FOM: Shelah talk (cardinal powers) JoeShipman at
Sun Aug 13 11:33:48 EDT 2000

In a message dated 8/13/00 10:44:45 AM Eastern Daylight Time, 
mfrank at 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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