[FOM] RE: FOM New Axioms

[Ralph Hartley]

Matt Insall wrote:
> I happen to think that the
> intuition that gave us the axioms of ZF was, or at least can be,
> grounded in the following principle:
> 
> (**)  When possible, sets behave as do finite sets.  Otherwise,
>       proofs or new axioms are required to explain why some sets
>       stray from behaving as if they are finite.
> 
> Thus, when a statement is consistent with previously chosen axioms
> and has a ``relativization'' to finite sets that is provable from those
> axioms, we should choose that statement to hold for all sets,a s a new
> axiom.

Does this principle always give unique results?

I can't off hand think of a statement that is true/provable for all finite 
sets which, if extended to all sets, implies CH or ~Choice, but can you 
prove there isn't one?

How hard is it to find a pair of statements that are both true for all 
finite sets, but which can't both be true for all sets (including infinite 
ones)?

How many statements are there that are provable/true for all finite sets, 
but who's extension to all infinite sets is independent of the axioms?

The more answers to the last question there are, and the harder it is to 
find answers to the second to last, the more productive (**) would be.

Ralph Hartley