Re: Aleph One Sets

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 11/05/04 

Date: Fri, 05 Nov 2004 06:30:56 -0600

On 4 Nov 2004 23:13:22 -0800, greeneg@cs.unc.edu (George Greene)
wrote:

>"Tim Peters" <tim.one@comcast.net> wrote in message news:<EaOdnS2TK_7p2hXcRVn-pQ@comcast.com>...
>
>> Sorry, I missed what Jesus has to do with this <wink>.
>> Outraged posturing is unattractive all the same.
>
>And unmerited scorn is infuriating, even when it's coming from idiots.

That's true, although I can't see the relevance to anything anyone's
said here...

>> > So what can you possibly even MEAN by
>> > "CH might be true"??
>>
>> Mike can speak for himself, but I know what it means to me: that we may
>> eventually discover that the consequences of ~CH are so contrary to what we
>> want from set theory that models wherein ~CH is true become uninteresting
>> except to set theory specialists.
>
>It does NOT mean that to you. Jesus.
>Every theorem of ZFC will STILL be true in these models,
>so there is NO WAY that they can become "so contrary" to what we want
>from set theory.

Here's a hint. It's an issue that seems to come up fairly often when
you're explaining the TRUTH to some IDIOT:

Hint. The axioms of ZFC were not handed down by God. It does seem
that they're sufficient to do most mathematics, and as far as
anyone can see they may very well be consistent. But it doesn't
follow from that that the theory of ZFC is the truth and the
whole truth regarding set theory. It could happen some day
that everyone agrees that something or other about sets
is true even though it doesn't follow from ZFC.

Honest, this does seem to be the source of the problem in a lot
of situations where you're debating things with some IDIOT who's
too STUPID to see that you're right (especially, hint, when the
dumbass is someone who knows more than you do about what is
and what is not a theorem of ZFC or a model of ZFC, etc): You're
taking "theorems of ZFC" to be the one true definition of "set
theory", while the DUMBASS who's spouting obvious nonsense
simply doesn't realize that this is the one and only possible
TRUE definition.

> THE WHOLE BABY will still be there. That you don't
>like the temperature or color of the surrounding bathwater will just be
>irrelevant. Non-standard arithmetic is of interest to more people than
>just number-theorists. It turns out to be important to why calculus
>even works at all.
>
>> Or we may find that the consequences of
>> CH are unacceptable. Or neither, or both.
>
>In none of the above cases will it become defensible to allege
>that "CH is true".
>
>> Or we may find a compelling
>> axiom that implies CH, or ~CH.
>
>How can anything POSSIBLY be "compelling" if the whole of ZFC
>remains provable BOTH with it AND without it??

How can you possibly know that "follows from ZFC" is the
only way something can possibly be compelling? How do you
know that Z and F got it absolutely right, capturing
everything there is to know about sets in those axioms?

>Nobody is going to
>be "compelled". At worst people will simply be intrigued.
>Fascism is unattractive even coming from effete academics.

"Facism"? Calling rhetoric like this "ouraged posturing" is
a perfectly accurate description, whether or not you feel
it's unmerited scorn.

>> Everyone in the field, from Cantor to
>> Woodin, has speculated about whether CH "is true" in this sense.
>
>Sez you. Observers of reasonable discernment do NOT agree.
>
>> Most after
>> Cantor seemed to think "it's false" (including Godel and Cohen, who was the
>> first person to whom the independence of CH wasn't news <wink>).
>
>OF COURSE they think it's false IF THEY CAN CONSTRUCT A MODEL IN
>which it's OBSERVABLY false!
>
>> As Mike said, Woodin's work is mentioned most often today in connection with
>> CH, and is often interpreted as casting strong doubt on "let's assume CH"
>> remaining a mainstream choice.
>
>Yet in the face of this, you STILL say we might FIND that CH
>"is true"?? That we might find a "compelling" reason to dismiss
>Woodin as irrelevant?? Even if fashions change, everything that
>Woodin did to "cast strong doubt on" CH will STILL be interesting
>in ITS own right!
>
>
>> Woodin himself is pretty non-committal; at
>> the end of "The Continuum Hypothesis, Part II"
>> (Notices of the AMS, Vol 48,
>> No 7) he says:
>>
>> So, is the Continuum Hypothesis solvable?
>
>This is what is known as A STUPID QUESTION.

Making _Woodin_ an IDIOT.

Do you _really_ think that there's anything about ZFC
that you understand that he doesn't? Do you _really_
not think it's even _possible_ that the point of
view of a guy like that could be reasonable, even
though it's not the same as yours?

I take back what I said. The accurate description
would be "arrogent outraged posturing".

>IT'S CONSISTENT EITHER WAY.
>There CANNOT BE ANYthing further to be SOLVED!
>ALL that CAN happen is that people can just DECREE
>that certain kinds of models are more interesting or
>SUBJECTIVELY Attractive than others!

The attitude you seem to take, that whether or not something
is true in every model of ZFC is the only way we are to
judge whether it's "true", is not just a DECREE? I imagine
not. How do you justify this non-decree?

>This decree is
>NOT going to stop the unattractive models from existing!
>It well MAY not even suffice to deter them from harassing you!
>
>> Perhaps I am not
>> completely confident the "solution" I have sketched is the
>> solution, but it is for me convincing evidence that there _is_
>> a solution.
>
>HOw exactly is one supposed to have convincing evidence that there
>is a solution, without having any idea which (given that there are only
>2 horns on this dilemma) way the coin flips?
>
>> Thus, I now believe the Continuum Hypothesis is
>> solvable,
>
>That is, I repeat, simply incoherent.

Or _poossibly_ you're simply misunderstanding what he's saying?

(Hint: yes, you're misunderstanding what Woodin is saying.
Because you're taking ZFC as the one true definition of set
theory, while, as one would think was obvious, he's not.)

>> which is a fundamental change in my view of set theory.
>
>Well, maybe if you were quoting a simple and straightforward presentation
>of "his view of set theory", as opposed to fallaciously appealing to authority,
>I might be more open-minded.

He _cited_ the article for you. They have a library where you are?

>> While most would agree that a clear resolution of the Continuum
>> Hypothesis would be a remarkable event,
>
>Well, call me contrarian, but this kind of thing cannot possibly be
>an "event". At best it will simply be an edict, unless the independence
>proofs are mistaken.
>
>> it seems relatively few
>> believe that such a resolution will ever happen.
>
>I wouldn't go that far. You just have to find some wholly new
>and impressive property of models that makes you want to deprecate
>all the models that lack it -- and that happens to decide the issue.
>THe problem is not that such a discovery is so distant and far off
>and heavenly. It is rather that we can't imagine ourselves being
>stupid enough to discard half the models. We're just not that bigoted
>any more.
>
>> Of course, for the dedicated skeptic there is always the "widget
>> possibility". This is the future where it is discovered that
>> instead of sets we should be studying widgets. Further, it is
>> realized that the axioms for widgets are obvious and, moreover,
>> that these axioms resolve the Continuum Hypothesis (and everything
>> else). For the eternal skeptic, these widgets are the integers (and
>> the Continuum Hypothesis is resolved as being meaningless).
>
>If you stick to first-order theories then there is ALWAYS a model
>where everything in the domain is an integer ANYhow, so claiming that sets and
>widgets are even DIFFERENT to begin with is kind of stupid. And they
>do NOT make CH "meaningless".
>
>> ...
>>
>> The universe of sets is a large place.
>
>There are no large places in first-order logic.
>Every theory has a countable model.
>
>> We have just barely begun
>> to understand it.
>
>Which ought very much to DISentitle us from pontificating!
>
>> > Even AFTER you think you have stumbled upon whatever
>> > allegedly conclusive evidence might "force" you to conclude
>> > that CH is true, The models in which it is false are NOT going
>> > to STOP existing!
>>
>> Models of ZF in which the axiom of choice is false didn't stop existing
>> either, although most working mathematicians are happy to ignore them
>> completely
>
>That is just plain ridiculous.
>Most working mathematicians are happy to ignore
>+AC as well. When they use it it is often unconsciously.
>Everybody who knows enough to care knows enough to care about
>the other models.
>
>> -- AC is too useful to give up,
>
>That doesn't mean that people don't ever investigate the models
>where it's false. THe people who care AT ALL are in fact much MORE
>likely to be dealing with those models, even if only exceptionally
>and occasionally.
>
>> and its odd consequences (like
>> Banach-Tarski, and that all sets can be well-ordered) aren't disturbing
>> enough to dissuade most workers from embracing ZFC wholeheartedly. If you
>> ask me what I mean by sets, then AC is "obviously true", and models of ZF in
>> which AC is false simply aren't models of what I mean by sets.
>
>That's overstating the case.
>All the things in all those models are all sets, even in your sense.
>The problem is that those models are leaving out some things
>(i.e. some choice-sets) that you would've also put in. You're
>deprecating models of ZF -AC as under-populated compared to models
>of ZF +AC. Even if THAT *is* "true", the models of ZF -AC are STILL
>populated with SETS and NOT with other things.
>
>> It remains possible that CH will meet a similar fate.
>
>Hardly. It's entirely too late. ZF plus AC minus CH has ALREADY
>been deemed to be a set theory. Everybody ALREADY concedes
>that since all its models are models of ZFC, the things in them
>are sets. They're GRANDFATHERED. They would have to be
>FORCIBLY excommunicated before what you are describing here
>could happen. That would entail people admitting that they
>were WRONG BEFORE about thinking these things were sets.
>Not bloody likely.
>
>> > No, if it's true, it will HAVE to be because you assumed it.
>> > It will HAVE to be because you assumed that, among the MANY
>> > available models of ZFC, one of the ones in which it
>> > HAPPENS to be true is THE "right" one.
>>
>> It would be enough that ~CH turns out to have consequences few will accept,
>> so that CH must be true in all models most will accept. Or vice versa. You
>> won't get a categorical model of ZFC by adding CH or ~CH either, so "THE
>> 'right' one" wouldn't necessarily apply.
>
>That completely misses the point; you will never get a categorical
>model by adding ANYthing to ANY 1st-order theory (that's consistent
>and halfway powerful). The categorical theory (with its unique, up
>to isomorphism, model) for PA is at *2nd*-order.
>Arguably, you DO get a single model for ZF by making that ascension,
>but that is almost vacuously conventional.

************************

David C. Ullrich

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