Fwd: Re: FOM: Inconsistency

[jshipman@bloomberg.net]

Forwarded by Joe Shipman from Lou van den Dries to FOM
---- Original Msg from: Lou van den Dries  <vddries at math.uiuc.edu> At: 11/19 17:
careful in what I said. In fact, it seems to me rather unlikely that


Well, in answer Shipman's questions, I realize I should have been more
careful in what I said. In fact, it seems to me rather unlikely that
ZFC is inconsisitent, and I don't really loose sleep over the possibility.
And objects like the power set of N (the set of reals, in other words)
feel okay, in the sense that I have quite a clear idea what an arbitrary
set of natural numbers looks like. But when you take the power set
of the set of reals, I already feel some discomfort. But somehow
I feel that all serious math can already be done below that level.
(and am happy to leave the details of this to others more interested
in such issues than I am). In my own work, I do not explicitly keep

track of how many iterations of the power set operation I am using.
(But I do strongly feel that this iteration is more a matter of
convenience than absolute need. I realize that actually showing this
to be so is a very time consuming (and for me unrewarding, compared
to other activities) business.) 

Anyway, to me it seems an inconsistency in ZFC is, first of all,
unlikely but conceivable, and secondly, if it happens, it will be
something easily repaired, and not affect anything "serious". 

By the way, the fact that taking the power set of the reals feels
uncomfortable to me is that it's hard to form a clear idea of a
completely arbitrary set of reals. It's taking set theory a bit
too seriously for my taste. (Thus questions like CH seem to have

nothing to do with the way I look at the continuum: what's important
about the continuum is that it supports the usual mathematical
objexts that naturally live on it, like the usual algebraic and
transcendental operations, manifolds, Lie groups, (separable)
Hilbertspace, even Borel and analytic sets, and measures, and
what have you.) -Lou van den Dries-