FOM: Large Cardinals Axioms

[JoeShipman@aol.com]

Avron:
>>I believe that the comparison with AC here is inappropriate. The big difference is that AC has implicitly been used by mathematicians (at least in a weak form, like DC) long before it was explicitly formulated by Zermelo and Russel... In contrast, I dont believe that any "core" (or other) mathematician has ever implicitly used  large cardinal axioms in her/his
work, and I dont believe one ever will (except perhaps set-theorists actually working in this particular area).<<

This is an important point.  On the other hand, AC was never even used IMPLICITLY before the 19th century; the development of mathematics led to a situation where the intuition of mathematicians changed and many of them began to think in ways which implicitly involved AC.  This may happen for large cardinal axioms too, especially if they are reformulated along more philosophical lines (as Friedman has suggested).  Because of the foundational self-awareness the mathematical community now has, it is less likely that mathematicians will simply use large cardinal principles [I mean "large cardinal principles" to include consistency statements as well as the ontologically committal large cardinal axioms themselves] without realizing it, as they once used AC unwittingly.  However, I find it plausible that they may increasingly begin to discern places where large cardinal principles are relevant and that this will further influence the development of their intuition that these princi!
ples are "true".

Arnon:
>>Let me add here that personally I cannot be persuaded by Friedman's arguments(and impressive achievements) because of my (old-fashioned?) criterion of accepting a statement as an AXIOM only if it is obviously, self-evidently TRUE.<<

The distinction between "assumption" and "axiom" (an assumption you also believe to be true) is valuable.  I think it ought to be enough if a (necessarily informal) case can be made that the axiom is true without it being "obviously, self-evidently" true.  If you can be persuaded somehow that an assumption is very likely true, reasoning from that assumption to reach other very likely true conclusions is still a fundamental enough mathematical activity that the assumption deserves the status of "axiom".  The consistency with ZFC of an inaccessible cardinal seems to me to be almost certainly true for a variety of reasons; but these reasons do not seem strong enough to me that I would call Con(ZFC+Inacc) "obvious" or "self-evident".  Maybe someday these reasons, or other ones, will seem strong enough that I would regard Con(ZFC+Inac) as self-evident, but the arguments for its truth are strong enough that it is fair to call it an "axiom" rather than merely an "assumption" or "hypo!
thesis".  


Avron:
>>Worse: I dont see what can possibly convince
me that large cardinal statements are meaningful - let alone true (I am not saying that they are not meaningful - I just dont know whether they are or not, and I cannot imagine what can settle this problem for me). What I can at most accept, on empirical ground, is that many of them are most probably consistent, and so what can be proved using them most probably cannot be refuted. But for this what is actually used is the consistency of these axioms, not the axioms themselves.<<

Agreed.  Both Friedman and Steel have argued (on differing grounds) that if you accept and use the consistency of a large cardinal axiom you might as well use the large cardinal axiom itself, but if one is concerned with Truth this attitude is problematic.

Illustration: I have a strong intuition that weakly inaccessible cardinals "exist" and a MUCH weaker sense that strongly inaccessible cardinals do.  But consistency-wise there is no difference.   As above, I would be willing to accept Con(Inacc) an "axiom", but not (Inacc) itself.  I would use (Inacc) as an eliminable "ideal" assumption, but if I could prove something ONLY from (Inacc) and not from Con(Inacc) [or other such principles like 1-Con(Inacc)] I would not have a strong sense that it was "true".

The existence (not simply consistency) of WEAK inaccessibles, on the other hand, is almost acceptable as an axiom for me because weak inaccessibles follow from the existence of a real-valued measure on the continuum, a statement I find intuitively highly plausible though not quite self-evident.

-- Joseph Shipman
(Affiliation: Software Developer, Savera Systems Inc.; Background: Math Ph.D. from Brandeis [Set Theory])