[FOM] DC vs. CC

Mitchell Spector spector at alum.mit.edu
Sat Aug 25 12:41:19 EDT 2012

Hi, Bob,

There's a proof that the Axiom of Countable Choice does not imply the Axiom of Dependent Choices in 
Jech's book The Axiom of Choice (Dover Publications, 1973).  See Theorem 8.12.

Jech credits this to a 1967 paper of Jensen (Consistency Models for ZF, Notices Am. Math. Soc. 14, 
137), but I haven't looked up Jensen's article.

Here's a related question to which I'd be interested in learning the answer. What is the consistency 
strength of:

(*) ZF + Countable Choice + "there exists a measurable cardinal kappa with a kappa-complete 
ultrafilter U such that the ultrapower kappa^kappa/U is not well-ordered in the usual ordering".

The Axiom of Dependent Choices fails in such a model, so a model of (*) is an example of what you 
were looking for (although requiring large cardinals, unlike Jensen's result).

In my Ph.D. thesis, I proved the consistency of (*) relative to the consistency of the infinite 
exponent partition relation kappa -> (kappa)^kappa (and countable choice).

However, I would imagine that (*) is weaker than that partition relation, and perhaps (*) is 
equiconsistent with ZFC + "there exists a measurable cardinal".  Or maybe not -- perhaps it implies 
the consistency of measurable cardinals of high order?

Does anybody know?


Mitchell Spector

Robert Lubarsky wrote:
> Does anyone have a reference for why Countable Choice does not imply Dependent Choice? I saw the
> construction of a model of CC + not DC once in an article in I believe an LNM volume, but have not
> been able to find it again. Are there any realizability models of such around?
> Bob Lubarsky
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