[FOM] Prejudice against "unnatural" definitions
Joe Shipman
JoeShipman at aol.com
Tue Mar 5 16:44:55 EST 2013
Mitchell, I understand that your interpretation is defensible if Con(ZF) is assumed, but my point is that the mathematicians I was quoting may not have had such a nuanced understanding. The problem with your interpretation is that the statement that I quoted could be literally false even though your interpretation is true. Define the following relation *<* on real numbers x and y:
******
If x and y are both in L, then x*<*y iff x is constructed before y in the well-ordering of L defined by Godel
If x is in L and y is not, then x*<* y and not y*<*x
Otherwise, x*<*y iff x<y in the usual ordering of the real numbers.
******
This is a definable relation on the reals. It is provably a definable total order. If and only if there are no nonconstructible reals, then it is a definable well-ordering. The only thing that is not provable is whether this provably definable ordering is "well-", but to assert the statement in the words I quoted
"there is no definable well-ordering of the reals"
is no more justified than to assert unqualifiedly
"there exist nonconstructible real numbers".
-- JS
Sent from my iPhone
On Mar 5, 2013, at 1:48 PM, Mitchell Spector <spector at alum.mit.edu> wrote:
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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