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Large cardinal structures below \(\aleph _{\omega}\). (English) Zbl 0634.03050

This paper studies large cardinals in the absence of AC (the axiom of choice), and their iterrelationship with AD (the axiom of determinateness) by examining what large cardinal structures are possible below \(\aleph_{\omega}\). Generalizing some unpublished work of H. Woodin it is shown that if “ZF \(+\) AC \(+\kappa_ 1<\kappa_ 2\) are supercompact cardinals” is consistent then so is “ZF \(+\) DC \(+\) the club filter on \(\aleph_ 1\) is a normal measure \(+\aleph_ 1\) and \(\aleph_ 2\) both are supercompact cardinals”. Another result states that if “ZF \(+\) AD” is consistent then so is “ZF \(+\aleph_ 1\), \(\aleph_ 2\) and \(\aleph_ 3\) are measurable cardinals which carry normal measures \(+\mu_{\omega}\) is not a measure on any of these cardinals”. A modification of the proof shows that we can have that \(\mu_{\aleph_ 1}\) is a normal measure on \(\aleph_ 2\). The proofs of these results use Radin forcing, Levy collapse and other notions of forcing.
Reviewer: U.Felgner

MSC:

03E60 Determinacy principles
03E55 Large cardinals
03E40 Other aspects of forcing and Boolean-valued models
03E35 Consistency and independence results
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