[FOM] questions re Axiom of Choice

Paul Blain Levy p.b.levy at cs.bham.ac.uk
Thu Feb 21 19:59:23 EST 2019

Hi, I have two questions regarding Choice.

Question 1:

I once read somewhere on the Web about the following axiom schema, which 
could be called "Unrestricted Ordinal Dependent Choice".

For any predicate phi(s,x), which as usual can contain additional free 
variables, it says:

For any ordinal alpha,

if for any beta < alpha and beta-sequence s there's an x such that phi(s,x),

then there's an alpha-sequence t such that for all beta < alpha we have 
phi (s restricted to beta, s(beta))

In ZF this schema is equivalent to Choice, but in the absence of 
Foundation or the presence of urelements, it is stronger than Choice, 
and including it achieves conservativity of the analogue of NBG.

I believe there was a reference to a PhD thesis in Czech.  Does anyone 
know about this?  I might be misremembering it.

A specific statement that can be proved by this schema but not by Choice 
(in ZF without Foundation or with urelements) would be welcome!

Question 2:

Is the following statement provable in ZF?

For any set A of Scott cardinals, there's a set B of sets, such that A = 
{card(X) | X in B}.


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