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ITERATING THE COFINALITY-
$\omega $ CONSTRUCTIBLE MODEL
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 88 / Issue 4 / December 2023
- Published online by Cambridge University Press:
- 05 January 2023, pp. 1682-1691
- Print publication:
- December 2023
-
- Article
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We investigate iterating the construction of
$C^{*}$, the L-like inner model constructed using first order logic augmented with the “cofinality 
$\omega $” quantifier. We first show that 
$\left (C^{*}\right )^{C^{*}}=C^{*}\ne L$ is equiconsistent with 
$\mathrm {ZFC}$, as well as having finite strictly decreasing sequences of iterated 
$C^{*}$s. We then show that in models of the form 
$L[U]$ we get infinite decreasing sequences of length 
$\omega $, and that an inner model with a measurable cardinal is required for that.
