We show first that it is consistent that Κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of \(H(\kappa^+)\). Then with further forcing we show that it is consistent that GCH fails at \(\aleph_\omega\), \(\aleph_\omega\) strong limit, while there is a lightface definable wellorder of \(H(\aleph_{\omega+1})\) ("definable failure" of the singular cardinal hypothesis at \(\aleph_\omega\)). The large cardinal hypothesis used is the existence of a \(\kappa^{++}\)-strong cardinal, where \(\kappa\) is \(\kappa^{++}\)-strong if there is an embedding \(j:V \to M\) with critical point \(\kappa\) such that \(H(\kappa^{++})\) is included in \(M\) (this is almost optimal). The fine structure of the canonical inner model \(L[E]\) for a \(\kappa^{++}\)-strong cardinal is used throughout. This is joint work with Sy D. Friedman.
The Definable Failure of the Singular Cardinal Hypothesis (SCH)
10.03.2011 15:00 - 16:30
Organiser:
KGRC
Location:
SR 101, 2. St.,