[FOM] Prejudice against "unnatural" definitions
Mitchell Spector
spector at alum.mit.edu
Wed Mar 6 12:48:25 EST 2013
Joe Shipman wrote:
> I disagree; I think that "naively" it could mean exactly what it says, and that the
> mathematicians who stated it, who were not logicians, didn't understand the distinctions and
> misinterpreted or misremembered the actual result they had read about.
Well, I can't argue with that, Joe. I was trying to come up with a charitable interpretation, but I
do agree that this is really not the same as the circumlocutions people often use in talking about
definability and such things.
Mitchell
--
Mitchell Spector
E-mail: spector at alum.mit.edu
>
> -- JS
>
> Sent from my iPhone
>
> On Mar 6, 2013, at 1:28 AM, Mitchell Spector <spector at alum.mit.edu> wrote:
>
> 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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