[FOM] An argument for V = L

Timothy Y. Chow tchow at alum.mit.edu
Sun Sep 7 18:22:38 EDT 2014

On Sat, 6 Sep 2014, Joseph Shipman wrote:
> 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.

The qualitative argument is that V=L has a tolerable chance of conquering 
the ordinary mathematician's phobia of set theory, by presenting a clear 
picture of a relatively "structureless" universe of sets, and eliminating 
much of the incompleteness phenomena that seem to make people queasy. 
(In particular, the argument *isn't* that V=L is currently very useful for 
ordinary mathematics that ordinary mathematicians already care about.)

*Consistency* of large cardinals remains untouched, of course, as you 
point out.  However, if these end up intruding into ordinary mathematics, 
then it's likely going to be via some sort of concrete combinatorial 
statements.  I believe that it's easier for ordinary mathematicians to 
wrap their minds around combinatorial axioms, even if they're "provably 
unprovable."  They don't (appear to) directly threaten the philosophical 
picture that "sets are structureless."  I can imagine people learning to 
handle the consistency (or 1-consistency) of large cardinals in much the 
way that the Riemann Hypothesis and P != NP are handled today---they're 
assumed whenever necessary.

At least, that's my prediction.  It could be that V=L would make no dent 
in the ordinary mathematician's distaste for set theory and it might not 
eliminate "enough" incompleteness.  But it still strikes me as worth a 
try.  In contrast, getting bogged down in arguments about whether V=L is 
"true" seems precisely the wrong strategy, from an evangelical perspective 
at least.


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