[FOM] Choice of new axioms 1

[Harvey Friedman]

On 2/14/06 3:30 PM, "Dmytro Taranovsky" <dmytro at MIT.EDU> wrote:

Friedman wrote:

>> What does Sigma-2 mean, and what examples are you talking about? Also which
>> of them are of concern for the working mathematician?
> 
> By general Sigma-2 statements, I mean statements like those claiming
> existence of uncountable, inaccessible, Mahlo, weakly compact, subtle,
> ineffable, omega-Erdos, and other cardinals.

So Sigma-2 is in the language of set theory. Thanks for the clarification.
Of course, none of these are part of normal mathematics, let alone core
mathematics. They won't play a role in the choice of new axioms by the
mathematical community under these circumstances.
> 
> Some of these statements are intuitively true.  An example (existence of
> an omega-Erdos cardinal) is that if kappa is a sufficiently large
> ordinal, then every predicate P on finite subsets of kappa has an
> infinite homogeneous set S, in the sense that whether P holds on a
> finite subset of S depends only on the number of elements.

This is not intuitively true and also is not intuitively false. I would
venture that 99% of people declaring it to be intuitively true would be set
theory scholars. So the burden of proof is on you to explain why this is
intuitively true. This would require a major breakthrough in the foundations
of set theory.
 
> Another example (existence of a subtle cardinal) is that every
> sufficiently large (having at least a certain number of elements)
> transitive set has elements x and y such that x is a proper subset of y
> and x is neither the empty set {} nor {{}}.
> 
Same response.

My observation is that set theorists have generally stopped trying to
justify the truth of even relatively small large cardinals such as these in
favor of justifying larger cardinals in terms of 1) utility, and 2) inner
model structure. This is quite different than declarations that they are
intuitively true.

Harvey Friedman