Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 49, Issue 2 (1984), 542-554.
Forcing with $\lbrack\kappa\rbrack^\kappa$ over a model of set theory with a strong partition cardinal, M. Spector produced a generic ultrafilter $G$ on $\kappa$ such that $\kappa^\kappa/G$ is not well-founded. Theorem. Let $G$ be Spector-generic over a model $M$ of $ZF + DC + \kappa \rightarrow (\kappa)^\kappa_\alpha, \kappa > \omega$, for all $\alpha < \kappa$. 1) Every cardinal (well-ordered or not) of $M$ is a cardinal of $M\lbrack G\rbrack$. 2) If $A \in M\lbrack G\rbrack$ is a well-ordered subset of $M$, then $A \in M$. Let $\Phi = \kappa^\kappa/G$. 3) There is an ultrafilter $U$ on $\Phi$ such that every member of $U$ has a subset of type $\Phi$, and the intersection of any well-ordered subset of $U$ is in $U$. 4) $\Phi$ satisfies $\Phi \rightarrow (\Phi)^\alpha_\beta$ for all $\alpha < \aleph_1$ and all ordinals $\beta$. 5) There is a linear order $\Phi'$ with property 3) above which is not "weakly compact", i.e., $\Phi' \nrightarrow (\Phi')^2$.
J. Symbolic Logic, Volume 49, Issue 2 (1984), 542-554.
First available in Project Euclid: 6 July 2007
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Henle, J. M. Spector Forcing. J. Symbolic Logic 49 (1984), no. 2, 542--554. https://projecteuclid.org/euclid.jsl/1183741554