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From: Jesse F. Hughes on 17 Jan 2010 13:34
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

> On Jan 16, 5:47 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> > I don't see these as analogous. We know already what a natural number
>> > should be. A set, though, is an abstraction that includes the sort of
>> > objects ZF deals with. So if ZFC includes any objects that are not
>> > sets, it is inconsistent.
>>
>> You're confused.
>>
>> ZFC is a theory.  It has various models.  Those models may consist of
>> sets or of porkpie hats.  ZFC is no different than N in this regard.
>
> - N is categorical and has only one model.

The standard first order theory for N is not categorical, of course.
Were you speaking of some other theory?

In any case, categorical theories have only one model *up to
isomorphism*. They have infinitely many distinct models.

> - What definition of 'set' are you using?

The usual informal notion of set. What definition of natural number
are you using?

It's not really essential that I answer this question more clearly
than that, it seems to me. After all, I need not define "natural
number" in order to observe that non-standard models of PA include
elements which are not natural numbers.

>> >> 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?
>>
>> > I don't know. I don't see any reason to believe one way or the
>> > other.
>>
>> You seem to believe that you have a notion of sets independent of
>> ZFC.  Is this a reasonably well-developed notion?  Can you not tell me
>> whether the universe of sets satisfies ZFC or not?
>
> Well, my definition is something like 'a definable, well-founded
> collection of definable objects'. This is not a mathematical concept,
> by intention.

And you have no reason to believe that the universe of such
collections satisfies ZF or not? You don't know whether you can
construct unordered pairs of "definable, well-founded collections of
definable objects? You don't know if they are distinguished by their
members (i.e., extensionality holds)? And so on?

>> > Why do countable models fail this test while the standard model
>> > succeeds? Does every model with a 'really' uncountable R contain all
>> > functions from its N to its R? Again, I don't know why I should
>> > believe this.
>>
>> The standard model contains every set and hence every function
>> N -> R.  A countable model does not contain every set and hence may
>> not (in fact, provably does not) contain every function N -> R.
>
> You are arguing in a circle. You are assuming that the standard
> model is 'every set', and therefore your statement is a
> tautology. That is my point.

The standard (i.e., intended) model *does* contain every set. What
could be more clear than that?

It seems clear to me that the universe of all sets is a model of ZF.
Is there any particular axiom of ZF that you doubt the universe of
sets satisfies?


--
Jesse F. Hughes
"Truth is common stuff, ready to your hand, but lies you have to make
yourself, and you can't be sure they are any good until you've
used them --- and then it's too late." John Steinbeck
From: David C. Ullrich on 18 Jan 2010 09:49
On Fri, 15 Jan 2010 10:14:01 -0800 (PST), Andrew Usher
<k_over_hbarc(a)yahoo.com> wrote:

>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.


You seem to have trouble following things. Someone said something
which was not a definition of "the sets in ZFC". You objected that
his "definition of the sets in ZFC" was circular. When I point out
it was no such thing you reply that it's the best we can do?

Try to keep up.


> 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.
>
>Andrew Usher

From: Aatu Koskensilta on 21 Jan 2010 05:44
MoeBlee <jazzmobe(a)hotmail.com> writes:

> TH(N) is categorical.

Nope. As Chris notes we can see this by invoking the upward
L�wenheim-Skolem theorem. Perhaps more elementarily, the
non-categoricity of Th(N) follows also by compactness. (Like Chris, I
surmise you in fact know all this already...)

On an unrelated note, ZFC does not have a standard model the way PA
has. Rather, it has an intended interpretation: the quantifiers range
over sets, 'in' means membership. (ZFC does have many non-isomorphic
standard models on a technical definition of "standard model of set
theory".) When discussing the Skolem paradox we needn't concern
ourselves with such technical details: we may just as well consider
e.g. third-order arithmetic or Zermelo set theory, where the intended
interpretation can unproblematically (in ordinary mathematics) be given
by a model.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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