[FOM] RE: New Pi01 Statements

[Dmytro Taranovsky]

This message offers some comments on known true arithmetical statements not
provable in ZFC.

Known examples of true natural arithmetical statements independent of second
order arithmetic tend to have consistency strength of finite order mahlo or
(less often) finite order subtle cardinals.  As such (and since the
arithmetical statements that are not Pi-0-1 are still provable from the
relevant large cardinal axioms), their truthfulness has been established beyond
a reasonable doubt.  However, for the purposes of reverse mathematics (finding
out which set theoretical principles are needed to obtain particular results)
and because, arguably, consistency of n-mahlo cardinals has not been
established with absolute certainty, the statements should continue to be
treated as propositions provable from such and such assumptions, rather than as
theorems.

The only known (at least, to me) natural arithmetical statements that have
consistency strength above that of zero sharp involve definability in systems
with elementary embeddings; they imply consistency of supercompact cardinals.

Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm