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Extensions with the approximation and cover properties have no new large cardinals. (English) Zbl 1066.03052
It is well known that ‘small’ forcings do not destroy large cardinals, e.g., adding a Cohen real will not make an uncountable measurable cardinal non-measurable. The author investigates the possible creation of large cardinals. The main result of the paper gives conditions when an elementary embedding of a larger universe induces a similar embedding in a smaller universe: if $$V\subseteq\overline{V}$$ and if $$\delta$$ is a cardinal such that 1) if $$A\subseteq V$$ belongs to $$\overline{V}$$ and if $$A\cap a\in V$$ whenever $$a\in V$$ has size less than $$\delta$$ then $$A\in V$$ ($$\delta$$-approximation), and 2) if $$A\subseteq V$$ belongs to $$\overline{V}$$ and $$| A| <\delta$$ in $$\overline{V}$$ then there is $$B\in V$$ with $$A\subseteq B$$ and $$| B| <\delta$$ in $$V$$ ($$\delta$$-covering). In this case an elementary embedding $$j:\overline{V}\to\overline{M}$$ with critical point above $$\delta$$ restricts to an elementary embedding $$j\upharpoonright V:V\to M$$, where $$M=\overline{M}\cap V$$.
This results is then applied to show that, above such a $$\delta$$, the universe $$\overline{V}$$ has no large cardinals not already in $$V$$; here ‘large’ can be weakly compact, ineffable, indescribable, strong, Woodin, supercompact, …; for strong compactness extra conditions are needed for the proof but the author conjectures that they are not needed for the result.
Reviewer: K. P. Hart (Delft)

MSC:
 3e+55 Large cardinals 3e+40 Other aspects of forcing and Boolean-valued models
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