[FOM] power of forcing/2

[Harvey Friedman]

THIS POSTING is to clarify my recent FOM posting "power of forcing"
where I made the following conjecture:

"CONJECTURE. The conjunctions that are consistent with ZFC are also
demonstrably treatable using forcing over an arbitrary countable model
of ZFC + V = L + "there exists a strongly inaccessible cardinal"."

I want to clarify this conjecture and point out some known results
that are in the direction of REFUTING this conjecture. So there is a
lot to clarify.

Recall that A_1,...,A_n is a listing of the statements in the set
theory literature such that each

A_i and not A_i are demonstrably forced

in the sense that, provably in ZFC, there is a notion of forcing
whereby, provably in ZFC, some condition forces A_i (not A_i) over any
ground model of ZFC + V = L + "there exists a strongly inaccessible
cardinal". We include the inaccessible cardinal because of certain
prominent set theoretic statements at that level (perfect sets in
uncountably analytic sets, projective sets Lebesgue measurable, for
instance).

This is to clarify that the conjunctions I was referring to in the
Conjecture, are the conjunctions of two or more of the A's and their
negations. NOT JUST length n conjunctions of the A's and either
negations.

I think that it is best to avoid class forcing issues here, and
consider only Boolean extensions - i.e., set forcing (where the
forcing conditions form a set). See below for some really relevant
information

CONTINUUM HYPOTHESIS AND SUSLIN'S HYPOTHESIS

CH (Cohen)
not CH (Cohen)
SH (Solovay Tennenbaum)
not SH.(Jech Tennenbaum).

CH + SH (Jensen)
CH + not SH (Jech Tennenbaum)
not CH + SH (Solovay Tennenbaum)
not CH + not SH (Jech Tennenbaum)

So it would seem that my Conjecture for conjunctions of length 2 is
already a challenge! How much more difficult is this for length 3?

GENERALIZED CONTINUUM HYPOTHESIS AND GENERALIZED SUSLIN'S HYPOTHESIS

If we only consider set forcing, then we can't include GCH or GSH in
the list A_1,...,A_n. However, if we allow class forcing, then
something interesting happens here.

It is not known if GCH + GSH is consistent. However, it has been shown
that "GCH + GSH implies the negation of the square principle at a
singular strong limit cardinal—in fact, at all singular cardinals and
all regular successor cardinals—it implies that the axiom of
determinacy holds in L(R) and is believed to imply the existence of an
inner model with a superstrong cardinal" according to
https://en.wikipedia.org/wiki/Suslin%27s_problem

So this situation suggests that certain combinations of "sidewise" set
theoretic statements may be strongly connected to "vertical" set
theoretic statements - which are not forceable.

WEAKER CONJECTURES

We have already alluded to working only with set forcing. Another
restriction is that the A's be required to be stated in third order
arithmetic.

Another approach, especially if it turns out that the Conjecture is
fundamentally wrong headed, is to flat out identify a few favorite set
theoretic statements, and work on the conjunctions of those statements
and their negations.

Harvey Friedman