FOM: New axioms/deepening our intuitions

Lee J. Stanley ljs4 at CS1.CC.Lehigh.EDU
Fri Dec 5 14:39:43 EST 1997

The following musings are prompted by the discussion of the
prospect or non-prospect of

finding new set theoretic principles which

 (a) come to command general acceptance,

 (b) settle CH.

A few questions here, for discussion.

(1) Omit (b).  Do the prospects improve?  If so, why?  My personal
view here is that I like to think that in the course of sharpening
our intuitions about sets

      (which are admittedly somewhat duller than our intuitions
       about natural numbers, but as a matter of degree, not
       as a matter of kind, so that in my view, they are both
       dismissed as fictions, or both admitted as some sort of

we will discover/formulate such principles.  But the view that we
will not is also interesting and makes ZFC an extremely compelling
and natural axiom system.

Further, there is a definite sociological character to (a).  Is it conceivable
that a principle which does settle CH for example (and of course such have
been found) might NEVER command the acceptance of some who have the conviction
that CH "should" go the other way?  Or that it is essentially undecidable,
and therefore any principle which settles it is suspect on that ground alone?

(2)  The following is relevant to the case where one answers "yes" to the
first question of (1) above.  I would just like to point out, though this
is clearly no mystery to anyone, that if we take CH in the form that implies
the wellorderability of the reals, then it can be rendered as a
\Sigma^2_1 statement:  the existence of a one-one function F from reals to reals
which code well orderings such distinct reals get F-values which code
non-isomorphic well-orderings.

Just one third order quantifier.  This represents some kind of essential
limit to human understanding?  Would it be hubris to say that that one
knows it doesn't?  Would it be hubris to say that one knows it does?
Would it be hubris that to suggest that even if it turns out that it does
(I would like to have a clearer idea of what sort of sociological
statement THAT turns out to be), such a turn of events might be sufficiently
important (a limitative result, after all) that we ought to try a bit harder
before giving up?

(3)  I want to argue against myself a bit here, and more specifically against
my parenthetical assertion in (1) above that there is a difference in degree
but not in kind in our intuitions about sets and our intuitions about
natural numbers.  Could one take the view that the absence of a standard
model (for sets) makes our intuitions one of kind after all?  But is this after
all just another way of saying there is a difference in degree?  We BELIEVE
we have a clear picture of a standard model in one instance, and we are
not prepared to make similar assertions in another?

(4)  Returning to new set theoretic principles, I admit that I have, at
present no good picture of what any might be.  But I remain convinced
that it is not only reasonable but IMPORTANT to continue to search
for them, regardless of the outcome and, at the level of gut feelings
(which I will refrain from attempting to erect into philosophy), my gut
feeling is that we will find some, provided that enough other people
feel that the search is worthwhile to allow research in set theory
to remain vigorous.  I would refer all to Steel's last post for the
details of what I mean here.

(5)  Returning to (2) again, I realize that one should not be overly
fixated on syntax, but I would be interested in hearing some ideas
about what, besides the way things out in ZFC, or your favorite system
of second order arithmetic, or whatever, makes the meaningfulness
of syntactically similar statements turn out so differently.  The point
being, that if there is no satisfactory answer here, then how do you know
before you look?  And when does it become ok to stop looking?

Best Wishes,

Lee Stanley

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