[FOM] Prejudice against "unnatural" definitions
Mitchell Spector
spector at alum.mit.edu
Wed Mar 6 01:28:09 EST 2013
Joe,
If the question is the actual status of the statement "There is no definable well-ordering of the
reals", the answer is that it's independent of ZFC. (There's one proviso here, but I'll hold off on
that for a minute.)
But if the question is what statement this could reasonably be a circumlocution for if it is claimed
to be true (either naively or very informally), I think the answer is the one I gave -- that there
is no definition that can be proven in ZFC to be a well-ordering of the reals.
My understanding of what you meant was the second interpretation.
* * *
Here's the proviso I was thinking of.
You can't express "There is no definable well-ordering of the reals" in ZFC, so it's not immediately
clear what it means to say that it's independent of ZFC. But you can clarify it like this (assuming
ZFC is consistent, of course):
(1) For the fact that it is consistent with ZFC, use the theorem that ZFC is consistent with "There
is no ordinal-definable well-ordering of the reals" (this is due to Levy, I think).
Ordinal-definability is expressible in ZFC, and every definable set is ordinal-definable, so this is
sufficient.
(2) To see that its negation is consistent with ZFC, note that the usual well-ordering of the
constructible reals is definable (Gödel gave an explicit definition), and it's consistent with ZFC
for this to be a well-ordering of all the reals. (Since this uses a particular specific definition
of a well-ordering, you don't need the general notion of definability.)
Mitchell
--
Mitchell Spector
E-mail: spector at alum.mit.edu
Joe Shipman wrote:
> 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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