[FOM] Platonists and large cardinals
Andreas Blass
ablass at umich.edu
Mon Jan 28 09:46:24 EST 2008
I consider myself a Platonist, and I'd like to comment on James
Hirschorn's view of what any reasonable Platonist should believe. I
agree with him when he writes:
> I should
> think that any reasonable Platonist believes that the concept of
> infinity, as
> being beyond the finite, has a reality independent of any
> formalization. Then
> from Cantor they know that some infinities are larger than others.
I even agree with him about
> not putting an artificial "ceiling" on infinities,
though this strikes me as a separate decision, not a consequence of
the previous one. But I disagree with the idea that, as a result,
> I suspect
> they believe that large cardinal axioms are true provided that they
> are
> consistent (at least for the currently known LC axioms).
The reason for my disagreement is that many of the large cardinal
axioms (including measurability, the one Hirschorn specifically
mentions in the next sentence) seem to me to be asserting
combinatorial properties of the cardinals that do not follow from the
intuition of large size. These combinatorial properties imply
largeness, but there's more to them than just largeness, and I don't
see that this "more" is justified by my Platonist intuitions. Some
years ago, I tried to see (by introspection) which large cardinals my
intuition would support. I ended up being happy with all sorts of
indescribable cardinals but not with subtle cardinals. So I was well
short of the domain where I could use large cardinals to refute V=L.
To avoid misunderstanding here, let me add that my intuition leads
me to deny V=L, but not for large-cardinal reasons. Rather, I think
that V=L postulates an unjustified similarity (or even dependence)
between two fundamentally different sorts of richness in the set-
theoretic universe, namely the notion of "arbitrary subset" (which
governs the formation of each stage of the cumulative hierarchy) and
the notion of "arbitrary length" (which governs the height of the
hierarchy). V=L says that all of the former richness can be
extracted from the latter; once you have the ordinals, you can get
all sets by explicit definitions. I see no such connection between
the two.
To avoid yet another misunderstanding, I add the obvious remark that
none of the preceding is intended to be mathematics. It's just a
statement of my opinion.
Andreas Blass
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