[FOM] Refuting the CH? (and a brief for aleph_2)

[Tom Dunion]

On July 5, 2016, Harvey Friedman said

<< Recall

[1] H. Friedman,
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#89
<https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/#89>
.

$. For all f:R into R, there exist two (distinct) reals, neither being
f at an integral shift of the other.

I now want to look at

$[k]. For all f:R into R, there exist k (distinct) reals, none being f
at an integral shift of the sum of the others.

Thus $ = $[2]. $[1] makes sense, and is equivalent to: for all f:R
into R, there exist a real which is not 0. Hence $[1] is provable.

(I always follow the convention that letters like "k" denote positive
integers, unless indicated otherwise).

I was expecting that for k >= 2, $[k] is equivalent to |R| >= omega_k.
However,, my proofs were completely blocked, which I found
frustrating. Here is the reason.

THEOREM A. $[k] s provable in ZC + notCH...>>

I suggest there may be grounds to be cautiously optimistic, and here's why.

Beyond reasonable doubt, a resolution of the CH (in whatever form or
fashion) has been a (the ?) holy grail of set theory from the beginning.

A big part of the "grailness" is the long-standing irresolution of Cantor's
claim.  But if we suppose (for the sake of argument) as a bit of
counterfactual history, that between the world wars someone had advanced an
argument against CH which met with near unanimous assent -- much as Choice
eventually did, or like large cardinals today -- then the new grail likely
would have been in determining the value of 2^{aleph_0}, not simply
affirming or refuting CH.

The independence results of Godel/Cohen might then have been primarily
understood as telling us about the limitations of ZFC, not about the
inherent vagueness of the CH, or the nebulousness of the Reals.  Even
Lusin's claim in the 1920s, re the cardinality of co-analytic sets ("We
don't know, and we shall never know") might have been seen in a similar
light, that is, we need additional axioms beyond ZFC.

There are now many trails leading to aleph_2; Martin's Maximum, Proper
Forcing Axiom, Open Coloring Axiom(s), other work of J.T. Moore and
collaborators... There looks to be a rich and coherent theory with this
value for the continuum, respecting many earlier results and concepts,
whether strong Fubini theorems, cardinal characteristics, or random real
forcing.  Interestingly, there don't seem to be principles or rules of
thumb of the form "Here's why aleph_2 is likely ruled out" or "If you can
get aleph_2, a little tweak always seems to get you to aleph_k"

Let me close with a remark intended to be provocative (in a good, not
antagonistic sense) -- Eighteenth century scientist Joseph Priestly
isolated something he considered "de-phlogisticated" air*;* it took
Lavosier to recognize there was actually a newly isolated element,
oxygen!   I wonder if we are not looking at more and more evidence, at a
level most scientists or detectives would find compelling.
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