[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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