[FOM] Prejudice against "unnatural" definitions
Mitchell Spector
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.
or, equivalently,
(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
that phi(x)."
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.
Mitchell
--
Mitchell Spector
E-mail: spector at alum.mit.edu
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