Abstract
We develop the theory of iterated ultrapowers, of \(0^\sharp \), and of short and long extenders.
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- 1.
For a proper class \(X\), we write \(\sigma ^{-1}(X)\) for \(\bigcup \{ \sigma ^{-1} (X\cap V_{\alpha }){:}\,X \cap V_\alpha \in \mathrm{ran }(\sigma )\}\).
- 2.
We here use the fact that the ultrapower construction may also be applied with transitive models of a sufficiently large fragment of ZFC. We leave the straightforward details to the reader.
- 3.
By our convention, cf. Definition 10.20, this is tantamount to saying that \({\fancyscript{M}}_{\alpha -1}\) is well-founded.
- 4.
i.e., \(X \in F_a\) or \(([\kappa ]^{\mathrm{Card} (a)}) \setminus X \in F_a\).
- 5.
Recall that the strength of an extender \(F\) is the largest ordinal \(\alpha \) such that \(V_\alpha \subset \mathrm{Ult}(V;F)\).
- 6.
If there is any. The current proof does not presuppose that there be some such branch. Rather, it will show the existence of some such \(b\) such that the direct limit along \(b\) is well-founded.
- 7.
The fact that \(F\) is certified implies that \(V_\alpha \subset \mathrm{Ult}(V;F)\).
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© 2014 Springer International Publishing Switzerland
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Schindler, R. (2014). Measurable Cardinals. In: Set Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-06725-4_10
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DOI: https://doi.org/10.1007/978-3-319-06725-4_10
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Online ISBN: 978-3-319-06725-4
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