[FOM] Davis/Friedman clarification

[Harvey Friedman]

In http://www.cs.nyu.edu/pipermail/fom/2016-July/019977.I wrote:

"At later, more advanced stages of the Concrete Incompleteness Program,
we would expect to have examples of mathematically interesting Pi01
sentences which are refuted in say 10 pages of mathematical
text,starting with the existence of large cardinal axioms, but in
order to be refuted in ZFC, would require at least 2^2^100 pages of
text. So the existence of counterexamples, in this interesting cases
expected to arise, is entirely useless in order to actually refute the
Pi01 statement."

What I meant was that in these expected perfectly natural examples,
the existence of the counterexamples is entirely useless in order to
actually refute the Pi01 statement in ZFC.

Since it appears that we are moving to a smooth transition from large
cardinals like SRP up to large cardinals like HUGE, we expect to have

"the existence of the counterexamples is entirely useless in order to
actually refute the Pi01 statement in HUGE".

This is much more powerful in the context of the current set theory
scene, since SRP is considered reliably consistent, or maybe certainly
consistent, whereas I know that some of, and even perhaps all of the
leading set theory players have shied away from endorsing HUGE as
consistent.

Furthermore, we can look forward to where HUGE gets replaced by any of
the intelligible hypotheses with or without choice (think V into V
over ZF and beyond).

Harvey Friedman