[FOM] Fwd: invitation to comment

[Timothy Y. Chow]

Voevodsky wrote:

> To put it very shortly I think that in-consistency of Peano arithmetic 
> as well as in-consistency of ZFC are open and very interesting problems 
> in mathematics. Consistency on the other hand is not an interesting 
> problem since it has been shown by Goedel to be impossible to proof.

I'd summarize this point of view as, "omega is sort of like an 
inaccessible cardinal."

I don't think anyone on FOM would wince if someone were to say that they 
felt that it was an interesting question whether (for example) measurable 
cardinals are inconsistent with ZFC, because even though Goedel has shown 
that we can't expect a "traditional" proof of their consistency, we can't 
rule out a proof of their inconsistency.  Scale this comment down to omega 
and we recover more or less what Voevodsky says here.

Of course, to be philosophically consistent, one should then back off from 
assuming omega in the rest of mathematics, but perhaps Voevodsky just 
hasn't gotten around to that yet.  It's perhaps slightly ironic that 
Voevodsky of all people would find finitism attractive, given that 
(according to Nath Rao) some of Voevodsky's work isn't even obviously 
formalizable in ZFC---

http://groups.google.com/group/sci.math.research/msg/a9279fa4411aaab3

---but I guess this is no more ironic than Brouwer proving his fixed point 
theorem and then later arguing hotly for intuitionism.

Tim