[FOM] V = L/crises

[Timothy Y. Chow]

Joe Shipman wrote:

> I disagree that ordinary mathematicians would regard V=L as a good 
> axiom, because it cannot even be STATED without bringing in much more 
> set theory than most mathematicians are comfortable with.

I agree that the precise statement of V = L involves more set theory than 
most mathematicians are comfortable with, but I don't think that this is 
necessarily a deal-breaker in this context.  Mathematicians are sometimes 
willing to accept an axiom that they don't understand in detail if they 
understand how it functions.  For example, most mathematicians couldn't 
state Replacement off the top of their heads but they accept it as part of 
the ZFC "package" since they understand that some collection of standard 
axioms is necessary.

Similarly, I could see V = L taking on the role of "that axiom that gets 
rid of almost all set-theoretic difficulties, should they arise."  If a 
mathematician finds the need arising, but is not comfortable with L, then 
he or she will just outsource the labor of figuring out what happens in L 
to the professional set theorist.

In fact, I've always been mildly surprised that this hasn't actually 
happened in the real world.  As far as I can tell, what seems to have 
happened is that the professional set theorists don't *believe* in V = L, 
and perhaps to some extent don't like V = L because it is boring.  So they 
don't want to include it in the ZFC package that they offer to the 
mathematical community.  I suspect, though, that the mathematical 
community might be O.K. with V = L even if they didn't *believe* it, if 
they found themselves bumping up against set-theoretical difficulties a 
lot and if V = L promised to provide a "standard" way of dealing with them 
without having to think too much.

Tim