FOM: Shelah talk (cardinal powers)
JoeShipman at aol.com
Sun Aug 13 11:33:48 EDT 2000
In a message dated 8/13/00 10:44:45 AM Eastern Daylight Time,
mfrank at math.uchicago.edu writes:
<< (aleph_omega) ^ (aleph_nought) < aleph_(aleph_4)) >>
Can you describe Shelah's proof, or did he not present one?
What I would like to see is a table summarizing everything that is known
about cardinal arithmetic (assuming AC). By Easton, powers of regular
cardinals can (simultaneously) be anything consistent with monotonicity and
cf(2^kappa)>kappa (which are the two properties of powers of regular
cardinals that can be proved outright). But with singular cardinals there
are a large number of diverse results, both outright proofs and consistency
Can anyone provide such a reference?
I'd also like to distinguish between proofs of (or proofs of consistency of)
individual equations or inequations like Shelah's above, and reults of the
form "If a set of conditions on cardinal powers holds, then an additional
condition holds", such as the result that if GCH holds below a supercompact
cardinal then it holds universally.
Where is the current state of knowledge on cardinal arithmetic summarized?
-- Joe Shipman
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