[FOM] Prejudice against "unnatural" definitions
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".
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.
(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.
E-mail: spector at alum.mit.edu
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