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From: MoeBlee on 6 Jul 2010 12:26
On Jul 5, 5:48 pm, RussellE <reaste...(a)gmail.com> wrote:
> On Jul 5, 10:47 am, MoeBlee <jazzm...(a)hotmail.com> wrote:

> > We define 'model of a theory'. Very roughly, a model of a theory T is
> > a model FOR the language of T such that every member of T is mapped to
> > the value 1 (1 for true, 0 for false) by a function that maps all
> > sentences of the language of T to either 0 or 1 and as stipulated by
> > the ordinary "Tarski method".
>
> Doesn't this mean the model is complete?

I'm going to stop you right there. What do you mean by "model is
complete"?

Do you very clearly understand what a model is as I have defined it in
my post and as my definition is found in Enderton's book?

> "All sentences" means any wff?

No, all sentences means all wffs that have no free variables. And in
this context I'm referring to all sentences in the language of PA.

I'll leave off the rest of your post for now.

As I know a great amount of your posting history, I can tell you that
you and I will not have a productive conversation if you don't
familiarize yourself with a basic textbook on this subject (preferably
Enderton, only because I used his particular formulations for some of
the basic terminology).

MoeBlee




From: Transfer Principle on 6 Jul 2010 15:55
On Jul 3, 2:05 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:
> Note: I hope to afford time to discuss this post with anyone who has
> informed, coherent, and rational comments. But I might not defend this
> post from uninformed, incoherent, and irrational criticisms,
> especially from posters who have shown themselves over a good period
> of time to be uneducable and hopeless cranks. Experience has shown
> that certain of these people will not allow themselves to be properly
> informed on certain matters in set theory and mathematical logic.

That's funny. In this paragraph, MoeBlee tells us with which type
of poster he wanted to have a discussion. And so rather than
continue the discussion in the existing threads, he started this
thread in the hopes that that other type of poster wouldn't
notice this thread at all.

But MoeBlee's plan failed. Nam Nguyen has noticed this thread,
Russell Easterly has noticed this thread, and now I have noticed
this thread. Three unwelcome posters whom MoeBlee was hoping
wouldn't discover this thread now participate in this thread.

Nice try, though, MoeBlee.

In this "secret" thread which he tried in vain to keep hidden,
MoeBlee provides us with a proof of Con(PA) in the theory "Z-R,"
or Z-Regularity. Sure enough, there isn't an "epsilon_0" to be
found anywhere in the proof, hearkening back to the other thread
in which I insisted that induction up to that ordinal was used
in the proof. (In my defense, I wasn't the only poster who had
mentioned epsilon_0 in that thread.)

Once again, I don't deny that Con(PA) is provable in ZFC, or
even Z-R, since Z-R proves the existence of omega (which MoeBlee
writes as "w"), which is the (domain of the) model of PA that
he constructs in the proof.

But my comments about epsilon_0 and all that refer back to the
mathematician Ed Nelson, who is working on a proof that PA is in
fact inconsistent. I know that I've mentioned that proof attempt
many times on sci.math, but now's as good a time as any to bring
the link to the proof up yet again:

http://www.math.princeton.edu/~nelson/papers/hm.pdf

Nelson writes:
"The goal is to produce an explicit superexponentially
long recursion and prove that it does not
terminate, thereby disproving Church’s Thesis from
below, demonstrating that finitism is untenable,
and proving that Peano Arithmetic is inconsistent."

"Superexponentiation"? Also known as "tetration," Nelson defines
this to be iterated exponentiation, in the same way that
exponentiation is iterated multiplication. But epsilon_0 is
defined to be the least infinite ordinal not obtainable from w
via finite additions, multiplications, and exponentiations -- so
superexponentiation is left out. To me, this leaves the door
open for a proof of ~Con(PA) involving superexponentiation.

A few other interesting Nelson quotes:

"There is no clear concept of the finite
in terms of which the infinite can be defined as not-finite."
(Nelson, "Completed vs. Incomplete Infinity in Arithmetic")

Hmmm. Doesn't that sound familiar? (Think AP.)

"Now I live in a world in which there are no numbers save those
that human beings on occasion construct."
(Nelson, "Mathematics and Faith")

Hmmm. Doesn't that sound familiar? (Think WM.)

And of course, Srinivasan often refers to the theory IST, a
theory created by -- you guessed it -- Ed Nelson.

MoeBlee, Aatu, and the others can say however many times they
want to about how PA is obviously consistent, and yet that
hasn't stopped Nelson from searching for a proof that PA is
instead inconsistent. And of course, since Z-R apparently
proves that PA is consistent, if Nelson can prove that PA is
inconsistent, then Z-R would be inconsistent as well.

In conclusion, Nelson's work has the potential to vindicate
those whom MoeBlee calls "hopeless cranks" across the board.
From: MoeBlee on 6 Jul 2010 16:03
On Jul 6, 2:55 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Jul 3, 2:05 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:
>
> > Note: I hope to afford time to discuss this post with anyone who has
> > informed, coherent, and rational comments. But I might not defend this
> > post from uninformed, incoherent, and irrational criticisms,
> > especially from posters who have shown themselves over a good period
> > of time to be uneducable and hopeless cranks. Experience has shown
> > that certain of these people will not allow themselves to be properly
> > informed on certain matters in set theory and mathematical logic.
>
> That's funny. In this paragraph, MoeBlee tells us with which type
> of poster he wanted to have a discussion. And so rather than
> continue the discussion in the existing threads, he started this
> thread in the hopes that that other type of poster wouldn't
> notice this thread at all.

You're LYING. I did not hope that the thread would not be noticed by
anyone at all. That would be a ridiculous hope. I've asked you many,
many times, please stop LYING about me.

> But MoeBlee's plan failed. Nam Nguyen has noticed this thread,
> Russell Easterly has noticed this thread, and now I have noticed
> this thread. Three unwelcome posters whom MoeBlee was hoping
> wouldn't discover this thread now participate in this thread.

You're CONTINUING your LIE.

> In this "secret" thread which he tried in vain to keep hidden,

> MoeBlee, Aatu, and the others can say however many times they
> want to about how PA is obviously consistent,

What SPECIFIC comment of mine do you have in mind where I mentioned
'obviousness'?

> In conclusion, Nelson's work has the potential to vindicate
> those whom MoeBlee calls "hopeless cranks" across the board.

WHAT? I've told you before what I mean by 'crank'. It's a posting
BEHAVIOR. It's not a matter of some OTHER mathematician devising this
or that formal proof.

And stop LYING about me.

MoeBlee

From: herbzet on 6 Jul 2010 16:08


Transfer Principle wrote:
> MoeBlee wrote:
> > Note: I hope to afford time to discuss this post with anyone who has
> > informed, coherent, and rational comments. But I might not defend this
> > post from uninformed, incoherent, and irrational criticisms,
> > especially from posters who have shown themselves over a good period
> > of time to be uneducable and hopeless cranks. Experience has shown
> > that certain of these people will not allow themselves to be properly
> > informed on certain matters in set theory and mathematical logic.
>
> That's funny. In this paragraph, MoeBlee tells us with which type
> of poster he wanted to have a discussion. And so rather than
> continue the discussion in the existing threads, he started this
> thread

So far, so good.

> in the hopes that that other type of poster wouldn't
> notice this thread at all.

Well, that's a ridiculous motive to impute to MoeBlee -- I hope
he won't bother with refuting such a patent absurdity.

--
hz
From: Transfer Principle on 6 Jul 2010 16:15
On Jul 5, 8:34 pm, William Hale <h...(a)tulane.edu> wrote:
> I understand that you [Aatu] don't like the term "work in ZFC" since
> mathematicians don't work in ZFC as such. But, I am using the term
> "work" as a shorthand for what is meant when we say that ZFC serves as a
> foundation for proving things in standard mathematics.
> By standard mathematics, I was limiting myself to the areas of real
> analysis, complex analysis, algebra, topology, and (differential)
> geometry. That is, areas of mathematics discussed before 1900. I didn't
> include logic since its main results were done after 1900.
> Let me give some more on what I mean by "work in ZFC". I think some
> non-mathematicians think that a mathematician comes up with an informal
> proof and that it may or may not be formalized.

I myself use the phrase "work in ZFC" all the time. But what
do I mean by "work in ZFC"?

Well, I consider those who _don't_ "work in ZFC" to be those
who contradict it. Posters who claim that only countably many
reals exist, such as Herc, don't "work in ZFC." Posters who
claim no infinite sets exist, such as Srinivasan, don't "work
in ZFC." Posters who claim that only finitely many naturals
exist, such as WM, don't "work in ZFC." Posters who propose
alternate theories and give such theories names, such as AP,
TO, and tommy1729, don't "work in ZFC."

By contrast, those who use ZFC to prove the above posters
wrong are those who _do_ "work in ZFC."

In a way, my definition of "work in ZFC" somewhat agrees with
Hale's definition. Working with LCM's (in N) constitutes
"working in PA" since PA proves the existence of LCM's, but
working with LCM's in general rings where AC (Zorn's lemma) is
needed constitutes "working in ZFC" instead. Doing classical
analysis constitutes "working in ZFC" since ZFC formalizes
classical analysis. On the other hand, working in a
non-classical analysis where there are only countably many
reals isn't "working in ZFC," since ZFC proves that there are
uncountably many reals.
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