[FOM] An argument for V = L
Joseph Shipman
JoeShipman at aol.com
Sat Sep 6 19:18:23 EDT 2014
V=L doesn't even prevent "large large cardinal hypotheses" from intruding into ordinary mathematics, because it only entails that large large cardinals don't exist, not that they aren't consistent with ZFC. V=L is completely irrelevant to Harvey's recent work on concrete mathematical statements related to consistency of huge cardinals, as well as anything covered by the Shoenfield Absoluteness Theorem.
Of course, V=L could still theoretically make proofs easier while being metamathematically eliminable; but the only example I know of where this occurred is the Ax-Kochen theorem, which is 50 years old. Are there any other examples of any significance?
If you want a new axiom that will help with the kind of set-theoretically absolute statement that mathematicians can be persuaded to care about, it's hard to avoid large cardinals--RVM is the best example I can think of because it's not really about large sets at all (although PD is worth considering for the same reason, it's hard to motivate as something intuitively plausible).
If you still like V=L but want to extend it to something with consequences for statements that aren't too abstract, V=M is the obvious generalization which says that the anti-maximization principle "only sets which must exist do exist" is true not only "horizontally" but also "vertically". It has no new arithmetical consequences but at least it gives you new Absolute consequences (such as "any countable model of ZFC is nonstandard").
-- JS
Sent from my iPhone
> On Sep 5, 2014, at 1:44 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:
>
> Colin McLarty wrote:
>
>>> Why is this a "monkey wrench"? Why is it not just a reason to continue pursuing low complexity sentences of clear mathematical interest equivalent to consistency of various formal systems?
>
> Friedman of course can speak for himself, but I interpreted him to mean that the "ordinary mathematician" who hopes that a single axiom (V = L) will eliminate all set-theoretical difficulties is perhaps being too optimistic, because "small large cardinal" hypotheses may still rear their head.
>
> But I agree that this shouldn't stop said ordinary mathematician from adopting V = L. It will do nicely for now, and if some day Mahlo cardinals or whatnot intrude into ordinary mathematics, then we can cross that bridge when we get to it.
>
> Tim
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