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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:
03E55 Large cardinals
03E40 Other aspects of forcing and Boolean-valued models
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