[FOM] Prejudice against "unnatural" definitions
spector at alum.mit.edu
Tue Mar 5 13:48:16 EST 2013
joeshipman at aol.com wrote:
> Example: "there is no definable well-ordering of the reals" presumes that there are
> nonconstructible reals, but I have seen that statement dozens of times and it is hardly ever
> qualified in a way that makes it both precise and correct.
> Can anyone give other examples of this, or attempt to repair the statements I have cited so that
> they state actual nontrivial theorems?
I've always interpreted statements like the one above to mean that there is no definition that can
be proven in ZFC to be a well-ordering of the reals. More precisely:
(1) There is no formula phi with two free variables in the language of ZFC with the property that
ZFC proves that the binary relation defined by phi is a well-ordering of the set of real numbers.
(2) There is no formula phi with one free variable in the language of ZFC with the property that ZFC
proves "Every x satisfying phi(x) is a well-ordering of the reals and there is exactly one x such
Of course, these statements can't be proven in ZFC since they imply Con(ZFC). They are provable in
ZFC + Con(ZFC) using a syntactic approach to forcing. (I'm thinking the forcing argument here is
due to Feferman, but I didn't look up the reference to verify that.)
A stronger statement is true: If ZFC is consistent, then there is a model of ZFC in which there is
no definable well-ordering of the reals. But I don't think that's the meaning that one would ascribe
to the statement you wrote, Joe.
E-mail: spector at alum.mit.edu
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