[FOM] Platonists and large cardinals
Timothy Y. Chow
tchow at alum.mit.edu
Tue Jan 29 17:06:03 EST 2008
I was going to reply to Hirschorn's "Platonist argument," but Andreas
Blass has already said much of what I was planning to say. I have a
couple of comments to add.
One does not have to interpret the question "Does such-and-such a large
cardinal exist?" as a question of the form, "Do cardinals get that large?"
One can instead interpret it as asking, "Are there any cardinals that have
such-and-such a property?"
Some of the smaller large cardinal axioms do have the flavor of "climbing
up the cardinal ladder from below" (as evidenced by their names) and one
might reasonably take the position that if there really are arbitrarily
large cardinals, then you must eventually hit some with the stated
property. But measurable cardinals aren't defined that way, and their
largeness is a non-trivial a posteriori inference. It's not hard to
envision cardinals going on forever, but just none of them happening to be
measurable. Similarly, if 0^# doesn't exist, it doesn't mean that if you
look on the real line where it's supposed to be, you'll see a gaping hole
because 0^# went AWOL. It's just that no real number has the requisite
Having said that, I agree that the different large cardinal axioms do fit
together with each other very nicely, and that if one contemplates the
picture long enough, one can come to believe that it's "real" and that if
we march our way up then we'll hit each kind of cardinal in turn. But
denying this picture doesn't necessarily mean positing a "ceiling"; it
just means denying the assumption (for example) that "if there are big
enough cardinals, then one of them must be measurable."
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