Edward Nelson Proves PA Inconsistent

[RussellE]

Edward Nelson has posted to FOM Archive
"inconsistency of P"
http://cs.nyu.edu/pipermail/fom/2011-September/015816.html

He provides these links:
http://www.math.princeton.edu/~nelson/books.html
http://www.math.princeton.edu/~nelson/papers/outline.pdf

From the first link go to works in progress.
The second link is a brief outline of the proof.

[Ross A. Finlayson]

On Sep 26, 5:44 pm, RussellE <reaste...@gmail.com> wrote:
> Edward Nelson has posted to FOM Archive
> "inconsistency of P"http://cs.nyu.edu/pipermail/fom/2011-September/015816.html
>

> He provides these links:http://www.math.princeton.edu/~nelson/books.htmlhttp://www.math.princeton.edu/~nelson/papers/outline.pdf

>
> From the first link go to works in progress.
> The second link is a brief outline of the proof.

Not unsensical.

[David Bernier]

The outline (not the 100-page book, which is a work in progress)
cites in the References an article in the Notices of the AMS,
volume 57, no. 11 by Shira Kritchman and Ran Raz.

The title of their article is "The Surprise Examination Paradox
and the Second Incompleteness Theorem".

From what I understand, Nelson says that there's a primitive
recursive function definition that gives a non-terminating
recursion, and that Church's Thesis is false: not because
there's an effectively everywhere-computable function N -> N
that isn't doable as a Turing Machine, but rather because
some PR function definition (which almost everybody thinks defines
valid functions) in fact gives a non-terminating recursion.

That is tied up with the proof of Goedel's 2nd Theorem in
the Kritchman & Raz article in the Notices.

That's pretty much all I understand so far about Nelson's approach,
in the big picture sense. The 100-page book sets up a
formal deductive system for arithmetic and, so far, the theorems
proved (which are in some way linked to qea, quod erat absurdum:
a computerized proving/proof verification program)
are such as commutativity of multiplication in N,
distributivity and a few more at a higher level.
But the proof of (say) "1 < 1" isn't in the
100-page book as of now...

David Bernier
--
true prophets are the gateway to true revelation
false prophets are the gateway to false revelation

[Bill Taylor]

I think we can safely say it's going to turn out to be rubbish!
Even though Ed Nelson has done good things in the past.

I am always open to refereed bets to this effect.

-- Withering William

[Bill Taylor]

I think we can safely say that it's going to turn out to be rubbish.
Even though he has done good things in the past.

I am always open to refereed bets to this effect.

-- Withering Willy

[Alan Smaill]

William, then Willy, eh?

"As if someone were to buy several copies of the morning paper to assure
himself that what it said was true."

--
Alan Smaill

[David Bernier]

As a virtual better in the "legendary" etale_drachma currency
(think N. Bourbaki), I hereby vouch to virtually pay you
ten etale_drachmas if it's rubbish, provided that you
promise to pay me one thousand etale_drachmas if it turns out
to not be rubbish. Line open to discuss method of arbitration/refereeing.

Am willing to gpg-sign the terms of an eventual accord with you
using gpg-key-pair generated by myself and that was ID'd by
its fingerprint in my sci.math .sig-line in the past year or two.

[David Bernier]

[...]

Over at http://forums.xkcd.com/ , user "skeptical scientist"
wrote an enlightening post in the thread:
"How legitimate is this?".

Cf.:
< http://forums.xkcd.com/viewtopic.php?t=74760&p=2764988 > .

It's the one with the date and time-stamp:
Tue Sep 27, 2011 10:35 am UTC .

[Transfer Principle]

On Sep 26, 5:44 pm, RussellE <reaste...@gmail.com> wrote:

> Edward Nelson has posted to FOM Archive
> "inconsistency of P"

I've mentioned Nelson's proof attempt in many threads before.

I'll wait to see whether Nelson really did prove ~Con(PA), and
whether Bernier or Taylor wins the wager in this thread. If
Bernier wins, and Nelson really did prove ~Con(PA), then we
know that if PA is inconsistent then so is ZFC.

And then the chickens would come home to roost. ZFC, the theory
that so many poster defend, would be inconsistent. It'd be
time for the insulters to become the insulted. The ex-ZFC users
would finally find out what it feels like to have everyone tell
them that their theory is inconsistent. And I'd be there to
point and laugh at them.

After the Destruction of the Temple of Mathematics, it would be
time to build a new temple -- a replacement theory for the
inconsistent PA and ZFC. Nelson recommends Q_0 or Robinson
Arithmetic, but what about set theory? Assuming that a set
theory still exists, it would have to be a theory strong enough
to prove Con(Q_0), but not strong enough to prove Con(PA).

But if Taylor wins and Nelson is rubbish, then it will be status
quo as usual...

[Jesse F. Hughes]

Thanks for the link, Russell.

--
"That's all the legacy I ever wanted, to have people remember me like
a shooting star streaking across their Life sky, illuminating, for
just one moment, unparalleled beauty unique to itself."
-- Weblogs are a particularly humble medium, unique to themselves.

[Jesse F. Hughes]

Transfer Principle <david.l...@lausd.net> writes:

> On Sep 26, 5:44 pm, RussellE <reaste...@gmail.com> wrote:
>> Edward Nelson has posted to FOM Archive
>> "inconsistency of P"
>
> I've mentioned Nelson's proof attempt in many threads before.
>
> I'll wait to see whether Nelson really did prove ~Con(PA), and
> whether Bernier or Taylor wins the wager in this thread. If
> Bernier wins, and Nelson really did prove ~Con(PA), then we
> know that if PA is inconsistent then so is ZFC.
>
> And then the chickens would come home to roost. ZFC, the theory
> that so many poster defend, would be inconsistent. It'd be
> time for the insulters to become the insulted. The ex-ZFC users
> would finally find out what it feels like to have everyone tell
> them that their theory is inconsistent. And I'd be there to
> point and laugh at them.

Let's suppose that ZFC is inconsistent. Should anyone here feel shame?
I don't see why.

Does it mean that the various arguments that ZFC is bad were right?
Of course not.

> After the Destruction of the Temple of Mathematics, it would be
> time to build a new temple -- a replacement theory for the
> inconsistent PA and ZFC. Nelson recommends Q_0 or Robinson
> Arithmetic, but what about set theory? Assuming that a set
> theory still exists, it would have to be a theory strong enough
> to prove Con(Q_0), but not strong enough to prove Con(PA).
>
> But if Taylor wins and Nelson is rubbish, then it will be status
> quo as usual...

If PA is inconsistent, then logic and mathematics will get mighty
interesting for a while.

But I don't understand why you think that it would be a victory for any
local denizens (or a defeat for others here). The crank arguments
against ZFC have been invalid, and Nelson's argument won't change that.
Nor should anyone here consider it a defeat that a theory they have
studied turns out to be inconsistent, although they didn't know or
expect it.

You really have an odd notion of mathematics.
c
--
Jesse F. Hughes
"Such behaviour is exclusively confined to functions invented by
mathematicians for the sake of causing trouble."
-Albert Eagle's _A Practical Treatise on Fourier's Theorem_

[Bill Taylor]

On Sep 28, 3:30 pm, Transfer Principle <david.l.wal...@lausd.net>
wrote:

> But if Taylor wins and Nelson is rubbish, then it will be status
> quo as usual...

Except that I'll be owed a lot of money that will never arrive.

-- Waiting Willy

[Frederick Williams]

"Jesse F. Hughes" wrote:

> "Such behaviour is exclusively confined to functions invented by
> mathematicians for the sake of causing trouble."
> -Albert Eagle's _A Practical Treatise on Fourier's Theorem_

Do you have Eagle's book on elliptic functions? Bizarre, isn't it?

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

[Frederick Williams]

Bill Taylor wrote:
>
> I think we can safely say that it's going to turn out to be rubbish.

Really? Not just wrong, but rubbish?

[Bill Taylor]

On Sep 28, 11:34 pm, Frederick Williams

<freddywilli...@btinternet.com> wrote:

> > I think we can safely say that it's going to turn out to be rubbish.
>
> Really?  Not just wrong, but rubbish?

In math most wrong is close to rubbish.

And even more so in math logic...

-- Withering Willy

[Jesse F. Hughes]

Frederick Williams <freddyw...@btinternet.com> writes:

> "Jesse F. Hughes" wrote:
>
>> "Such behaviour is exclusively confined to functions invented by
>> mathematicians for the sake of causing trouble."
>> -Albert Eagle's _A Practical Treatise on Fourier's Theorem_
>
> Do you have Eagle's book on elliptic functions? Bizarre, isn't it?

Never seen it. I don't recall where I got that quote.

--
Jesse F. Hughes

"Hey look, Captain, next time someone wants to tie us up, let's put up
a fight." --Adventures by Morse

[WM]

On 28 Sep., 05:26, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Transfer Principle <david.l.wal...@lausd.net> writes:

> > And then the chickens would come home to roost. ZFC, the theory
> > that so many poster defend, would be inconsistent. It'd be
> > time for the insulters to become the insulted. The ex-ZFC users
> > would finally find out what it feels like to have everyone tell
> > them that their theory is inconsistent. And I'd be there to
> > point and laugh at them.
>
> Let's suppose that ZFC is inconsistent.  Should anyone here feel shame?
> I don't see why.

Because there cannot be more infinite paths in the Binary Tree than
points where paths get distinct, i.e, nodes where they split. It is a
very simple calculation:
|
o
/ \

Every point increases the number of distinct paths by 1.
A countable number of points makes a countable number of distinct
paths.

Therefore a set of uncountable paths cannot be distinct.
But you feel not ashamed to believe in that rubbish?

Further the subset of paths without a finite definition does not allow
to chooses a certain element from it. It cannot be defined. That makes
Zermelo's axiom obsolete - and his "proof" of well-ordering every set
too.

Quite a lot of simple mistakes to be ashamed.

Regards, WM

[Jesse F. Hughes]

WM <muec...@rz.fh-augsburg.de> writes:

> On 28 Sep., 05:26, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Transfer Principle <david.l.wal...@lausd.net> writes:
>
>> > And then the chickens would come home to roost. ZFC, the theory
>> > that so many poster defend, would be inconsistent. It'd be
>> > time for the insulters to become the insulted. The ex-ZFC users
>> > would finally find out what it feels like to have everyone tell
>> > them that their theory is inconsistent. And I'd be there to
>> > point and laugh at them.
>>
>> Let's suppose that ZFC is inconsistent.  Should anyone here feel shame?
>> I don't see why.
>
> Because there cannot be more infinite paths in the Binary Tree than
> points where paths get distinct, i.e, nodes where they split. It is a
> very simple calculation:
> |
> o
> / \
>
> Every point increases the number of distinct paths by 1.
> A countable number of points makes a countable number of distinct
> paths.
>
> Therefore a set of uncountable paths cannot be distinct.
> But you feel not ashamed to believe in that rubbish?

Are you alleging that you can prove (in ZFC) that the set of infinite
paths of a binary tree is countable?

If not, then I don't see why I should care about your intuitions (that
include an unjustified principle involving commuting limits and
cardinality).

If so, simply present a valid proof of this in ZFC.

Otherwise, I'm just not interested in another dull conversation with
you. The fact is that you've shown no competence in mathematical
reasoning.

--
"[I]f I could go back, [...] I would tell myself not to step into a position
where the fate of the entire world could rest in my hands. I would [avoid
this] path to a nightmarish and surreal world, a topsy-turvy world, where
everything changes." -- James S. Harris cannot escape his destiny.

[MoeBlee]

On Sep 27, 9:30 pm, Transfer Principle <david.l.wal...@lausd.net>
wrote:

> I'll wait to see whether Nelson really did prove ~Con(PA),

~Con(PA) proved with what axioms and rules?

> And then the chickens would come home to roost. ZFC, the theory
> that so many poster defend, would be inconsistent. It'd be
> time for the insulters to become the insulted.

"defend" in what sense? I hope you don't think that studying ZFC and
criticizing certain ignorant and irrational arguments against it
obligates one not to allow that there may be an epistemologically
relevant proof that ZFC is inconsistent. Certain people have claimed
to a certainty that ZFC is consistent, but the mere fact that one
studies ZFC and criticizes ignorant and irrational arguments against
it doesn't entail that one claims to a certainty that ZFC is
consistent.

> The ex-ZFC users
> would finally find out what it feels like to have everyone tell
> them that their theory is inconsistent.

Now you're referring not just to "defenders" but to "users". So, for
example, I study ZFC. I use it in that sense and in the sense that I
use ZFC to study certain areas of mathematics. That does not obligate
me to claim to a certainty that ZFC is inconsistent. If an
epistemologically relevant proof of the inconsistenty of ZFC were
given, then that would be exciting and fascinating news to me.

> And I'd be there to
> point and laugh at them.

Why? Just by the fact that one uses ZFC, one does not obligate oneself
to a claim of certainty that it is consistent.

Don't conflate different things:

(1) Continuing to espouse a theory that has ALREADY been shown to be
inconsistent. And continuing to do that in an irrational and/or
incoherent way in the face of numerous clear demonstrations and
explantions. Or continuing to espouse a view of mathematics that is
not even coherent enough to have a formalization but that is still
inconsistent in the everyday informal sense of 'inconsistent' and
continuing to do that in the face of numerous clear demonstrations and
explanations.

(2) Studying a theory that has not been shown to be inconsistent but
also hasn't yet undergone very much scrutiny for consistency.

(3) Studying a theory, such as ZFC, that has not been shown to be
inconistent, and has undergone an enormous amount of use without yet
discovery of inconsistency, as well as having certain reasonable
arguments that the theory is consistent and taking all of that as
reasonable evidence that ZFC is consistent.

(4) Studying ZFC and taking certain arguments to provide a certainty
that ZFC is consistent.

> a replacement theory for the
> inconsistent PA and ZFC.

The literature of mathematical foundations is already (and has been
for many decades) brimming with many different proposals.

MoeBlee

[Virgil]

In article
<5d0bcf12-5056-434e...@q26g2000vby.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 28 Sep., 05:26, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > Transfer Principle <david.l.wal...@lausd.net> writes:
>
> > > And then the chickens would come home to roost. ZFC, the theory
> > > that so many poster defend, would be inconsistent. It'd be
> > > time for the insulters to become the insulted. The ex-ZFC users
> > > would finally find out what it feels like to have everyone tell
> > > them that their theory is inconsistent. And I'd be there to
> > > point and laugh at them.
> >
> > Let's suppose that ZFC is inconsistent.  Should anyone here feel shame?
> > I don't see why.
>
> Because there cannot be more infinite paths in the Binary Tree than
> points where paths get distinct, i.e, nodes where they split. It is a
> very simple calculation:
> |
> o
> / \

Then WM is claiming the the number of subsets of a countably infinite
set must be countably infinite, even there are proofs that this is false.

>
> Every point increases the number of distinct paths by 1.
> A countable number of points makes a countable number of distinct
> paths.

A finite set of points has a finite set of subsets so that the number of
paths in a finite tree must be finite.

A countably infinite set of points has an uncountably infinite set of
subsets, so that nothing restricts the set of paths in an infinitie tree
to being countable, and the set of paths of a countably infinite tree
can easily be bijected to the set of subsets of that same tree, so is
equally uncountable, despite WM's paranoid refusal to understand this.

>
> Therefore a set of uncountable paths cannot be distinct.
> But you feel not ashamed to believe in that rubbish?

I would feel ashamed to believe a set that is so provably uncountable to
be countable.
--

[FredJeffries]

On Sep 26, 5:44 pm, RussellE <reaste...@gmail.com> wrote:

> Edward Nelson has posted to FOM Archive
> "inconsistency of P"http://cs.nyu.edu/pipermail/fom/2011-September/015816.html
>

> He provides these links:http://www.math.princeton.edu/~nelson/books.htmlhttp://www.math.princeton.edu/~nelson/papers/outline.pdf

>
> From the first link go to works in progress.
> The second link is a brief outline of the proof.

There's a discussion at the n-Category cafe, initiated by John Baez
and including at least one remark by Nelson:
http://golem.ph.utexas.edu/category/

Did anyone happen to read the second paragraph of the announcement?
"The outline begins with a formalist critique of finitism, making the
case that there are tacit infinitary assumptions underlying finitism"

[WM]

On 28 Sep., 16:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 28 Sep., 05:26, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> >> Transfer Principle <david.l.wal...@lausd.net> writes:
>
> >> > And then the chickens would come home to roost. ZFC, the theory
> >> > that so many poster defend, would be inconsistent. It'd be
> >> > time for the insulters to become the insulted. The ex-ZFC users
> >> > would finally find out what it feels like to have everyone tell
> >> > them that their theory is inconsistent. And I'd be there to
> >> > point and laugh at them.
>
> >> Let's suppose that ZFC is inconsistent.  Should anyone here feel shame?
> >> I don't see why.
>
> > Because there cannot be more infinite paths in the Binary Tree than
> > points where paths get distinct, i.e, nodes where they split. It is a
> > very simple calculation:
> >  |
> >  o
> > /  \
>
> > Every point increases the number of distinct paths by 1.
> > A countable number of points makes a countable number of distinct
> > paths.
>
> > Therefore a set of uncountable paths cannot be distinct.
> > But you feel not ashamed to believe in that rubbish?
>
> Are you alleging that you can prove (in ZFC)

You can't think without ZFC?
ZFC does not distinguish between cause and result?
The cause is the splitting of two paths.
The result is that they can be distinguished.

> that the set of infinite
> paths of a binary tree is countable?

The set of paths cannot be larger than the set of tails of paths. And
the set of tails of paths cannot be larger than the set of origins of
tails of paths, i.e., nodes.

>
> If not, then I don't see why I should care about your intuitions (that
> include an unjustified principle involving commuting limits and
> cardinality).

There is no commuting in saying that first comes a commnon node and
then comes a first pair of different nodes, by which two tails of
paths can be distinguished.

>
> If so, simply present a valid proof of this in ZFC.  

ZFC is not able to describe the whole field of mathematics. Even if it
was free of inconsistencies, it is not suitable as a foundation of
mathematics, in particular because it can't be used to describe the
Binary Tree and the fact that no tail of a path can be constructed
without a node where it deviates from another path.

>
> Otherwise, I'm just not interested in another dull conversation with
> you.  The fact is that you've shown no competence in mathematical
> reasoning.

Mathematics is very different from ZFC. In particular because ZFC
assumes the existence of God and rejects the sober mathematical
statement of Gauss that there is no finished infinity.

>
> --
> "[I]f I could go back, [...] I would tell myself not to step into a position
> where the fate of the entire world could rest in my hands.

or in the silly assumption of a finished infinity?

Regards, WM

[FredJeffries]

On Sep 27, 7:30 pm, Transfer Principle <david.l.wal...@lausd.net>
wrote:

> On Sep 26, 5:44 pm, RussellE <reaste...@gmail.com> wrote:
>
> > Edward Nelson has posted to FOM Archive
> > "inconsistency of P"
>
> I've mentioned Nelson's proof attempt in many threads before.
>

Then why don't you make some intelligent comment about it instead of
your chortling/whining? I don't pretend to be able to understand it
and would appreciate someone's "Nelson for Dummies" explanation.

[Virgil]

In article
<bc383ca3-c9dd-4b12...@c1g2000yql.googlegroups.com>,

The "tail" of a path in a finite tree, and indeed the path itself, can
be identified by a single node, its last, or terminal node.

But in an infinite binary tree, paths do not have terminal nodes and no
path can be identified by a single node nor even by a finite set of
nodes, but requires a set of at least infinitely many nodes.

> And
> the set of tails of paths cannot be larger than the set of origins of
> tails of paths, i.e., nodes.

Claimed but not proved, and, in fact, disproved.
Since each path has at least one "tail" unique to it, and there are more
paths than nodes, there are also more tails than nodes.

>
> >
> > If not, then I don't see why I should care about your intuitions (that
> > include an unjustified principle involving commuting limits and
> > cardinality).
>
> There is no commuting in saying that first comes a commnon node and
> then comes a first pair of different nodes, by which two tails of
> paths can be distinguished.
> >
> > If so, simply present a valid proof of this in ZFC.  
>
> ZFC is not able to describe the whole field of mathematics.

Binary trees are hardly the whole of mathematics.

> Even if it was free of inconsistencies

It is certain free from any known self-contradictions, and what you
claim are "inconsistencies" are hardly universally so regarded, so your
only possible basis for objection is that you cannot prove your case in
a system so unambiguous s ZFC.

> it is not suitable as a foundation of
> mathematics, in particular because it can't be used to describe the
> Binary Tree

Any system adequate as a foundation to the set of naturals and
containing powersets is capable of modeling a complete infinite binary
tree.
For example: take the set of naturals, N = {1,2,3,...} as the set of
nodes, with node n having 2*n and *n+1 as left and right children
respectively, and each subset of N determining the set of levels of a
path at which it branches left.

> and the fact that no tail of a path can be constructed
> without a node where it deviates from another path.

Irrelevant, since each path deviates from uncountaly many other paths at
each of is nodes.

And all of this works well in ZFC.

>
> >
> > Otherwise, I'm just not interested in another dull conversation with
> > you.  The fact is that you've shown no competence in mathematical
> > reasoning.
>
> Mathematics is very different from ZFC. In particular because ZFC
> assumes the existence of God and rejects the sober mathematical
> statement of Gauss that there is no finished infinity.

Which of ZFC's axiom alleges any gods?

Which axiom even mentions gods?


I have read through those ZFC axioms quite carefully without seeing any
mention of gods.

> >
> > --
> > "[I]f I could go back, [...] I would tell myself not to step into a position
> > where the fate of the entire world could rest in my hands.
>
> or in the silly assumption of a finished infinity?

Any axiom system which does not within itself allow proof of something
of the form "P and not P" is suitable for mathematical investigation.

And the religious objections of fools like WM are irrelevant to them.
--

[LudovicoVan]

"Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
news:87pqikr...@phiwumbda.org...

> WM <muec...@rz.fh-augsburg.de> writes:
>
>> Every point increases the number of distinct paths by 1.
>> A countable number of points makes a countable number of distinct
>> paths.
>>
>> Therefore a set of uncountable paths cannot be distinct.
>> But you feel not ashamed to believe in that rubbish?
>
> Are you alleging that you can prove (in ZFC) that the set of infinite
> paths of a binary tree is countable?

No, we have rather and repeatedly alleged that your proof (in ZFC or else)
that the set of infinite paths in the binary is not countable, while the set
of nodes is, is (of course!) rubbish. As well as the vase ending up empty,
etc. etc. Just rubbish.

Clearer?

Whatever the outcome on Nelson findings...

-LV

[LudovicoVan]

"LudovicoVan" <ju...@diegidio.name> wrote in message
news:j5vvjo$q1k$1...@speranza.aioe.org...

> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
> news:87pqikr...@phiwumbda.org...
>> WM <muec...@rz.fh-augsburg.de> writes:
>>
>>> Every point increases the number of distinct paths by 1.
>>> A countable number of points makes a countable number of distinct
>>> paths.
>>>
>>> Therefore a set of uncountable paths cannot be distinct.
>>> But you feel not ashamed to believe in that rubbish?
>>
>> Are you alleging that you can prove (in ZFC) that the set of infinite
>> paths of a binary tree is countable?

That is as easy to prove as a counting argument. (Not in ZFC? Then ZFC is
unsound, if not plainly inconsistent.)

-LV

[Jesse F. Hughes]

I am asking you what you are alleging.

> ZFC does not distinguish between cause and result?
> The cause is the splitting of two paths.
> The result is that they can be distinguished.

Ah. I see. It has nothing to do with ZFC or any other rigorous
reasoning.

Well, of course, I expected as much.

No point in carrying on with this discussion. I'll leave the rest of
your post intact so that your truly pathetic attempts at forming a
coherent argument are clear.

--
"Am I am [sic] misanthrope? I would say no, for honestly I never heard
of this word until about 1994 or thereabouts on the Internet reading a
post from someone who called someone a misanthrope."
-- Archimedes Plutonium

[Ross A. Finlayson]

On Sep 26, 8:59 pm, David Bernier <david...@videotron.ca> wrote:
> Ross A. Finlayson wrote:
> > On Sep 26, 5:44 pm, RussellE<reaste...@gmail.com>  wrote:
> >> Edward Nelson has posted to FOM Archive
> >> "inconsistency of P"http://cs.nyu.edu/pipermail/fom/2011-September/015816.html
>

> >> He provides these links:http://www.math.princeton.edu/~nelson/books.htmlhttp://www.math.princ...

With the basic premise that the Platonist's numbers have their
concomitant continuum, the abstract is sensical.

[FredJeffries]

On Sep 28, 11:37 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> Mathematics is very different from ZFC. In particular because ZFC
> assumes the existence of God and rejects the sober mathematical
> statement of Gauss that there is no finished infinity.

For those able to read German, the 1831 correspondence with Schumacher
which prompted above reference to the oft-quoted passage may be found
in Band 8 of his collected works beginning at page 210

http://resolver.sub.uni-goettingen.de/purl?PPN236010751
and go to page 216:210 in the dropdown

The passage itself appears on page 222:216, second paragraph

See also William C. Waterhouse "Gauss on Infinity" Historia
Mathematica
Volume 6, Issue 4, November 1979, Pages 430-436

Abstract: In opposing the use of completed infinity in mathematics,
Gauss was making a valid criticism of one particular kind of
argument. His celebrated statement has no connection with the
set theory to which it was later applied.

[FredJeffries]

On Sep 28, 11:37 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>

> Mathematics is very different from ZFC. In particular because ZFC
> assumes the existence of God and rejects the sober mathematical
> statement of Gauss that there is no finished infinity.

It is hilarious that someone would choose a thread discussing the work
of Ed Nelson to ring up the old fallacy that ZFC and Cantor's
transfinite
sets are in some way dependent on ones belief in God. Nelson is one
of the few current mathematicians to openly state that his belief in
God
affects his mathematics: he uses it to reject platonism and completed
infinities.

Contrary to the flippant superficiality above, it was Cantor, Dedekind
and the other founders of set theory who clearly distinguished between
mathematical and theological infinities. If the mathematical
transfinite
is so dependent on theology, why wasn't it produced during the Age of
Faith instead of having to wait until the Enlightenment? Why was
Giordano
Bruno burned at the stake for believing in an infinity of worlds?

Why was it Kronecker who invoked the deity in his argument against
Cantor?
"God created the integers".

And why are the religious beliefs of Gauss never brought up when
referring
to the famous out-of-context quote?

[WM]

On 28 Sep., 22:13, Virgil <vir...@ligriv.com> wrote:

> > The set of paths cannot be larger than the set of tails of paths.
>
>  The "tail" of a path in a finite tree, and indeed the path itself, can
> be identified by a single node, its last, or terminal node.

You can take any node as the starting point. Any sequence of nodes
following this node can be defined as the tail defined by this node.
Other tails, branching off from this one can be defined by the nodes
where this happens.

>
> > And
> > the set of tails of paths cannot be larger than the set of origins of
> > tails of paths, i.e., nodes.
>
> Claimed but not proved, and, in fact, disproved.
> Since each path has at least one "tail" unique to it, and there are more
> paths than nodes, there are also more tails than nodes.

At every finite level n you can distinguish 2^n paths.

There is no other possibility, there are no other than finite levels.
This excludes more than countably many paths.

Regards, WM

[WM]

On 28 Sep., 22:21, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in messagenews:87pqikr...@phiwumbda.org...
>

> > WM <mueck...@rz.fh-augsburg.de> writes:
>
> >> Every point increases the number of distinct paths by 1.
> >> A countable number of points makes a countable number of distinct
> >> paths.
>
> >> Therefore a set of uncountable paths cannot be distinct.
> >> But you feel not ashamed to believe in that rubbish?
>
> > Are you alleging that you can prove (in ZFC) that the set of infinite
> > paths of a binary tree is countable?
>
> No, we have rather and repeatedly alleged that your proof (in ZFC or else)
> that the set of infinite paths in the binary is not countable, while the set
> of nodes is, is (of course!) rubbish.  As well as the vase ending up empty,
> etc. etc.  Just rubbish.
>

There is a simple proof that should even be possible in ZFC:
At every level n of the Binary Tree the number of paths that can be
distinguished is 2^n.

Paths do not enter levels with infinite n. They are confined to finite
levels. Therefore you cannot distinguish more than countably many
paths in the complete Binary Tree.

Regards, WM

[WM]

ZFC is a system of 9 axioms and FOPL. There are no natural numbers, no
real numbers and no Binary Trees in ZFC. All this can at most be
defined by public consensus, but not by strict rules. Therefore it is
easy to prevent contradictions in ZFC. When the danger raises, then
the argument will be rejected.

Otherwise it would be very easy to see:

At every level n of the Binary Tree you can distinguish 2^n paths.
There are no levels with infinite index. Hence you cannot distinguish
more than countably many paths.

Regards, WM

[WM]

His argument in particular concerned a geometric application but
covers, according to his statement, every infinity.

It is not surprising that set theorists will blatantly lie about this
topic. (Compare the Binary Tree. Its paths exist only at finite
levels, where only countably many can be distinguished. Nevertheless
set theorists lie that there were uncountably many paths.) Cantor
himself accepted Gauss statement as what it is and rejected it with
respect to God knowing all numbers:

" ... und daß die Alten keine Ahnung vom Transfiniten gehabt zu haben
scheinen, deren Möglichkeit sogar von Aristoteles und seiner Schule
heftig bestritten wird, wie auch in der neueren Zeit von d'Alembert,
Lagrange, Gauß, Cauchy und deren Anhängern."
1895, 21. Sep. Cantor to Peano

heftig bestritten, that means violently disputet.

Regards, WM

[WM]

On 29 Sep., 10:47, FredJeffries <fredjeffr...@gmail.com> wrote:
> On Sep 28, 11:37 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
> > Mathematics is very different from ZFC. In particular because ZFC
> > assumes the existence of God and rejects the sober mathematical
> > statement of Gauss that there is no finished infinity.
>
> It is hilarious that someone would choose a thread discussing the work
> of Ed Nelson to ring up the old fallacy that ZFC and Cantor's
> transfinite
> sets are in some way dependent on ones belief in God.

Who else would know all nunber, in particular numbers that nobody can
define?

> Contrary to the flippant superficiality above, it was Cantor, Dedekind
> and the other founders of set theory who clearly distinguished between
> mathematical and theological infinities. If the mathematical
> transfinite
> is so dependent on theology, why wasn't it produced during the Age of
> Faith instead of having to wait until the Enlightenment?

The actual infinite was produced before Cantor on many occasions, for
instance by Grosseteste:
So ist die Reihe der geraden und die der ungeraden Zahlen unendlich,
aber die der geraden ist größer um 1. - Oder die unendlichen
Zahlenreihen, die durch geometrische Progression durch Verdoppelung
oder Verdreifachung zustande kamen, sind größer, als die unendlichen
Zahlenreihen, die durch fortgesetzte Halbierung oder Drittelung
zustande kommen. [Ludwig Baur: "Die Philosophie des Robert
Grosseteste, Bischofs Von Lincoln", Aschendorff, Münster (1917)]
http://www.archive.org/stream/diephilosophiede00baur/diephilosophiede00baur_djvu.txt

Cantor was proud to find arguments for his *mathematics* in the holy
bible, with St. Augustin, and with St. Thomas. If you understand
German, you may google my Kalenderblatt for those keywords. You will
find some paragraphs.
http://www.hs-augsburg.de/~mueckenh/KB/

Schwarz deplored:
Heute erhielt ich durch die Post einen Separatabdruck der
„Mittheilungen zur Lehre vom Transfiniten", mit der handschriftlichen
Widmung: H. A. Schwarz in Erinnerung an die alte Freundschaft
zugeeignet vom Verf. Nachdem ich so Gelegenheit erhalten habe, diesen
Aufsatz mit Muße anzusehen, kann ich nicht verhehlen, daß mir derselbe
als eine krankhafte Verirrung erscheint. Was haben denn in aller Welt
die Kirchenväter mit den Irrationalzahlen zu thun?! Möchte sich doch
die Befürchtung nicht bewahrheiten, daß unser Patient auf derselben
schiefen Ebene angelangt sei
(1887, 17. Okt. H.A. Schwarz an Carl Weierstraß)

> Why was
> Giordano
> Bruno burned at the stake for believing in an infinity of worlds?

Easy to see. Men should be the only proud creation of God.

>
> Why was it Kronecker who invoked the deity in his argument against
> Cantor?
> "God created the integers".
>
> And why are the religious beliefs of Gauss never brought up when
> referring
> to the famous out-of-context quote?

In these cases the beliefs are not interesting for the mathematics of
these people. I would think that Kronecker was rather joking.

Regards, WM

[WM]

On 29 Sep., 10:47, FredJeffries <fredjeffr...@gmail.com> wrote:

> On Sep 28, 11:37 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>

Remark:
I do not oppose to belief in God.
I do oppose to involve God in mathematics.

Regards, WM

[Bill Taylor]

On Sep 29, 9:47 pm, FredJeffries <fredjeffr...@gmail.com> wrote:

> Nelson is one of the few current mathematicians to openly state
> that his belief in God

Great Scott! I never heard that before!

These are damaging admissions.
The plot becomes a little...

- b

[MoeBlee]

On Sep 29, 5:03 am, WM <mueck...@rz.fh-augsburg.de> wrote:

> defined by public consensus, but not by strict rules.

Apparently WM is unfamiliar with formal definition in first order
theories.

MoeBlee

[Jesse F. Hughes]

Bill Taylor <wfc.t...@gmail.com> writes:

> On Sep 29, 9:47 pm, FredJeffries <fredjeffr...@gmail.com> wrote:
>
>> Nelson is one of the few current mathematicians to openly state
>> that his belief in God
>
> Great Scott! I never heard that before!
>
> These are damaging admissions.

No, they aren't.

> The plot becomes a little...

Your bigotry shows. So what?

Either Nelson's proof works or it doesn't. His motivation is completely
irrelevant.

So far, the most obvious fallacies in this thread occur in your post.
(To be fair, I suppose WM's posts have pretty obvious fallacies as
well...)

--
Jesse F. Hughes
"She moaned, in pain and pleasure, as, in a confused whirlwind, she
glimpsed an image of Saint Sebastian riddled with arrows, crucified
and impaled." --Mario Vargas Llosa on category theory

[Frederick Williams]

Bill Taylor wrote:
>
> On Sep 29, 9:47 pm, FredJeffries <fredjeffr...@gmail.com> wrote:
>
> > Nelson is one of the few current mathematicians to openly state
> > that his belief in God
>
> Great Scott!

Which one? Dana?

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

[Virgil]

In article
<e862cedc-0fa3-4966...@g29g2000yqh.googlegroups.com>,

There is a set in ZFC which has all of the necessary mathematical
properties of the natural numbers and from which a complete infinite
binary tree can be constructed within ZFC.

> All this can at most be
> defined by public consensus, but not by strict rules. Therefore it is
> easy to prevent contradictions in ZFC. When the danger raises, then
> the argument will be rejected.
>
> Otherwise it would be very easy to see:
>
> At every level n of the Binary Tree you can distinguish 2^n paths.

Of which binary tree? In a binary tree of n+1 or more levels one can at
level n distinguish between 2^(n+1) SETS of paths, but at level n one
cannot distinguish any one path from all others.

> There are no levels with infinite index. Hence you cannot distinguish
> more than countably many paths.

Non sequitur.

Note that for an infinite binary tree, every subset of N = {1,2,3,...}
determines a unique path, the path that branches left at just the levels
in that set and branches right at all the levels not in that set.

But it is well known that the power set of that N is uncountable.
--

[Virgil]

In article
<dcee1324-277e-426b...@de2g2000vbb.googlegroups.com>,

WRONG!

In a complete infinite binary tree there is NO path that is confined to
a finite set of levels.

Only a tree in which every path has a terminal node has the properties
WM demands, but in a Complete Infinite Binary Tree no path ever has any
terminal node.


WM's excessively finite mind is out of its league here.
--

[Virgil]

In article
<d9b62cbc-77e9-48b5...@p11g2000yqe.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 28 Sep., 22:13, Virgil <vir...@ligriv.com> wrote:
>
> > > The set of paths cannot be larger than the set of tails of paths.
> >
> >  The "tail" of a path in a finite tree, and indeed the path itself, can
> > be identified by a single node, its last, or terminal node.
>
> You can take any node as the starting point. Any sequence of nodes
> following this node can be defined as the tail defined by this node.

The such tails are themselves binary trees, and in the case of a
Complete Infinite Binary Tree, will be another Complete Infinite Binary
Tree.

> Other tails, branching off from this one can be defined by the nodes
> where this happens
>
> >

> > > And
> > > the set of tails of paths cannot be larger than the set of origins of
> > > tails of paths, i.e., nodes.

But the tail of a node is a tree, not a path and for a Complete Infinite
Binary Tree, each such tail is also a Complete Infinite Binary Tree.

> >
> > Claimed but not proved, and, in fact, disproved.
> > Since each path has at least one "tail" unique to it, and there are more
> > paths than nodes, there are also more tails than nodes.
>
> At every finite level n you can distinguish 2^n paths.

Presuming you start with a 0 level, at level n you can distinguish only
those sets of paths passing through a given level n node from other such
sets of paths passing through other n level nodes. And unless level n is
all terminal nodes, those are not singleton sets of paths.

WRONG! Those are sets of paths, not individual paths, being
distinguished, unless level n is a terminal level.
>
> There is no other possibility

Since your analysis is not even a possibility itself, there must be
others.

> This excludes more than countably many paths.

Which is silly, because it is easy to see that there are as many paths
in a Complete Infinite Binary Tree as there are subsets of N.
--

[Virgil]

In article
<0f9d47d2-8fd1-4797...@db5g2000vbb.googlegroups.com>,

You too often seem to invoke the supernatural to derive your results, as
they do not follow from anything less.
--

[Virgil]

In article
<ef2bf23c-01e4-4e09...@u13g2000vbx.googlegroups.com>,

If one allows that there is a set with the well-ordering properties of
natural numbers and that for each set there is a power set, the
uncountable sets are inevitable.

The only alternative to the naturals as a set is to deny all inductive
proofs.
--

[Bill Taylor]

> > Great Scott!  
>
> Which one?  Dana?

VERY TRUE!

One of the most underrated logicians of our times!

b

[Bill Taylor]

On Sep 30, 4:13 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Either Nelson's proof works or it doesn't.

It doesn't.

>  His motivation is completely irrelevant.  

It is when it doesn't!

> "She moaned, in pain and pleasure, as, in a confused whirlwind, she
> glimpsed an image of Saint Sebastian riddled with arrows, crucified
> and impaled."

AHA. Nuff said.

wfct

[WM]

On 29 Sep., 21:48, Virgil <vir...@ligriv.com> wrote:

> > There are no levels with infinite index. Hence you cannot distinguish
> > more than countably many paths.
>
> Non sequitur.

Draw an infinite path through every node of the infinite Binary Tree.
Then you have countably many paths. You have no chance to distinguish
further paths by nodes at finite levels.

> But it is well known that the power set of that N is uncountable.

In an inconsistent theory you csan prove everything.

Regards, WM

[WM]

On 29 Sep., 22:15, Virgil <vir...@ligriv.com> wrote:

> > > > And
> > > > the set of tails of paths cannot be larger than the set of origins of
> > > > tails of paths, i.e., nodes.
>
> But the tail of a node is a tree, not a path

The tail of a node K is an infinite path. You can choose it from the
tree that is defined by this node K, if you choose it at root-node
K_0.

Regards, WM

[Jesse F. Hughes]

Bill Taylor <wfc.t...@gmail.com> writes:

> On Sep 30, 4:13 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
>> Either Nelson's proof works or it doesn't.
>
> It doesn't.

I suppose not also, though I'll wait and see (more specifically, wait
for others' who understand the material better to report on its status).

>>  His motivation is completely irrelevant.  
>
> It is when it doesn't!

I suppose you mean his motivation is relevant when the proof doesn't
work.

But, no, I don't see why that should be so. Why should we care about
Nelson's reasons for studying PA's consistency? All that matters is
what is produced.

You are simply showing your own bigotry here.

--
Jesse F. Hughes
"I'm ruler", said Yertle, "of all that I see.
But I don't see enough. That's the trouble with me."
-- Yertle the Turtle, by Dr. Suess

[Jesse F. Hughes]

"Jesse F. Hughes" <je...@phiwumbda.org> writes:

> Bill Taylor <wfc.t...@gmail.com> writes:
>
>> On Sep 30, 4:13 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>>
>>> Either Nelson's proof works or it doesn't.
>>
>> It doesn't.
>
> I suppose not also, though I'll wait and see (more specifically, wait
> for others' who understand the material better to report on its

^^^^^^^
> status).

Nothing like an errant apostrophe to make one look stoopid.

--
"A recruitment consultant I know thinks the most important quality in
a winner is to be lucky. To avoid wasting his time with unlucky
applicants, he takes half the resumes piled on his desk and throws
them straight in the bin." -- John Ramsden

[Bill Taylor]

On Sep 30, 11:26 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> >> Either Nelson's proof works or it doesn't.
> > It doesn't.
>
> I suppose not also, though I'll wait and see (more specifically, wait
> for others' who understand the material better to report on its status).

We've already seen some (on FoM), and Nelson's replies to it were
insufficient.

> >>  His motivation is completely irrelevant.  
> > It is when it doesn't!
> I suppose you mean his motivation is relevant when
> the proof doesn't work.

Got it in one!

> But, no, I don't see why that should be so

I would have happily explained further but....

> You are simply showing your own bigotry here.

...I don't take kindly to gratuitous insults.
So now you will never know what may be the relevance of
personal motivations when assessing wrong work.

-- Touchy Taylor

[Jesse F. Hughes]

Right.

You call mention that Nelson is partially motivated by religious faith
"a damaging admission", but when I call this bigotry, I'm being
gratuitously insulting.

Sorry I failed to live up to your high standards.

--
Jesse F. Hughes

"It is a brilliant proof you, you math haters!!!"
-- James S. Harris

[LudovicoVan]

"WM" <muec...@rz.fh-augsburg.de> wrote in message
news:dcee1324-277e-426b...@de2g2000vbb.googlegroups.com...

> On 28 Sep., 22:21, "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in
>> messagenews:87pqikr...@phiwumbda.org...
>> > WM <mueck...@rz.fh-augsburg.de> writes:
>>
>> >> Every point increases the number of distinct paths by 1.
>> >> A countable number of points makes a countable number of distinct
>> >> paths.
>>
>> >> Therefore a set of uncountable paths cannot be distinct.
>> >> But you feel not ashamed to believe in that rubbish?
>>
>> > Are you alleging that you can prove (in ZFC) that the set of infinite
>> > paths of a binary tree is countable?
>>
>> No, we have rather and repeatedly alleged that your proof (in ZFC or
>> else)
>> that the set of infinite paths in the binary is not countable, while the
>> set
>> of nodes is, is (of course!) rubbish. As well as the vase ending up
>> empty,
>> etc. etc. Just rubbish.
>
> There is a simple proof that should even be possible in ZFC:
> At every level n of the Binary Tree the number of paths that can be
> distinguished is 2^n.

Something along that lines (and also along the lines of what you are saying
above) is in fact what I had in mind when I said that a simple counting
argument is enough.

> Paths do not enter levels with infinite n. They are confined to finite
> levels. Therefore you cannot distinguish more than countably many
> paths in the complete Binary Tree.

I am not convinced by this argument, paths are countable even when allowing
actually infinite objects (I think).

Anyway, a question: you do not believe actual infinities can exist as
mathematical objects. If I am not mistaken, such take entails that
irrationals cannot be point-like numbers, they must be intervals: so the
"continuum" cannot exist either. Correct?

-LV

[Virgil]

In article
<7cfc3095-ac96-4bf4...@p11g2000yqe.googlegroups.com>,

The "tail" of a node in a tree must, if the word is to make sense,
contain all the children of that node and all their children and so on,
and in a Complete Infinite Binary Tree so on ad infinitum.
--

[Virgil]

In article
<8dfaf32e-654d-4062...@d28g2000vby.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 29 Sep., 21:48, Virgil <vir...@ligriv.com> wrote:
>
> > > There are no levels with infinite index. Hence you cannot distinguish
> > > more than countably many paths.
> >
> > Non sequitur.
>
> Draw an infinite path through every node of the infinite Binary Tree.

For a complete binary tree with more that two nodes no path goes through
all nodes.

> Then you have countably many paths. You have no chance to distinguish
> further paths by nodes at finite levels.

But you can distinguish paths of a Complete Infinite Binary Tree by
bijecting the set of them with the set of subsets of N.

>
> > But it is well known that the power set of that N is uncountable.
>
> In an inconsistent theory you csan prove everything.

But since you cannot prove ZFC is inconsistent you have no case for it
being inconsistent.

Thus, at least until you CAN prove ZFC inconsistent,
any Complete Infinite Binary Tree in ZFC has uncountably many paths.
--

[WM]

On 30 Sep., 21:20, "LudovicoVan" <ju...@diegidio.name> wrote:
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>

> > There is a simple proof that should even be possible in ZFC:
> > At every level n of the Binary Tree the number of paths that can be
> > distinguished is 2^n.
>
> Something along that lines (and also along the lines of what you are saying
> above) is in fact what I had in mind when I said that a simple counting
> argument is enough.
>
> > Paths do not enter levels with infinite n. They are confined to finite
> > levels. Therefore you cannot distinguish more than countably many
> > paths in the complete Binary Tree.
>
> I am not convinced by this argument, paths are countable even when allowing
> actually infinite objects (I think).

I would agree. In particular because infinity is potential and
therefore cannot be surpassed.

>
> Anyway, a question: you do not believe actual infinities can exist as
> mathematical objects.  If I am not mistaken, such take entails that
> irrationals cannot be point-like numbers, they must be intervals: so the
> "continuum" cannot exist either.  Correct?

That is a problem.

First, why do I not believe in actua infinity?
Because if there was an actual infinite sequence
0.11111111111and.so.on, then it should be longer than every finite
sequence
0.1
0.11
0.111
and.so.on

But that is not the case. You can distinguish 0.11111111111and.so.on
from every finite sequence, but you cannot distinguish it from all
(taken together) finite sequences. This shows, in my opinion, what in
general is overlooked: 0.111... is not an infinite expression but only
a finite formula that allows to calculate every digit (but not all).

Second, as to irrational numbers:
I think a number is an entity that mathematicians must be able to talk
about and to identify. In this respect, SUM1/2^n and 0.090909... are
numbers. These numbers can also be expressed as pi^2/6 or 1/11.
Therefore I think that irrational numbers exist. But they have no
decimal representation. Why should they? 1/11 has no decimal
representation either.

Third: Uncountability is nonsense and has nothing to do with
mathematics. Every number ever thought belongs to a countable set.

Regards, WM

[Virgil]

In article
<cf718c83-8a99-4ddb...@k6g2000yql.googlegroups.com>,

WM may not be able to, but almost everyone else can easily distinguish
between a single object and a collection of infinitely many objects.

> This shows, in my opinion, what in
> general is overlooked: 0.111... is not an infinite expression but only
> a finite formula that allows to calculate every digit (but not all).

It allows one to represent decimally the rational number 1/9, which none
of those infinitely many finite decimals can represent.

>
> Second, as to irrational numbers:
> I think a number is an entity that mathematicians must be able to talk
> about and to identify. In this respect, SUM1/2^n and 0.090909... are
> numbers. These numbers can also be expressed as pi^2/6 or 1/11.
> Therefore I think that irrational numbers exist. But they have no
> decimal representation. Why should they? 1/11 has no decimal
> representation either.

It has a repeating decimal representation, as do all rationals which do
not have terminating ones.

>
> Third: Uncountability is nonsense and has nothing to do with
> mathematics.

It may have nothing to do with WM's very limited notion of what should
be allowed to be considered mathematics, but there are more things in
mathematics, WM, than are dreamt of in your philosophy.
--

[LudovicoVan]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:cf718c83-8a99-4ddb...@k6g2000yql.googlegroups.com...

> On 30 Sep., 21:20, "LudovicoVan" <ju...@diegidio.name> wrote:
>> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>

>> > Paths do not enter levels with infinite n. They are confined to finite
>> > levels. Therefore you cannot distinguish more than countably many
>> > paths in the complete Binary Tree.
>>
>> I am not convinced by this argument, paths are countable even when
>> allowing
>> actually infinite objects (I think).
>
> I would agree. In particular because infinity is potential and
> therefore cannot be surpassed.

How can you agree if you again state that infinity is never actual?

-LV

[abo]

On Sep 27, 2:44 am, RussellE <reaste...@gmail.com> wrote:
> Edward Nelson has posted to FOM Archive
> "inconsistency of P"http://cs.nyu.edu/pipermail/fom/2011-September/015816.html
>
> He provides these links:http://www.math.princeton.edu/~nelson/books.htmlhttp://www.math.princeton.edu/~nelson/papers/outline.pdf
>
> From the first link go to works in progress.
> The second link is a brief outline of the proof.

Although one cannot be sure of these things because this is the
Internet, but it looks from a comment on n-Category Cafe that Nelson
has withdrawn his claim, having actually committed the error that
Terrence Tao thought he had. There's nothing yet up on the fom
archives, however.

There seems to have been far more reaction to Nelson's claim, than
Kiselev's claim about inaccessible cardinals, which seems to me to be
far more likely to be true (not that I think that PA must be
consistent, just that I would expect it to be inconistent either: (1)
by a far simpler proof; or (2) a proof that is too long to think of).
Anyway, have I missed something, or is the status of Kiselev's claim
still active?

[LudovicoVan]

"WM" <muec...@rz.fh-augsburg.de> wrote in message

news:cf718c83-8a99-4ddb...@k6g2000yql.googlegroups.com...

> On 30 Sep., 21:20, "LudovicoVan" <ju...@diegidio.name> wrote:
>> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>

>> > Paths do not enter levels with infinite n. They are confined to finite
>> > levels. Therefore you cannot distinguish more than countably many
>> > paths in the complete Binary Tree.
>>
>> I am not convinced by this argument, paths are countable even when
>> allowing
>> actually infinite objects (I think).
>
> I would agree. In particular because infinity is potential and
> therefore cannot be surpassed.

[Frode Bjørdal]

On 1 Okt, 15:35, abo <dkfjd...@yahoo.com> wrote:
> On Sep 27, 2:44 am, RussellE <reaste...@gmail.com> wrote:
>
> > Edward Nelson has posted to FOM Archive
> > "inconsistency of P"http://cs.nyu.edu/pipermail/fom/2011-September/015816.html
>

> > He provides these links:http://www.math.princeton.edu/~nelson/books.htmlhttp://www.math.princ...

>
> > From the first link go to works in progress.
> > The second link is a brief outline of the proof.
>
> Although one cannot be sure of these things because this is the
> Internet, but it looks from a comment on n-Category Cafe that Nelson
> has withdrawn his claim, having actually committed the error that
> Terrence Tao thought he had.  There's nothing yet up on the fom
> archives, however.
>
> There seems to have been far more reaction to Nelson's claim, than
> Kiselev's claim about inaccessible cardinals, which seems to me to be
> far more likely to be true (not that I think that PA must be
> consistent, just that I would expect it to be inconistent either: (1)
> by a far simpler proof; or (2) a proof that is too long to think of).
> Anyway, have I missed something, or is the status of Kiselev's claim
> still active?

Dana Scott called for experts to scrutinize Kiselev's claims, as he
found Kiselev's writing competent and well founded in the literature:
http://www.cs.nyu.edu/pipermail/fom/2011-August/015699.html

I cannot see that this important call for expertise verdict has been
met. Does anyone know more?

[Frederick Williams]

RussellE wrote:
>
> Edward Nelson has posted to FOM Archive
> "inconsistency of P"

And now withdrawn it:

Terrence Tao, at
http://golem.ph.utexas.edu/category/2011/09/
and independently Daniel Tausk (private communication)
have found an irreparable error in my outline.
In the Kritchman-Raz proof, there is a low complexity
proof of K(\bar\xi)>\ell if we assume \mu=1, but the
Chaitin machine may find a shorter proof of high
complexity, with no control over how high.

My thanks to Tao and Tausk for spotting this.
I withdraw my claim.

The consistency of P remains an open problem.

Ed Nelson

[David Bernier]

David Bernier wrote:
<snipped>

Following Edward Nelson's withdrawal of his claim to a proof,
I am posting today a digitally signed message (below) that references
my bet offer to William Taylor:

-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA256

On Tuesday, 27 September 2011 11:30:33 -0400, I wrote in reply to Bill Taylor:
>
> As a virtual better in the "legendary" etale_drachma currency
> (think N. Bourbaki), I hereby vouch to virtually pay you
> ten etale_drachmas if it's rubbish, provided that you
> promise to pay me one thousand etale_drachmas if it turns out
> to not be rubbish. Line open to discuss method of arbitration/refereeing.
>
> Am willing to gpg-sign the terms of an eventual accord with you
> using gpg-key-pair generated by myself and that was ID'd by
> its fingerprint in my sci.math .sig-line in the past year or two.
>


David Bernier
- ----------------------------------------------------------------------------
pub 2048D/653721FF 2010-09-16
Key fingerprint = D85C 4B36 AF9D 6838 CC64 20DF CF37 7BEF 6537 21FF
uid David Bernier (Biggy) <davi...@videotron.ca>
sub 2048g/9088FD03 2010-09-16
- ----------------------------------------------------------------------------
-----BEGIN PGP SIGNATURE-----
Version: GnuPG v1.4.11 (GNU/Linux)

iF4EAREIAAYFAk6HtqsACgkQzzd772U3If/wggEA2E7J8sI7jIra/w+kexbboSLE
A7iGvSOuwoYxvxpg6IMA/0+7ZF1eD3TRj5ULmFEvt8541vRSZLGV0EuFwNGsajsA
=hE0/
-----END PGP SIGNATURE-----

[Transfer Principle]

On Oct 1, 8:57 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:

> RussellE wrote:
> > Edward Nelson has posted to FOM Archive
> > "inconsistency of P"
> And now withdrawn it:

> My thanks to Tao and Tausk for spotting this.
> I withdraw my claim.
> The consistency of P remains an open problem.
> Ed Nelson

And so it appears that Taylor has won his bet, and the status quo
lives on.

And so it's time for me to bid this thread adieu. I see a post
above me in this thread about a claim that ZF proves that no
inaccessibles exist. But I'm not as excited about that
possibility as I am with PA, since the upper bounds of those
posters who oppose ZFC (from AP's 10^603 all the way up to
tommy1729's aleph_aleph_0) are all far short of the first
inaccessible cardinal.

So, goodbye thread.

[David Bernier]

David Bernier wrote:
> David Bernier wrote:
> <snipped>
>
> Following Edward Nelson's withdrawal of his claim to a proof,
> I am posting today a digitally signed message (below) that references
> my bet offer to William Taylor:
>
> -----BEGIN PGP SIGNED MESSAGE-----
> Hash: SHA256
>
> On Tuesday, 27 September 2011 11:30:33 -0400, I wrote in reply to Bill
> Taylor:

[...]

I think Docendi.org has a pretty attractive Web interface for
reading threads (as opposed to Google Groups).

Among other things, It's easier to keep track of who said what,
because each level of "quote recursion"
has its own colour.

So
Andy wrote: Bob wrote:

and
Andy wrote:

will give (say) orange and blue text, respectively.
Changing pages is easy, with a widget at page bottom.

As an extra, they provide related threads in a separate area below
the examined thread. However, the archived news is at least
24 to 36 hours old, approximately.

But with a view by discussion thread by default, it's pretty darn good
compared to reading at GoogleGroups.

Cf.:
< http://science.niuz.biz/edward-t443056.html > [ page 1 ].

David Bernier
--
true prophets are the gateway to true revelation
false prophets are the gateway to false revelation

[Alan Smaill]

Transfer Principle <david.l...@lausd.net> writes:

> On Oct 1, 8:57 am, Frederick Williams <freddywilli...@btinternet.com>
> wrote:
>> RussellE wrote:
>> > Edward Nelson has posted to FOM Archive
>> > "inconsistency of P"
>> And now withdrawn it:
>> My thanks to Tao and Tausk for spotting this.
>> I withdraw my claim.
>> The consistency of P remains an open problem.
>> Ed Nelson
>
> And so it appears that Taylor has won his bet, and the status quo
> lives on.

And so Nelson has shown that he can accept he got something wrong.
Namely, this particular attempted proof.

> And so it's time for me to bid this thread adieu.

Why?

It's not the end of the story, is it?


--
Alan Smaill

[Bill Taylor]

On Oct 2, 3:53 pm, David Bernier <david...@videotron.ca> wrote:

> > Following Edward Nelson's withdrawal of his claim to a proof,
> > I am posting today a digitally signed message (below) that
> > references my bet offer to William Taylor:

I'll take it as read, and thanks. You're a star.

> Among other things,  It's easier to keep track of who said what,
> because each level of "quote recursion"
> has its own colour.

That's VERY cute, isn't it! And pretty to read.
_ _ _ _ _ _ _

One thing that this thread highlights, and is worth noting,
concerns people's reactions to alleged theory-busting news.
Not just theory-busting - that happens often enough, but
theory-busting by (what amounts to) LOGICAL CONTRADICTIONS.

A similar thing arose back in the early 70's regarding Uri Geller.
Several people in my department were wide-eyed with wonder
and excitement by this alleged confirmation of "psychic power".
Geez. OC, no-one was *quite* willing to come out and say
"Yes wow, this is probably true" - they hid behind weasel words
like "not yet disconfirmed" and "our theories will always have gaps"
and so forth, much as has been done here, (and similarly on
the faster-than-light threads). But it is/was easy to tell from
the tone of their comments that they're really *hoping* and
*wanting* it to be true. My opinion of several of my colleagues
dropped irrepairably over Uri Geller. (As did my opinion of "Nature"
vs "New Scientist", who ran simultaneous articles pro & con
Geller respectively). I dare say similar disrespect will result
over these latest two idiocies.

Furthermore, the tangent thread of people's motivation for
promulgating obviously wrong results, is now seen to be
HIGHLY relevant, IMHO. Nelson has been striving for years
to find an inconsistency in PA - it might not be going too far
to say that his whole life's work is now irretrievably wound up in it.
Such a motivation (forget the silly religious comments) is
highly dubious, and will now surely make us doubt his next
pronouncement on the matter, just as it should have made us
doubt this one!

So it is a good thing that such nonsense flairs up from time
to time - it helps sort the sheep from the goats. But what is
REALLY sad in all this, is that such wistful hopers must have
very dull intellectual lives - indeed I feel a little sorrow for them
mixed in amongst my scorn. They seem to be unaware, or
insufficiently so, of the already incredible wonder of the physical
7 abstract worlds, (as seen through so-far-standard science & math.)

It is sad that they have to leap at these false hopes, being unable
to see the wonder in what we already have, & what we don't know
but still hope to make LOGICAL sense of in future.

In conclusion, I say once again:- NYAH!!

-- Wondering Willy

[Bill Taylor]

> My thanks to Tao and Tausk for spotting this.
> I withdraw my claim.

!Que sorpresa!

Now where is all the virtual cash I won on bets people were
too chicken to take me up on!

NYAH NYAH NYAH!!!

> The consistency of P remains an open problem.

It does not. We have at least two proofs of it.
Proofs that for any other topic would be accepted beyond doubt.
But for some reason Consis(PA) seems to get people's hackles up.
Or rather, gets their common sense down.

-- Told-you-so Taylor

P.S. We may have to wait a little longer to get confirmation
of my firm skepticism regarding the other theory-destroying
announcement of recent times - the alleged exceeding of light speed.

NYAH.

[Nam Nguyen]

On 01/10/2011 11:15 PM, Bill Taylor wrote:
>> My thanks to Tao and Tausk for spotting this.
>> I withdraw my claim.
>
> !Que sorpresa!
>
> Now where is all the virtual cash I won on bets people were
> too chicken to take me up on!
>
> NYAH NYAH NYAH!!!
>
>> The consistency of P remains an open problem.
>
> It does not. We have at least two proofs of it.
> Proofs that for any other topic would be accepted beyond doubt.

Proofs are _proven_ - based on definitions - not "accepted"!

> But for some reason Consis(PA) seems to get people's hackles up.
> Or rather, gets their common sense down.
>
> -- Told-you-so Taylor
>
> P.S. We may have to wait a little longer to get confirmation
> of my firm skepticism regarding the other theory-destroying
> announcement of recent times - the alleged exceeding of light speed.
>
> NYAH.
>

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

[k...@kymhorsell.com]

In sci.math Nam Nguyen <namduc...@shaw.ca> wrote:
...

> Proofs are _proven_ - based on definitions - not "accepted"!

...

They're only accepted after they've been checked & rechecked!

--
>Why is it relevant that the 'chief scientist' is a woman?
Because women are easier prey for scams such as The Great Global Warming Hoax!
-- BONZO@27-32-240-172 [daily nymshifter], 7 Feb 2011 11:28 +1100

[Nam Nguyen]

On 02/10/2011 12:05 AM, k...@kymhorsell.com wrote:
> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
> ...
>> Proofs are _proven_ - based on definitions - not "accepted"!
> ...
>
> They're only accepted after they've been checked& rechecked!

In this particular case, what are they checked & rechecked against:
rules of inferences that are clearly defined? or some "principles"
that must be _accepted_?

[k...@kymhorsell.com]

In sci.math Nam Nguyen <namduc...@shaw.ca> wrote:
> On 02/10/2011 12:05 AM, k...@kymhorsell.com wrote:
>> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
>> ...
>>> Proofs are _proven_ - based on definitions - not "accepted"!
>> ...
>> They're only accepted after they've been checked& rechecked!
> In this particular case, what are they checked & rechecked against:
> rules of inferences that are clearly defined? or some "principles"
> that must be _accepted_?

LOL. Almost as if you haven't read any tricky proofs. :)

It was a throw away line. But I can defend the comment until bored.

Proofs can involve ideas that need to be vetted before a proof can
be "accepted".

I'll have to mention physics on sci.math, but renormalisation once was not
accepted by almost everyone as a method for proving/solving certain problems.
Nowadays it's just looked down on and we wish it would just go away.

Or you could ask Wiles about tricky proofs. Not only did it take a few
100s years to come up with it, but it took a good couple of years before
anyone was really sure it proved FLT and -- hence -- could be "accepted".

No doubt in some proof domains whether or not something *is* a (valid) proof
is not decidable. Fortunately for my spleen, I have not needed to verify that. ;)

--
This proposal on this beautiful part of Australian coastline is a monstrosity
that, above all, is not needed.
-- AUS Greens leader Bob Brown, Kilcunda, 19 Nov 2007

[WM]

On 1 Okt., 15:30, "LudovicoVan" <ju...@diegidio.name> wrote:
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> news:cf718c83-8a99-4ddb...@k6g2000yql.googlegroups.com...
>
> > On 30 Sep., 21:20, "LudovicoVan" <ju...@diegidio.name> wrote:
> >> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> >> > Paths do not enter levels with infinite n. They are confined to finite
> >> > levels. Therefore you cannot distinguish more than countably many
> >> > paths in the complete Binary Tree.
>
> >> I am not convinced by this argument, paths are countable even when
> >> allowing
> >> actually infinite objects (I think).
>
> > I would agree. In particular because infinity is potential and
> > therefore cannot be surpassed.
>
> How can you agree if you again state that infinity is never actual?

Sorry, I have not formulated carefully enough. More precisely:
I agree that all paths belong to a set that is not larger than a
countable set because I believe that actually infinite paths cannot
exist even when you allow them.

Regards, WM

[David Bernier]

> vs "New Scientist", who ran simultaneous articles pro& con

> Geller respectively). I dare say similar disrespect will result
> over these latest two idiocies.
>
> Furthermore, the tangent thread of people's motivation for
> promulgating obviously wrong results, is now seen to be
> HIGHLY relevant, IMHO. Nelson has been striving for years
> to find an inconsistency in PA - it might not be going too far
> to say that his whole life's work is now irretrievably wound up in it.
> Such a motivation (forget the silly religious comments) is
> highly dubious, and will now surely make us doubt his next
> pronouncement on the matter, just as it should have made us
> doubt this one!
>
> So it is a good thing that such nonsense flairs up from time
> to time - it helps sort the sheep from the goats. But what is
> REALLY sad in all this, is that such wistful hopers must have
> very dull intellectual lives - indeed I feel a little sorrow for them
> mixed in amongst my scorn. They seem to be unaware, or
> insufficiently so, of the already incredible wonder of the physical
> 7 abstract worlds, (as seen through so-far-standard science& math.)

[...]

I read a book by Bertrand Russell (a sort of autobiography)
many years ago.
It seems discovering his paradox was a great setback for him.

From a true autobiography (see below), he says he met Peano in
Paris at the 1900 ICM, and that was great for his
intellectual life: he studied Peano's work intensively.

He got working hard on Principia Mathematica, and found the
paradox ~ May 1901. Of course, he wanted a Foundations work
where Russell's Paradox couldn't be derived. He says
that proved to be really hard, and they had classes, etc.
I'm not sure how things turned out for the Principia.

Could someone confirm/deny or add to that version of events? It was
a long time ago now.

<
http://books.google.ca/books?id=SlMrmmrNuEoC&lpg=PP1&dq=Bertrand%20Russell&pg=PA150#v=onepage&q&f=false>

[Frederick Williams]

David Bernier wrote:

> I think Docendi.org has a pretty attractive Web interface for
> reading threads (as opposed to Google Groups).

Thank you for mentioning it. Let's hope that Google Groups now dies the
death it deserves to.

[Jesse F. Hughes]

Well, for my part, bah.

I can't imagine, personally, what an inconsistency in PA would mean.
But I recognize that my own imagination is limited and I have heard that
Nelson is a respected mathematician who has done good work previously,
and so I will wait for expert review on his alleged proof, although I
will doubt that it is correct.

I see nothing irrational in this response. And I surely don't know who
you refer to as "hoping" the proof works, aside from TP and the usual
gaggle of cranks. Oh, if PA is inconsistent, then mathematical logic
probably gets a lot more interesting, but where have you seen an
otherwise reasonable poster turn this observation into wishful thinking?

Seems to me the wishful thinking going on here is your judgment of your
own intellectual superiority. Par for the course, really.

As far as Nelson's failure causing us to pre-judge his next
pronouncement, surely so. And this is reasonable, too. But let's
recall that his announcement was to FOM, not the general press. I'm
sure he wanted feedback on his outline and he got it. Although his
argument was invalid, it seems to me that he has behaved as a
respectable mathematician should throughout the event.

--
"There's lots of things in this old world to take a poor boy down.
If you leave them be, you can save yourself some pain.
You don't have to live in fear, but you best have some respect,
For rattlesnakes, painted ladies and cocaine." -- Bob Childers

[Nam Nguyen]

>> vs "New Scientist", who ran simultaneous articles pro& con

>> Geller respectively). I dare say similar disrespect will result
>> over these latest two idiocies.
>>
>> Furthermore, the tangent thread of people's motivation for
>> promulgating obviously wrong results, is now seen to be
>> HIGHLY relevant, IMHO. Nelson has been striving for years
>> to find an inconsistency in PA - it might not be going too far
>> to say that his whole life's work is now irretrievably wound up in it.
>> Such a motivation (forget the silly religious comments) is
>> highly dubious, and will now surely make us doubt his next
>> pronouncement on the matter, just as it should have made us
>> doubt this one!
>>
>> So it is a good thing that such nonsense flairs up from time
>> to time - it helps sort the sheep from the goats. But what is
>> REALLY sad in all this, is that such wistful hopers must have
>> very dull intellectual lives - indeed I feel a little sorrow for them
>> mixed in amongst my scorn. They seem to be unaware, or
>> insufficiently so, of the already incredible wonder of the physical

>> 7 abstract worlds, (as seen through so-far-standard science& math.)

>>
>> It is sad that they have to leap at these false hopes, being unable

>> to see the wonder in what we already have,& what we don't know

>> but still hope to make LOGICAL sense of in future.
>>
>> In conclusion, I say once again:- NYAH!!
>
> Well, for my part, bah.
>
> I can't imagine, personally, what an inconsistency in PA would mean.

It would simply mean there's a theorem of the form (F ^ ~F) or
alternatively there's no model for it, which consequently, in this
theoreatically but still possible case, what we've alluded to as
its "standard" model would NOT be its model (though it might still
a model of the underlying language).

This is trivial of course.

> But I recognize that my own imagination is limited and I have heard that
> Nelson is a respected mathematician who has done good work previously,
> and so I will wait for expert review on his alleged proof, although I
> will doubt that it is correct.
>
> I see nothing irrational in this response.

As far as foundation of mathematics is concerned, "irrational" could be
in any camp, not just the "crank" one!

> And I surely don't know who
> you refer to as "hoping" the proof works, aside from TP and the usual
> gaggle of cranks.

I for one don't have any reason for such a hope and so wouldn't insist
on it as (some of) the cranks would do. That said, in meta level,
there's a _technical_ third position to be considered: it's impossible
to know whether or not PA is consistent.

What's technically wrong with the third?

> Oh, if PA is inconsistent, then mathematical logic
> probably gets a lot more interesting,

That's somewhat of a fallacy.

The theory T = (Axy[x=y] /\ ~Axy[x=y]) is inconsistent but the model,
say M, with U = {{}} is one in which _infinitely many theorems_ of T
are true!

It's technically possible that N, as the _perceived_ "the standard"
model of the language L(PA), would be in the same fate of M, viz-a-viz
the theory PA.

There's such necessary tradition in mathematical reasoning that things
have to stop somewhere, and infinite regression should cease.
The axiomatic approach is the core of that tradition where we have
to start the reasoning with axioms.

The weakness of the cranks though is that they would claim, say, PA is
inconsistent without producing a syntactical proof.

The weakness of some of the "orthodox" is that they would _claim_
PA is consistent _without acknowledging_ the third possibility.

[One may as well simplify the whole situation by assuming the
principle that PA be consistent and then proving other things,
including some of other principles that they've assumed!]

[Nam Nguyen]

On 02/10/2011 2:36 AM, k...@kymhorsell.com wrote:
> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
>> On 02/10/2011 12:05 AM, k...@kymhorsell.com wrote:
>>> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
>>> ...
>>>> Proofs are _proven_ - based on definitions - not "accepted"!
>>> ...
>>> They're only accepted after they've been checked& rechecked!

>> In this particular case, what are they checked& rechecked against:

>> rules of inferences that are clearly defined? or some "principles"
>> that must be _accepted_?
>
> LOL. Almost as if you haven't read any tricky proofs. :)

Save your laughing.

In the context of say sci.logic, what do you think the definition of
_proof_ be?

>
> It was a throw away line. But I can defend the comment until bored.
>
> Proofs can involve ideas that need to be vetted before a proof can
> be "accepted".
>
> I'll have to mention physics on sci.math, but renormalisation once was not
> accepted by almost everyone as a method for proving/solving certain problems.
> Nowadays it's just looked down on and we wish it would just go away.
>
> Or you could ask Wiles about tricky proofs. Not only did it take a few
> 100s years to come up with it, but it took a good couple of years before
> anyone was really sure it proved FLT and -- hence -- could be "accepted".
>
> No doubt in some proof domains whether or not something *is* a (valid) proof
> is not decidable. Fortunately for my spleen, I have not needed to verify that. ;)
>


--

[k...@kymhorsell.com]

In sci.math Nam Nguyen <namduc...@shaw.ca> wrote:
> On 02/10/2011 2:36 AM, k...@kymhorsell.com wrote:
>> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
>>> On 02/10/2011 12:05 AM, k...@kymhorsell.com wrote:
>>>> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
>>>> ...
>>>>> Proofs are _proven_ - based on definitions - not "accepted"!
>>>> ...
>>>> They're only accepted after they've been checked& rechecked!
>>> In this particular case, what are they checked& rechecked against:
>>> rules of inferences that are clearly defined? or some "principles"
>>> that must be _accepted_?
>> LOL. Almost as if you haven't read any tricky proofs. :)
> Save your laughing.
> In the context of say sci.logic, what do you think the definition of
> _proof_ be?

...

You are asking what principles some group "accepts"? :)

More LOL.

--
[Excess rainfall as the origin of SLR:]
The slow rise of sea level is caused by rain. Water transfer the soil to see.
The acceleration during the last 50 years is caused by using gas and oil
instead of coal. Gas and oil are changed into water during combustion.
So the slow or the accelerated rise of sea level is not a problem.
-- Szczepan Bialek <sz.b...@wp.pl>, 28 May 2011 09:50 +0200

[Jesse F. Hughes]

Nam Nguyen <namduc...@shaw.ca> writes:

>> I can't imagine, personally, what an inconsistency in PA would mean.
>
> It would simply mean there's a theorem of the form (F ^ ~F) or
> alternatively there's no model for it

[...]

Well, yes, that is what it would literally mean.

I was speaking more loosely. What I can't imagine is how or why PA
could be inconsistent.

Even with that rephrasing, I suspect one can misunderstand my comment so
let me just say instead: I would be mighty surprised if PA were proved
inconsistent. Exceptionally bumfuzzled, even.

--
Jesse F. Hughes
"If this novel could be compared to sculpture I'd have to compare it
to the Sistine Cieling."
-- An insightful review of Ryskamp's "Nature Studies"

[MoeBlee]

On Oct 1, 9:35 pm, Transfer Principle <david.l.wal...@lausd.net>
wrote:

> And so it's time for me to bid this thread adieu.

It wouldn't be so bad if you responded to my post to you on Sept. 28.

MoeBlee

[Nam Nguyen]

On 02/10/2011 12:04 PM, k...@kymhorsell.com wrote:
> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
>> On 02/10/2011 2:36 AM, k...@kymhorsell.com wrote:
>>> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
>>>> On 02/10/2011 12:05 AM, k...@kymhorsell.com wrote:
>>>>> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
>>>>> ...
>>>>>> Proofs are _proven_ - based on definitions - not "accepted"!
>>>>> ...
>>>>> They're only accepted after they've been checked& rechecked!
>>>> In this particular case, what are they checked& rechecked against:
>>>> rules of inferences that are clearly defined? or some "principles"
>>>> that must be _accepted_?
>>> LOL. Almost as if you haven't read any tricky proofs. :)
>> Save your laughing.
>> In the context of say sci.logic, what do you think the definition of
>> _proof_ be?
> ...
>
> You are asking what principles some group "accepts"? :)

You don't seem to understand the fine technical details of what we
call proof in the context of mathemnatical reasoning.

It's true that it's all relative when it comes to the choice of
reasoning framework you would accept to make inferences. E.g.,
I might have in mind FOL= where some theorems are accepted as valid
inference; but you might have FOL without equality where the same
"theorems" aren't theorems! So yes, it's all relative to what we'd
accept and this part you do seem to understand.

What you don't seem to understand though is there's is such a thing as
layering (Shoenfield called "edifice") within the context of of an
accepted reasoning framework, say FOL=, in which as you progressively
move down from the upper layers you'd be more constraint and would
_NOT be free to introduce new principles or what you'd wish at will_ .

For instance, again, you don't have to accept FOL= but if you do,
then there's strict definition of inference rules and theorems
you have to abide in claiming what is a proof; and there's strict
definition of what a model is so that people could check to see if
a thing could be called a model. (And I dare you, in your own wording,
"checked & rechecked" N as a model of PA based upon the technical
definition of model! Read: nobody can!).

So, despite that reasoning framework has to be accepted - and could
be accepted at will - the consistency of PA is something you can NOT
accept, _without undermining the upper layer definitions_ of, say, FOL=.

And if you do undermine the very own framework you've accepted,
why bother to accept it in the first place?

(One may as well accept the reasoning framework that everything "kym"
says be true, and a lot of "interesting" conclusions are still _valid_
within this _accepted_ framework!)

>
> More LOL.
>

The more you'd have to save.

[k...@kymhorsell.com]

In sci.math Nam Nguyen <namduc...@shaw.ca> wrote:
...

> You don't seem to understand the fine technical details of what we
> call proof in the context of mathemnatical reasoning.

...

LOL.

I'm sorry, I don't accept your "proof" that proofs don't rely on
principles that need to be accepted since you seem to reference the
principles accepted by readers of a certain USENET newsgroup and need
me to accept them.

Alternatively, if you can prove your claim without reference to
what principles I need to accept then please feel free to continue
your proof.

--
Scientists [and kooks] are always changing their story and as a Conservative,
I have no tolerance for ambiguity. It proves that all science is lies and
the only thing we can trust is right wing rhetoric.
-- BONZO@27-32-240-172 [daily nymshifter], 14 Jan 2011 14:46 +1100

CORRECTION: True science, (remember that?) can be trusted, but this
"science" is ALL LIES!
-- BONZO@27-32-240-172 [daily nymshifter], 19 Feb 2011 14:46 +1100

[Nam Nguyen]

On 02/10/2011 11:54 AM, Jesse F. Hughes wrote:
> Nam Nguyen<namduc...@shaw.ca> writes:
>
>>> I can't imagine, personally, what an inconsistency in PA would mean.
>>
>> It would simply mean there's a theorem of the form (F ^ ~F) or
>> alternatively there's no model for it
> [...]
>
> Well, yes, that is what it would literally mean.
>
> I was speaking more loosely. What I can't imagine is how or why PA
> could be inconsistent.

All that would need is a syntactical proof. But you do have a
clarification below.

>
> Even with that rephrasing, I suspect one can misunderstand my comment so
> let me just say instead: I would be mighty surprised if PA were proved
> inconsistent. Exceptionally bumfuzzled, even.

Ignoring physics, biology, or what have we, it'd not be the first time
one would be surprised about something of that nature. I strongly
suspect that Quinne might have been surprised when it turned out his
ML theory is inconsistent.

If that's still not satisfactory, why don't you consider what I've
said about the theory T:

> The theory T = (Axy[x=y] /\ ~Axy[x=y]) is inconsistent but the model,
> say M, with U = {{}} is one in which _infinitely many theorems_ of T
> are true!

[Nam Nguyen]

On 02/10/2011 12:52 PM, k...@kymhorsell.com wrote:
> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
> ...
>> You don't seem to understand the fine technical details of what we
>> call proof in the context of mathemnatical reasoning.
> ...
>
> LOL.

Sorry. I can't continue discussing technical issues with one
who's just keep laughing - no matter what!

Bye.

>
> I'm sorry, I don't accept your "proof" that proofs don't rely on
> principles that need to be accepted since you seem to reference the
> principles accepted by readers of a certain USENET newsgroup and need
> me to accept them.
>
> Alternatively, if you can prove your claim without reference to
> what principles I need to accept then please feel free to continue
> your proof.
>


--

[Virgil]

In article
<01f4ca96-bbcd-4e18...@dd6g2000vbb.googlegroups.com>,

Thus WM believes in a system which allows both A and not_A to be
simultaneously true.

No wonder he has so much trouble with math.
--

[k...@kymhorsell.com]

In sci.math Nam Nguyen <namduc...@shaw.ca> wrote:
> On 02/10/2011 12:52 PM, k...@kymhorsell.com wrote:
>> In sci.math Nam Nguyen<namduc...@shaw.ca> wrote:
>> ...
>>> You don't seem to understand the fine technical details of what we
>>> call proof in the context of mathemnatical reasoning.
>> ...
>> LOL.
> Sorry. I can't continue discussing technical issues with one
> who's just keep laughing - no matter what!

...

LOL.

--

Proofs are _proven_ - based on definitions - not "accepted"!

-- Nam Nguyen <namduc...@shaw.ca>, 01 Oct 2011 23:57 -0600

[Transfer Principle]

On Oct 2, 12:05 pm, k...@kymhorsell.com wrote:

> In sci.math Nam Nguyen <namducngu...@shaw.ca> wrote:> On 02/10/2011 12:52 PM, k...@kymhorsell.com wrote:
> > Sorry. I can't continue discussing technical issues with one
> > who's just keep laughing - no matter what!

> LOL.
> --
> Proofs are _proven_ - based on definitions - not "accepted"!

> -- Nam Nguyen <namducngu...@shaw.ca>, 01 Oct 2011 23:57 -0600

And here we see that this poster has resorted to using a .sig
randomizer, eerily similar to that of Jesse Hughes. For just like
Hughes's, this one contains his opponents' quotes.

I don't know whether this one is as random as Hughes's. It probably
is, for I'd think that one wouldn't intentionally choose a quote
made by the very poster to which one is replying (i.e. Nam Nguyen).

[Transfer Principle]

It seems as if I can never leave a thread cleanly, can I? OK,
here's that other post:

> > I'll wait to see whether Nelson really did prove ~Con(PA),
> ~Con(PA) proved with what axioms and rules?

Irrelevant now that Nelson has withdrawn his proof.

> > And then the chickens would come home to roost. ZFC, the theory
> > that so many posters defend, would be inconsistent. It'd be
> > time for the insulters to become the insulted.
> "defend" in what sense? I hope you don't think that studying ZFC and
> criticizing certain ignorant and irrational arguments against it
> obligates one not to allow that there may be an epistemologically
> relevant proof that ZFC is inconsistent. Certain people have claimed
> to a certainty that ZFC is consistent, but the mere fact that one
> studies ZFC and criticizes ignorant and irrational arguments against
> it doesn't entail that one claims to a certainty that ZFC is
> consistent.

I notice that you (MoeBlee) distinguish between "ignorant and
irrational
arguments against" ZFC and "an epistemologically relevant proof"
against
the theory.

I distinguish between them as well, but mainly in terms of the
direction
of the _insults_ in each case. In particular:

I/I arguments against ZFC -> you will insult the others
some ER proof against ZFC -> the others will insult you

And if one considers the insults in one direction to be undeserved,
then
they should hold the others to be underserved as well.

I myself consider both directions to be undeserved, _but_ I note that
it
would be _poetic_justice_ for the second insults to occur.

> > And I'd be there to point and laugh at them.
> Why? Just by the fact that one uses ZFC, one does not obligate oneself
> to a claim of certainty that it is consistent.

Here's my view of how mathematicians feel about ZFC:

_Every_ mathematician believes that it's _theoretically_ possible that
ZFC is inconsistent. But _no_ mathematician believes that it's
_actually_
possible that ZFC is inconsistent.

Now as long as no ER proof of ~Con(ZFC) has been found (and it
hasn't), I
want to find a way for those who are opposed to some of ZFC's results
to
avoid insults. For example, the arguments of one such poster (Herc)
basically boils down to "I don't like uncountable sets." While of
course
this is an I/I argument against ZFC, it can be made into a _rational_
argument via a rigorous theory. And, just as I predicted, zuhair has
posted a theory, and Herc is looking at it. I won't post in that
thread
just yet, for I want to see how that thread develops naturally without
my interference. Of course, zuhair's theory, even if it does prove
~Con(ZFC), still won't be considered ER, but at least it will give the
posters something to discuss without resorting to _insults_.

That's all I want -- to reduce/eliminate the _insults_.

> > a replacement theory for the inconsistent PA and ZFC.
> The literature of mathematical foundations is already (and has been
> for many decades) brimming with many different proposals.

Of course, a theory strong enough to prove the consistency of PA might
not survive Nelson's proof.

Nelson himself provides an alternative to PA, and that alternative is
a form of Robinson arithmetic. I'm not sure what an alternative to set
theory like ZFC would be lke. But once again, this is all irrelevant
as Nelson's proof fails.

Since I'm back in this thread, I might as well make a few more
responses to various posters.

[Transfer Principle]

On Oct 1, 10:15 pm, Bill Taylor <wfc.tay...@gmail.com> wrote:
> > The consistency of P remains an open problem.
> It does not.  We have at least two proofs of it.
> Proofs that for any other topic would be accepted beyond doubt.
> But for some reason Consis(PA) seems to get people's hackles up.
> Or rather, gets their common sense down.

Here's why Con(PA) gets my own hackles up:

We know that PA proves the existence of arbitrarily large
naturals (via the successor axiom). But many posters object to
the existence of arbitrarily large naturals -- in particular
those that are larger than any number that could possibly arise
in physics. For example, AP claims that 10^603 is larger than any
finite natural, since larger numbers have little physical use.

> P.S. We may have to wait a little longer to get confirmation
>         of my firm skepticism regarding the other theory-destroying
> announcement of recent times - the alleged exceeding of light speed.

It's interesting that you mention the recent controversy involving
the exceeding of c here. Since the mainstream theory is that no
particle with mass can exceed c, some posters believe that there
should be an upper bound to the finite naturals as rigid as c is
as a speed. Even measuring c in the units of, say, Planck-lengths
per proton-halflife gives a number falling far short of 10^603.

The large number mentioned in the abstract of Nelson's attempted
proof is much larger than 10^603 -- it's more like 2^^^16 (i.e., 2
_pentated_ to the 16). One would be hard-pressed to find a number
relevant to physics as large as 2^^^16.

So my interest in Con(PA) is mainly geared towards ultrafinitists
and others who are insulted for arguing that there exists a largest
finite natural. That's why I was "wistfully hoping" that PA would
turn out inconsistent, so that the _insulters_ would get a taste of
their own medicine.

> NYAH.

I have no problem with Taylor gloating here -- after all, he did
just become a little richer due to his wager -- as long as he does
so without insults. (I caught only one minor insult, the word
"idiocies," appearing in another post of his.)

[Joshua Cranmer]

On 10/2/2011 9:59 PM, Transfer Principle wrote:
> So my interest in Con(PA) is mainly geared towards ultrafinitists
> and others who are insulted for arguing that there exists a largest
> finite natural. That's why I was "wistfully hoping" that PA would
> turn out inconsistent, so that the _insulters_ would get a taste of
> their own medicine.

I have no problems with people who object to, say, Cantor's proof on the
basis of ultrafinitism. Even if they had a nonstandard theory for their
beliefs, I wouldn't object to it, if they at least are willing to accept
the following:

1. The theory must be coherent. Definitions should be precise, clear,
and consistent; a participant should also be willing to offer more
information on the theory if things are unclear.
2. The theory must not be immediately evident to be inconsistent. PA,
after all, may be inconsistent, but its inconsistency is at least not
obvious to most people.
3. They must be willing to graciously retract their statements if it is
unambiguously shown to be false using their theory.

Most of the people recently attacking Cantor's proof have failed to
uphold all 3 of these statements; Mr. Nelson, on the other hand, appears
to have satisfied all 3 of these: he retracted his proof when a fatal
flaw was pointed out. Compare that to, say, Mr. Cooper, most of whose
recent posts appear to be incoherent babble. To be fair, some of those
who respond to him have also resorted to incoherent babble.

From my perspective, anyone who objects to "settled" matters like
Godel's theorems or the uncountability of real numbers on mostly the
fact that the result is counterintuitive would amount to a crank, much
as those who have in the past attempted to trisect an angle or square
the circle given only a straightedge and compass. Either they should
realize that the math they believe in is counter to their intuition, or
they should construct a math that follows their intuition better. Or, at
the very least, cease to believe that it is their job to convince
everyone else that their proof is better than the giants of mathematics.

--
Beware of bugs in the above code; I have only proved it correct, not
tried it. -- Donald E. Knuth

[k...@kymhorsell.com]

In sci.math Transfer Principle <david.l...@lausd.net> wrote:

> On Oct 2, 12:05?pm, k...@kymhorsell.com wrote:
>> In sci.math Nam Nguyen <namducngu...@shaw.ca> wrote:> On 02/10/2011 12:52 PM, k...@kymhorsell.com wrote:
>> > Sorry. I can't continue discussing technical issues with one
>> > who's just keep laughing - no matter what!
>> LOL.
>> --
>> Proofs are _proven_ - based on definitions - not "accepted"!
>> -- Nam Nguyen <namducngu...@shaw.ca>, 01 Oct 2011 23:57 -0600

...

> I don't know whether this one is as random as Hughes's. It probably
> is, for I'd think that one wouldn't intentionally choose a quote
> made by the very poster to which one is replying (i.e. Nam Nguyen).

Tin's sig selector is random but not uniform. It seems to sort
the files in the .Sig directory into an order selected from the
file meta-data, and the date of creation therefore figures.

--
[Feel the meta-evidence, Luke:]
The great thing about science is that once you understand it you tend
to defend it, especially against pretenders to science like the agw
activists here and at various institutions like the CRU, GISS, Penn
State and against political activists at the IPCC and Greenpeace.
-- Tunderbar <tdco...@gmail.com>, 8 Jul 2011 11:05 -0700 (PDT)

[David Libert]

Frederick Williams (freddyw...@btinternet.com) writes:
> RussellE wrote:
>>
>> Edward Nelson has posted to FOM Archive
>> "inconsistency of P"
>
> And now withdrawn it:
>
> Terrence Tao, at
> http://golem.ph.utexas.edu/category/2011/09/
> and independently Daniel Tausk (private communication)
> have found an irreparable error in my outline.
> In the Kritchman-Raz proof, there is a low complexity
> proof of K(\bar\xi)>\ell if we assume \mu=1, but the
> Chaitin machine may find a shorter proof of high
> complexity, with no control over how high.

>
> My thanks to Tao and Tausk for spotting this.
> I withdraw my claim.
>

> The consistency of P remains an open problem.
>

> Ed Nelson

> --
> When a true genius appears in the world, you may know him by
> this sign, that the dunces are all in confederacy against him.
> Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

Now that the dust is starting to settle around the proof attempt, I
will collect some references relating to points about it, such as what
stronger results it had also been claiming.

Edward Nelson's note to FOM that Frederick quoted is from Nelson's
FOM post

[1] Edward Nelson [FOM] inconsistency of P
Sat Oct 1 08:36:43 EDT 2011
http://www.cs.nyu.edu/pipermail/fom/2011-October/015832.html


Early in this thread

[2] David Bernier " Re: Edward Nelson Proves PA Inconsistent"
sci.logic, sci.math, comp.theory Sep 27, 2011
http://groups.google.com/group/sci.logic/msg/e332cd0482f3f340

noted a reference:

>Over at http://forums.xkcd.com/ , user "skeptical scientist"
>wrote an enlightening post in the thread:
>"How legitimate is this?".
>
>Cf.:
>< http://forums.xkcd.com/viewtopic.php?t=74760&p=2764988 > .
>
>It's the one with the date and time-stamp:
>Tue Sep 27, 2011 10:35 am UTC .


That xkcd reference has near its end:

>9) This result can be pulled back to Q0, so Robinson arithmetic
> (and hence PA) are inconsistent.


(I broke a long line).

So this actually would seem to have been getting Robinson
arithmetic. Though he also notes after

>I could easily be misunderstanding Nelson somewhere, so please
> let me know if I went wrong!

(Broke a long line again.)

This is the only thing I have noticed pulling it down to Robinson's
arithmetic.

And that's something to think about. What would it mean for
Robinson's arithmetic to be inconsistent?

[3] http://en.wikipedia.org/wiki/Robinson%27s_Arithmetic

There is no induction in Robinson's arithmetic Q. More or less Q
just gives the definitions to form terms as we know them in
0, S, +, * and has 2 axioms that sort of say these terms are
everything: no 0 predecessor and everything is either 0 or a
succssor. Ie a couple of cases where you can't think of a term then
there is indeed no such number.

Back to other parts.

Nelson's FOM quote above mentions Terence Tao, and also private
communication from Daniel Tausk. Nelson gives a url for Tao's which
is comments on a topic of his proof at blog The n-Category Cafe:

[4] The Inconsistency of Arithmetic
The n-Category Cafe Sep 27 - Oct 1 , 2011 so far
http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#comments


Early in [3] is a threaded exchange between Tao and Nelson leading
up to Nelson withdrawing the claim:

[5] http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039590


[5] is the withdrawal, you can scroll backwards to find the
discussion leading up to that in threaded view. (4 articles zigzag
Tao, Nelson).

There are several other threads by Tao on his same comments, in more
or less detail.

Nelson had also posted to FOM about personal communication from
Daniel Tausk.

In [4], Daniel Tausk posted an email of his to Nelson:

[6] http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039590


I have not noticed anything in [4] about the result also applying to
Robinson's Q, as in the xkcd reference from [2].

But there was something in [4] which seemed to say this proof would
also be claiming to prove the inconsistency of PRA:

[7] http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039541

[8] http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic




--
David Libert ah...@FreeNet.Carleton.CA

[Frederick Williams]

David Libert wrote:
>
> [...]

>
> I have not noticed anything in [4] about the result also applying to
> Robinson's Q, as in the xkcd reference from [2].

Um... I know nothing about these things, but I would say that
cartoonists and mathematicians are two disparate groups of people and
the claims of the former place no obligations on the latter. (It used
to be though impossible that someone could do something on one day and
thereby place an obligation on someone else some days earlier, but those
pesky neutrinos have scotched that idea.)

[Patricia Shanahan]

On 10/2/2011 7:59 PM, Transfer Principle wrote:
> On Oct 1, 10:15 pm, Bill Taylor <wfc.tay...@gmail.com> wrote:
>>> > > The consistency of P remains an open problem.
>> > It does not. We have at least two proofs of it.
>> > Proofs that for any other topic would be accepted beyond doubt.
>> > But for some reason Consis(PA) seems to get people's hackles up.
>> > Or rather, gets their common sense down.
> Here's why Con(PA) gets my own hackles up:
>
> We know that PA proves the existence of arbitrarily large
> naturals (via the successor axiom). But many posters object to
> the existence of arbitrarily large naturals -- in particular
> those that are larger than any number that could possibly arise
> in physics. For example, AP claims that 10^603 is larger than any
> finite natural, since larger numbers have little physical use.

...

I just don't see the connection between wanting to have a finite upper
bound on numbers, or for that matter wanting any other property, and the
specific question of the consistency of PA.

PA could be consistent and still be inferior, in your opinion, to some
other system that has properties unrelated to consistency that you
consider important.

Patricia

[WM]

On 3 Okt., 06:08, Joshua Cranmer <Pidgeo...@verizon.invalid> wrote:

> 1. The theory must be coherent. Definitions should be precise, clear,
> and consistent; a participant should also be willing to offer more
> information on the theory if things are unclear.
> 2. The theory must not be immediately evident to be inconsistent. PA,
> after all, may be inconsistent, but its inconsistency is at least not
> obvious to most people.
> 3. They must be willing to graciously retract their statements if it is
> unambiguously shown to be false using their theory.
>
> Most of the people recently attacking Cantor's proof have failed to
> uphold all 3 of these statements;

Would you let it go through as a coherent argument, that an infinite
triangle

o
oo
ooo
...

cannot have a completed left L or right R edge without also having a
completed basis B?

In every finite case we have L = R = B. But in the infinite limit we
get L = R = omega whereas B < omega?

I gladly will offer more information if things are unclear.

Regards, WM

[R. Srinivasan]

On Oct 2, 7:35 am, Transfer Principle <david.l.wal...@lausd.net>
wrote:
> On Oct 1, 8:57 am, Frederick Williams <freddywilli...@btinternet.com>

> wrote:
>
> > RussellE wrote:
> > > Edward Nelson has posted to FOM Archive
> > > "inconsistency of P"
> > And now withdrawn it:

> > My thanks to Tao and Tausk for spotting this.
> > I withdraw my claim.

> > The consistency of P remains an open problem.

> > Ed Nelson
>
> And so it appears that Taylor has won his bet, and the status quo
> lives on.
>
> And so it's time for me to bid this thread adieu. I see a post
> above me in this thread about a claim that ZF proves that no
> inaccessibles exist. But I'm not as excited about that
> possibility as I am with PA, since the upper bounds of those
> posters who oppose ZFC (from AP's 10^603 all the way up to
> tommy1729's aleph_aleph_0) are all far short of the first
> inaccessible cardinal.
>
> So, goodbye thread.
>
>
I do not think that the final word has been said on Nelson's approach.
Nelson did withdraw his paper citing an "irreparable" error. But the
error may *possibly* turn out to be reparable, as per my (very
incomplete) understanding.

Terence Tao found the error in Nelson's proof outline, namely, that
the use of an unrestricted proof verifier in what Nelson calls the
"Chaitin machine" is unacceptable. Nelson eventually accepted Tao's
analysis and withdrew his paper. The possibility of fixing this error
by using a restricted proof verifier was also ruled out by Tao in the
following post:

http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039547

Here Tao says, summing up his argument:

"Basically, I think the conceptual error here is to believe that the
quantity l=l(T) provided by Chaitin’s theorem is monotone in the sense
that if a given l works for a theory T, then it would also work for
all restricted subtheories of T. This is not the case, because a
subtheory can in fact be much more complicated than the original
theory in the sense that it requires a much longer proof verifier."

But Panu Raatikainen has apparently disagreed with what seems to be
the essence of Tao's argument above. See the following posts by Panu:

http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039598

http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039619

http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039621

http://www.cs.nyu.edu/pipermail/fom/2011-October/015839.html

http://www.cs.nyu.edu/pipermail/fom/2011-October/015847.html

It seems to me from Panu's posts that Nelson's argument could possibly
be repaired after all, by the use of a restricted proof verifier. At
least it seems to me that Tao's argument against such use has been
challenged by Panu, and if this challenge stands up, Nelson should be
back in business.

I do have a lot to say on the idea that PA may be inconsistent, but
for the moment I will be content with pointing out just one fact.
Nelson's proof strategy to find an inconsistency in currently accepted
finitary reasoning, if it does eventually succeed, has an essential
dependence on the validity of Godel's / Turing's / Chaitin's
incompleteness theorems. My logic NAFL rejects these as infinitary,
and the finitary reasoning that NAFL describes is not susceptible to
the kind of inconsistency that Nelson hopes to nail down (and I do
hope that Nelson will eventually succeed).

RS

[WM]

On 3 Okt., 16:18, "R. Srinivasan" <sradh...@in.ibm.com> wrote:

> I do have a lot to say on the idea that PA may be inconsistent, but
> for the moment I will be content with pointing out just one fact.
> Nelson's proof strategy to find an inconsistency in currently accepted
> finitary reasoning, if it does eventually succeed, has an essential
> dependence on the validity of Godel's / Turing's / Chaitin's
> incompleteness theorems.

Goedel's incompleteness theorem is based upon the infinite hierarchy
and therefore it is false:
"Der wahre Grund für die Unvollständigkeit, welche allen formalen
Systemen der Mathematik anhaftet, liegt, wie im II. Teil dieser
Abhandlung gezeigt werden wird darin, daß die Bildung immer höherer
Typen sich ins Transfinite fortsetzen läßt [...] während in jedem
formalen System höchstens abzählbar viele vorhanden sind."
[Kurt Gödel: "Über formal unentscheidbare Sätze der Principia
Mathematica und verwandter Systeme I", Monatshefte für Mathematik und
Physik 38 (1931) p. 191]

> My logic NAFL rejects these as infinitary,
> and the finitary reasoning that NAFL describes is not susceptible to
> the kind of inconsistency that Nelson hopes to nail down (and I do
> hope that Nelson will eventually succeed).

Whether he will succeed or not: For all natural numbers that can be
used in an uninterrupted sequence, arithmetic and induction hold. If
something is contradicted, then it is the axiom system. But axiom
systems are of secondary importance.

Regards, WM

[R. Srinivasan]

<Sorry if this post appears twice. Not sure if my first attempt went
through>

On Oct 3, 8:27 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 3 Okt., 16:18, "R. Srinivasan" <sradh...@in.ibm.com> wrote:
>
> > I do have a lot to say on the idea that PA may be inconsistent, but
> > for the moment I will be content with pointing out just one fact.
> > Nelson's proof strategy to find an inconsistency in currently accepted
> > finitary reasoning, if it does eventually succeed, has an essential
> > dependence on the validity of Godel's / Turing's / Chaitin's
> > incompleteness theorems.
>
> Goedel's incompleteness theorem is based upon the infinite hierarchy
> and therefore it is false:
> "Der wahre Grund für die Unvollständigkeit, welche allen formalen
> Systemen der Mathematik anhaftet, liegt, wie im II. Teil dieser
> Abhandlung gezeigt werden wird darin, daß die Bildung immer höherer
> Typen sich ins Transfinite fortsetzen läßt [...] während in jedem
> formalen System höchstens abzählbar viele vorhanden sind."
> [Kurt Gödel: "Über formal unentscheidbare Sätze der Principia
> Mathematica und verwandter Systeme I", Monatshefte für Mathematik und
> Physik 38 (1931) p. 191]
>
>

From the point of view of my logic NAFL, "arithmetization of syntax",
which is used to formulate Godel's reasoning in very weak systems of
arithmetic, is infinitary and therefore unacceptable. The usual
argument given for justifying arithmetization is that the syntax just
consists of meaningless finite strings and there is nothing wrong with
encoding these as numbers and then quantifying over them.. But in
NAFL, the syntax (e.g. a function symbol f(x) in the language of a
theory) can never be divested of its intrinsic meaning which must
unavoidably be assigned within the theory itself. In NAFL, f(x), when
no value has been assigned to the variable x, is an infinite
superposition of all possible values of the form {f(0), f(1), .....}
and is therefore an infinite class. You can also consider <x, f(x)> as
in infinite class of ordered pairs when no value has been assigned to
x. f(x) becomes a finite object only when a specific value has been
assigned to x. Since quantification over infinite (proper) classes is
not allowed in NAFL, that effectively throws out Godel's reasoning via
"arithmetization of syntax".

>
>
> > My logic NAFL rejects these as infinitary,
> > and the finitary reasoning that NAFL describes is not susceptible to
> > the kind of inconsistency that Nelson hopes to nail down (and I do
> > hope that Nelson will eventually succeed).
>
> Whether he will succeed or not: For all natural numbers that can be
> used in an uninterrupted sequence, arithmetic and induction hold. If
> something is contradicted, then it is the axiom system. But axiom
> systems are of secondary importance.
>
>

Here I must disagree with you partially. Axioms are actually of
primary importance in NAFL. But as you say, one must make honest use
of the axioms, which in this case are just addition, multiplication,
induction, etc. *within* the natural numbers. Surely no contradiction
can arise from these. The contradictions start to appear only when one
stops being honest and starts pretending that numbers "encode"
infinite objects like functions, and quantifies over these to derive
conclusions about the theory that describes these numbers (as Godel
did).

RS