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From: Jesse F. Hughes on 14 Jan 2010 09:36
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

> On Jan 12, 10:36 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> > This is wrong, I think. The proof only shows that there is no
>> > surjection within ZFC; I don't see why we have to believe that the
>> > standard model includes all possible mappings, given that non-standard
>> > models don't.
>>
>> The so-called standard model is supposed to include all the sets that
>> "really" exist.
>
> Is this provable? If not, how is it not just blind faith?

That's what the standard model *is*. It's the universe of all sets.
(Well, there may be minor issues regarding the definition of model
here, since what I call "the standard model" is class-sized, but let
us ignore these technicalities.)

>> If so, then it includes all functions which "really"
>> exist and hence there is no surjection from the "real" N onto the
>> "real" R.
>
>> > The reason we know that N < R is the informal or non-
>> > firstorderisable argument; a formal argument within set theory can
>> > only be less certain.
>>
>> I don't know what you mean by "the non-firstorderisable argument"
>> (though I'm mighty fond of the adjective).
>
> I didn't invent it. Anyway, what I mean is that the 'real' N and R
> can't be defined by first-order logic (an immediate consequence of
> LS), so the informal proof (e.g. Cantor's argument) that N < R can't
> be, either. What I mean is that seeing it formalised, or knowing it
> can be formalised, in ZFC does not make one any more certain that R is
> not countable; on the other hand, it is open to objections (such as
> mine, above) while the informal argument is not.

I don't know the details, but it seems to me that if there is a
universe of sets, then it is a model of ZFC and in that model of ZFC,
the interpretation of N is the "real" set of natural numbers (and R
the "real" set of reals). The interpretation of N and R in ZFC
involves quantification over all the objects in its model and hence
over all sets.

But I'll defer to someone who knows more about ZFC than I do for this
argument.

> So what is set theory really good for, anyway?

If you're not interested in studying the use and limits of formal
arguments, not so much.

--
"Britney thought the idea of a pre-nup was vile, because she is
loved-up with Kevin and cannot envisage breaking up. However, [...] no
one in Hollywood these days get married without brokering a
deal. [...] She had a long chat with Kevin and he was cool about it."
From: David C. Ullrich on 15 Jan 2010 08:28
On Fri, 15 Jan 2010 05:20:16 -0800 (PST), Andrew Usher
<k_over_hbarc(a)yahoo.com> wrote:

>On Jan 14, 8:41�am, "H. J. Sander Bruggink" <brugg...(a)uni-due.de>
>wrote:
>
>> You have to start somewhere, yes. You have to make some basic
>> assumptions about the "real" sets. These basic assumptions are called
>> axioms. The whole point about using a formal system is that it is
>> completely clear what the axioms are. If the axioms of ZFC are actually
>> true of sets, then the model of "real" sets is indeed a model of ZFC.
>
>This is circular. Defining the 'real sets' in terms of ZFC makes no
>sense because, clearly, the 'real sets' do exist independently of any
>formal theory we may write.

How in the world can you read the previous paragraph as a supposed
_definition_ of the "real sets"?

>> > I didn't invent it. Anyway, what I mean is that the 'real' N and R
>> > can't be defined by first-order logic (an immediate consequence of
>> > LS), so the informal proof (e.g. Cantor's argument) that N< �R can't
>> > be, either. What I mean is that seeing it formalised, or knowing it
>> > can be formalised, in ZFC does not make one any more certain that R is
>> > not countable; on the other hand, it is open to objections (such as
>> > mine, above) while the informal argument is not.
>>
>> The informal argument needs axioms also. It just doesn't spell them out.
>> So you can still ask: is it provable that the informal argument is about
>> "real sets"?
>
>Those axioms, though, are simply definition. We all agree, even when
>not defining them formally, that N is the unique structure satisfying
>Peano's axioms (or equivalent) and that R is the unique complete
>Archimedean ordered field. Those definitions are not provable because
>they are definition. But the formalisation in set theory requires
>those _and_ that ZFC correctly models them - a strictly stronger
>assumption.
>
>Andrew Usher

From: Jesse F. Hughes on 15 Jan 2010 11:53
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

> On Jan 14, 8:36 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> >> The so-called standard model is supposed to include all the sets that
>> >> "really" exist.
>>
>> > Is this provable? If not, how is it not just blind faith?
>>
>> That's what the standard model *is*.  It's the universe of all sets.
>> (Well, there may be minor issues regarding the definition of model
>> here, since what I call "the standard model" is class-sized, but let
>> us ignore these technicalities.)
>
> So, in other words, it's just the largest possible model of ZFC?

I don't know whether there are larger possible models of ZFC. My only
(informal) claim is that the universe of all sets is a model of ZFC.

>> I don't know the details, but it seems to me that if there is a
>> universe of sets, then it is a model of ZFC and in that model of ZFC,
>> the interpretation of N is the "real" set of natural numbers (and R
>> the "real" set of reals).  The interpretation of N and R in ZFC
>> involves quantification over all the objects in its model and hence
>> over all sets.
>
> Granting that there is a model of ZFC with the 'real' N and R, though,
> how do we know that that same model also includes all possible
> functions from N to R? And of course we can't 'quantify over all
> sets'.

We are not merely granting that some model has sets (isomorphic to)
the real N and R, but rather that

(1) The universe of all sets contains sets (isomorphic to?) the real N
and R and

(2) The universe of all sets is a model of ZFC.

Now, any function from N to R is a subset of N x R and hence an
element of the universe of all sets. Thus, this same model has all
possible functions N -> R.

(Again, we are ignoring the technical difficulty that models are
usually defined as having a carrier set, while here we are speaking of
a class-sized carrier. In any case, the set of all functions N -> R
exists in certain set-sized models of ZFC, if I'm not mistaken.)

As far as your last remark (we can't 'quantify over all sets'), I'm
mighty confused. ZFC does just that. Perhaps you have never seen the
axioms of ZFC?

--
"Now I'm informing all of you that the people arguing against me are EVIL,
yes they are real, live EVIL people as mathematics is that important, so
it's important enough for Evil itself to send minions like them."
-- James Harris on Evil's interest in Algebraic Number Theory
From: Jesse F. Hughes on 15 Jan 2010 13:45
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

> On Jan 15, 10:53 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> > So, in other words, it's just the largest possible model of ZFC?
>>
>> I don't know whether there are larger possible models of ZFC. My only
>> (informal) claim is that the universe of all sets is a model of ZFC.
>
> And I suppose such a larger model, which you claim might exist,
> includes sets not in the universe of all sets??

I made no claim that a larger model exists. Of course, if a larger
model exists, then it would include objects which are not sets (in the
same way that non-standard models of PA include objects which are not
natural numbers).

"I don't know whether X is true," does not mean "X might be true".

>> > Granting that there is a model of ZFC with the 'real' N and R, though,
>> > how do we know that that same model also includes all possible
>> > functions from N to R? And of course we can't 'quantify over all
>> > sets'.
>>
>> We are not merely granting that some model has sets (isomorphic to)
>> the real N and R, but rather that
>>
>> (1) The universe of all sets contains sets (isomorphic to?) the real N
>> and R and
>>
>> (2) The universe of all sets is a model of ZFC.
>
> These axioms assume that we have some definition of 'the universe of
> all sets' independent of ZFC.

I don't really want to make any claims regarding Platonism/Realism.
My comments on the universe of all sets is really just a matter of
explaining my argument in your terms.

For my own part, what I know about sets comes from the axioms of ZFC.
One consequence of those axioms is that, in any model of ZFC, there is
no surjection N -> R. That is all I mean when I say that R is
uncountable.

Admittedly, some models of ZFC are countable. That's pretty odd, no
doubt. But it certainly does not make the claim "R is uncountable"
suspect, since R *is* uncountable in the requisite sense in *every*
model.

You, on the other hand, are motivated by the view that there is a
"real" set R and a "real" set N (and, one supposes, a real universe of
sets containing R and N). Let us suppose so. I reckon that this
universe is a model of ZFC -- do you disagree? From these
suppositions, it follows that there is no surjection N -> R in the
real universe of sets either.

Do you see anything at all controversial about these claims from your
perspective?

>> Now, any function from N to R is a subset of N x R and hence an
>> element of the universe of all sets. Thus, this same model has all
>> possible functions N -> R.
>
> But wouldn't this argument apply to all models, not only the standard
> one. In that case, countable models could not exist (because N < R
> would then be false in them), and you have acknowledged they do.

No, because countable models do not include every function N -> R.

>> As far as your last remark (we can't 'quantify over all sets'), I'm
>> mighty confused. ZFC does just that. Perhaps you have never seen the
>> axioms of ZFC?
>
> I meant, we can't do it in a model-independent way.

I'm not sure what that means, but if you agree that there is a
universe of sets and this universe is a model of ZFC, then the formal
proof that R is uncountable shows that there is no surjection N -> R
in the universe of sets.

Isn't that what you were interested in?

--
Jesse F. Hughes
"Radicals are interesting because they were considered 'radical' by
the people who played with them who wrote a lot of math work that
modern mathematics depends on." --Another JSH history lesson
From: Jesse F. Hughes on 15 Jan 2010 13:46
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

> On Jan 15, 7:28 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote:
>
>> >This is circular. Defining the 'real sets' in terms of ZFC makes no
>> >sense because, clearly, the 'real sets' do exist independently of any
>> >formal theory we may write.
>>
>> How in the world can you read the previous paragraph as a supposed
>> _definition_ of the "real sets"?
>
> It's the best we can do, given the inherent logical concept. I take
> Goedel's theorem as meaning that no formal system can characterise
> everything that's real, because it would have to be consistent and
> complete to do that.

Are you sure you read David's question? Because I don't see how this
was any sort of an answer.

--
"Sexual love makes of the loved person an Object of appetite; as soon
as that appetite has been stilled, the person is cast aside as one
casts away a lemon which has been sucked dry." -- Immanuel Kant
"Squeeze my lemon til the juice runs down my leg." -- Robert Johnson
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Prev: Mathematics can only cover a validity-range from 10^500 to 10^-500 Re: #272 mathematics ends at about 10^500 Re: Powerset
Next: definition of ellipsis to define Finite in the old math and why the Peano Axioms are inconsistent Re: #281 mathematics ends at about 10^500 Re: Powerset