a model $M$ in which $P_1 (M)$ is $\kappa_1$-like ordered, $P_2(M)$ is $\kappa_2$-like ordered $\ldots$, and $Q_1 (M)$ if of power $\lambda_1, \ldots$, then $T$ has a model $N$ in which $P_1(M)$ is $\kappa_1'$-like ordered $\ldots, Q_1(N)$ is of power $\lambda_1,\ldots$. (In this article $\kappa$ is a strong-limit singular cardinal, and $\kappa'$ is a singular cardinal.) We also sometimes add the condition that $M, N$ omits some types. The results are seemingly the best possible, i.e. according to our knowledge about $n$-cardinal problems (or, more precisely, a certain variant of them).

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J. Symbolic Logic, Volume 37, Issue 2 (1972), 247-267.

First available in Project Euclid: 6 July 2007

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Shelah, Saharon. On Models with Power-Like Ordering. J. Symbolic Logic 37 (1972), no. 2, 247--267.

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