[FOM] Re: Choice of new axioms 1

Dmytro Taranovsky dmytro at MIT.EDU
Tue Feb 14 15:30:42 EST 2006

Harvey Friedman wrote, 
> > *  Many general Sigma-2 statements, including a number of partition
> > properties, are undecidable in ZF + V=L.  They are generally resolved
> > with the help of zero sharp.
> 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.

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.

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 {{}}.

Dmytro Taranovsky

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