[FOM] Solving set theoretic problems?

Harvey Friedman hmflogic at gmail.com
Tue May 10 11:12:59 EDT 2016

In my recent FOM posting, Refuting the Continuum Hypothesis?/3, I proposed

EXISTENTIAL TRUTH. ET. If it is consistent with ZC that a given simple
existential property holds of all f:R into R, then the given simple
existential property holds of all f:R into R.

EXISTENTIAL TRUTH PROGRAM. ETP. Show that ET is consistent with ZFC when
stated for existential properties in more and more expressive
languages.

ET quickly leads to a "refutation" of the continuum hypothesis.

DIGRESION. There is another principle that we have been investigating.
See http://www.cs.nyu.edu/pipermail/fom/2016-April/019775.html

BOREL TRANSFER. BT. If a given simple existential property holds of
all Borel f:R into R, then it holds of all projective f:R into R.

BOREL TRANSFER PROGRAM. BTP. Show that BT is provable from large cardinals,
when stated for existential properties in more and more expressive
languages.

BT quickly leads to a "proof" of PD.
http://www.cs.nyu.edu/pipermail/fom/2016-April/019775.html

Obviously, we also have this formulation:

BOREL TRANSFER PROGRAM. BTP. Show that BT is provable from PD,
when stated for existential properties in more and more expressive
languages.

*****************************

We can view the EXISTENTIAL TRUTH PROGRAM as a special case of a yet
wider program that needs to be explored.

SIMPLE POSITIVE TRUTH. SPT. Any "simple positive" statement in a
fundamental environment that is consistent with ZC is true.

SIMPLE POSITIVE TRUTH PROGRAM. SPTP. Explore CT in various fundamental
environments.

Now obviously we expect that if we pick different fundamental
environments, then SPT for those environments are going to be
incompatible with each other.

So it is not clear that SPT can be used to decide set theoretic
problems without making a case that one or another fundamental
environment is a preferred fundamental environment, and that we have
an appropriate notion of "simple positive" statement. .

NEVERTHELESS, it is clear that it is important to develop SPTP for
various fundamental environments. ETP is our original case of SPTP.

It also seems reasonable that SPT can be used in an interesting way to
"justify" all of the standard FOM systems such as PA, Z_2, Z, RTT, ZF,
also with the axiom of choice, and various large cardinal axioms.

ALSO NOTE that the experience with ETP in Refuting the Continuum
Hypothesis?/3 (and in
http://www.cs.nyu.edu/pipermail/fom/2016-April/019775.html) definitely
indicate the likely essential use of COMPUTER TECHNOLOGY to carry out
these programs. So we vividly see the role of computer technology in
fleshing out a new approach to set theoretic problems and set
theoretic foundations.

Harvey Friedman