[FOM] CH and mathematics32

Timothy Y. Chow tchow at alum.mit.edu
Mon Jan 28 10:22:54 EST 2008

Arnon Avron wrote:
> I am not arguing about that. The only point I wanted to make at the 
> above quoted paragraph is that it is intelectually dishonest to reject 
> one axiom (V=L) because it seems to be FALSE, but to argue for another 
> one (-CH) because there are reasons to view it as more productive than 
> its negation.

This reminds me of another issue that has bothered me, which is the status 
of the axiom of regularity, which I'll write as "V = WF."  How is it that 
V = WF has managed to get accepted so widely with so little fuss?  It is 
not obviously "true" and it restricts the universe, so shouldn't the usual
criticisms of V = L also apply to it?

The answer seems to be that V = WF does not exclude anything that most 
mathematicians care about.  In Kunen's set theory text he says that V = WF 
is "totally irrelevant" to mathematics (but adopts it anyway because it is 
convenient!).  But I suspect that if non-well-founded sets were to become 
of central interest to mathematicians then V = WF would come under fire.

In light of this observation, my feeling is that people who reject V = L 
reject it not because it is "false" (even if that's what they say), but 
because adopting it eliminates a lot of phenomena that they find very 
interesting.  (I suppose they could, in principle, develop the theory of 
such phenomena *within* ZFC + V=L, but that would be cumbersome.)

And getting back to ~CH, I think the argument from "productivity" is 
unlikely to carry the day by itself.  Instead, what has to happen is for 
some ("productive") axiom to (1) imply ~CH and (2) exclude nothing of 
"interest" if it is adopted.


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