[FOM] Platonists and large cardinals

[Timothy Y. Chow]

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 
property.

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."

Tim