[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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