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Aug 16, 2008, 11:30:56 AM8/16/08

to

I've come across James R. Meyer's website and taken a look at his

argument at http://jamesrmeyer.com/pdfs/FFGIT_Meyer.pdf

argument at http://jamesrmeyer.com/pdfs/FFGIT_Meyer.pdf

Basically he claims there is a confusion between meta-language and

object-language in the statement of Gödel's theorem V in the 1931

paper:

"For every recursive relation R(x1, ..., xn) there is an n-ary

RELATION SIGN r (with FREE VARIABLES u1, ..., un) such that for all

numbers x1, ..., xn we have:

R(x1, ..., xn) -> Bew(Sb(u1, ..., un, r, Z(x1), ..., Z(xn)))

~R(x1, ..., xn) -> Neg(Bew(Sb(u1, ..., un, r, Z(x1), ..., Z(xn))))"

Meyer claims that Gödel refers to some object-language in which

recursive relations are expressed, so that 'x1, ..., xn' are to be

variables in the meta-language (in 'for all numbers x1, ..., xn') and

also in that purported object-language (in 'R(x1, ..., xn)'). He

claims that the purported confusion invalidates the theorem.

I've argued with him that Gödel doesn't refer to expressions of an

object-language in which recursive relations would be expressed, that

Gödel is actually referring to recursive relations themselves; that

there is no meta- and object-language in the theorem but only ordinary

English (German) extended with mathematical notation; that Gödel is

USING the expression 'R(x1, ..., xn)' as a variable for n-ary

recursive relations, not MENTIONING it.

I have even constructed some versions in which such an object-language

actually appears, in order to show Meyer that the theorem can be

clearly stated even if made about an object-language able to express

all recursive relations.

As I see it, Meyer's claim amounts to contending that statements like:

"For all constant functions f and all numbers x, y:

f(x) = f(y)"

are ill-formed, which is absurd.

Can you see any point in Meyer's contention?

Aug 16, 2008, 12:35:37 PM8/16/08

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On Aug 16, 4:30 pm, LauLuna <laureanol...@yahoo.es> wrote:

> I've argued with him that Gödel doesn't refer to expressions of an

> object-language in which recursive relations would be expressed, that

> Gödel is actually referring to recursive relations themselves; that

> there is no meta- and object-language in the theorem but only ordinary

> English (German) extended with mathematical notation

Exactly.

The essential claim here that (primitive) recursive functions can be

represented in (weak) formal arithmetics is elementarily provable (as

any modern textbook will teach Meyer).

Aug 17, 2008, 8:36:51 PM8/17/08

to

There is a deeper issue here

> > there is no meta- and object-language in the theorem but only ordinary

> > English (German) extended with mathematical notation

>

> Exactly.

> > there is no meta- and object-language in the theorem but only ordinary

> > English (German) extended with mathematical notation

>

> Exactly.

OK, fine, exactly, but this IS NOT the point!

> The essential claim here that (primitive) recursive functions can be

> represented in (weak) formal arithmetics is elementarily provable (as

> any modern textbook will teach Meyer).

Weak formal arithmetic simply is not where any of this is trying to

be.

The deeper issue here is, WTF is ANYbody doing in the original in

1931??

That was being GROPED toward. SINCE then, everybody has figured out

what is REALLY going on, as a result of which, ANY reasonably strong

PURELY FORMAL

metatheory for this realm has the property that Godel's theorem (and

Loeb's theorem,

for that matter) CAN BE PROVED IN IT ,*FORMALLY*.

And the deeper issue is simply that you canNOT ARGUE with a FORMAL

proof.

The conclusion either follows by the rules from the axioms OR IT

DOESN'T, and

this is PRIMITIVELY recursively checkable, MECHANICALLY.

So anybody WHO IS EVER trying to "argue" with it

I S A F U C K I N G I D I OT.

The deeper point is that REGARDLESS of what was going on in German in

the original in 1931, Godel's theorem is provable here&now TODAY in

ZFC as a meta-theory for PA (one simply uses ZFC sets as models for

PA), FORMALLY.

Thinking that 1931 even MATTERS AT ALL just proves you're

intellectually unfit to even be participating in the discussion.

Aug 17, 2008, 10:14:38 PM8/17/08

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george <gre...@email.unc.edu> writes:

> So anybody WHO IS EVER trying to "argue" with it

> I S A F U C K I N G I D I OT.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

I gotta say, it's the details that matter. Strange spacing adds that

extra spittle, when ALL CAPS just aren't enough.

Bravo, George!

--

"[Y]ou never understood the real role of mathematicians. The

position is one of great responsibility and power. [...] You people

have no concept of what it means to be an actual mathematician versus

pretending to be one, dreaming you understand." -- James S. Harris

Aug 18, 2008, 10:06:43 AM8/18/08

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Meyer is marketing a novel about Gödel's theorem in which a lot of

misunderstandings and awkward claims are conveyed. In the preface the

author claims with all seriousness to be the first who has really

understood Gödel's theorem.

And there is the other guy, Jeff Kegler, and his novel about Gödel's

ontologocal argument. Unfortunately, Kegler does not restrict himself

to the 'God Proof', he also says quite a deal about Gödel's

incompleteness results. Actually he condenses these two results into a

statement like:

if the world is consistent (sic), then there is a Liar sentence (sic;

Gödel's) that is true but that we cannot prove so, and we cannot prove

the world consistent either.

This, Kegler writes, is not that bad, for if the world cannot be

proven consistent, well, that's a proof that it is indeed consistent.

Does anyone knows of any decent fiction work about Gödel, Cantor,

etc. ? I enjoy that stuff whenever no gross inaccuracies make me

angry.

Regards

Aug 18, 2008, 6:14:21 PM8/18/08

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On Aug 18, 3:06 pm, LauLuna <laureanol...@yahoo.es> wrote:

> Does anyone knows of any decent fiction work about Gödel, Cantor,

> etc. ? I enjoy that stuff whenever no gross inaccuracies make me

> angry.

>

> Regards

Mathematical / logical fiction:

I like Smullyans books, (all his mathematical Puzzle books except the

last one mentions it.) My favorite is "Forever Undecided"

on Cantor there is "Satan Cantor and Infinity" (I guess it also

mentions Godel, sometimes i think Snmullyan is a bit addicted to

Godel)

Non Mathematical fiction :

Godel Esher Bach? (I am not fond of it)

Aug 18, 2008, 7:03:17 PM8/18/08

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On Aug 18, 7:06 am, LauLuna <laureanol...@yahoo.es> wrote:

> Does anyone knows of any decent fiction work about Gödel, Cantor,

> etc. ?

I like Roeg's film 'Insignificance', which has a beautiful scene in

which Einstein (Michael Emil) meets Marilyn Monroe (Theresa Russell),

also with Tony Curtis as Joe McCarthy and Gary Busey as Joe DiMaggio.

MoeBlee

Aug 18, 2008, 11:50:04 PM8/18/08

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I'm pleased that my novel about Gödel's ontological proof got

mentioned on this list. I was not very surprised that LauLuna had

reservations about my approach to Gödel. _God Proof_ is a venture

into hard SF at the boundary of math, religion & philosophy.

mentioned on this list. I was not very surprised that LauLuna had

reservations about my approach to Gödel. _God Proof_ is a venture

into hard SF at the boundary of math, religion & philosophy.

Combining math & philosophy is dicey. Throw in religion and things

get really treacherous. Add to that a writer who has decided to

deal with the whole business in the form of a novel, and even a

generous person might wonder if he's not dealing with a crank or

madman.

I took it from LauLuna's remarks that he'd not actually read _The

God Proof_, but that he had done me the favor of looking it over

to see if, despite everything, there might be something there. I

hope I don't presume on sci.logic's patience if I explain a bit why

I dealt with this material in the way I did.

Let me say that my research into modal logic

and Gödel was quite serious. I became proficient enough in the

math to assist two of the professionals then studying Gödel's

ontological proof. Melvin Fitting

(http://comet.lehman.cuny.edu/fitting/errata/book_errors/

godelbookerrors/godelerrata.pdf)

and Jordan Howard Sobel (by letter) were kind enough to acknowledge

my minor assistance. Assistance to another researcher at the

boundaries of philosophy and mathematics landed me a mention in

_Mind_ (V115, N459, p. 692). An acknowledgment does not compare

with a publication, but just the same seeing my name in the same

pages which have carried articles by Turing, Freud and James gives

me a shiver. On my own, I'm a published mathematician of minor note.

(Communications of the ACM, V29 #6, June 1986, pp. 556-558).

Far more serious qualifications than mine would be no guarantee

against error. LauLuna presents some paraphrases as evidence that

I've got Gödel wrong on the Incompleteness Theorems. For example,

LauLuna says,

[ a proof that the world cannot be proved consistent ], Kegler

writes, is not that bad, for if the world cannot be proven

consistent, well, that's a proof that it is indeed consistent.

While my original langauge was carefully chosen and I prefer it,

the paraphrase above is close enough for this purpose. LauLuna seems

to be saying that statements like this clearly demonstrate that I've

gone off the rails. I can't for the life of me see where.

Let's leaving aside my use of the philosophically-loaded term "world"

for the moment, the math is not only correct, but downright boringly

orthodox testbook stuff. Not very formally,

the argument goes like this:

1. An inconsistent system is, by definition, one with a logical

contradiction.

2. From a logical contradiction, you can deduce any statement

whatsoever. (This is the principle of explosion, very well-

established

in classical logic.)

3. Conversely, if there is any statement at all which cannot be

deduced in a system, the system must be consistent.

4. Gödel noticed that arithmetic cannot prove its own consistency.

This (the fact that there is something arithmetic cannot prove) is

a meta-proof that arithmetic is consistent. That's because an

inconsistent system proves everything, including both its own

consistency and its inconsistency.

I expect the above will be very familiar to a lot of you on sci.logic.

Now, OK, what's the point of my talking about the consistency of

worlds instead of logical systems?

If a novel is centered on math, it needs to be made real. Usually,

in math courses all the philosophical baggage is stripped off and

ignored. How good an idea this is I won't address, but it does

free up the syllabus for a march through the formal systems and

their consequences.

But Gödel felt, as I do, and as a novelist of math must, that math

is about real things of real concern to real people. That means

my narrator (Josh Bryant) is fated to tackle those philosopical

issues head-on.

Josh is taking the position that logic underlies the world of the

senses. This is not beyond debate, but it is very mainstream.

You're very hard put to justify why it's even worth an attempt to

do science unless the world of appearances is logically coherent.

And how do you do this without treating the basic laws and results

of logic as facts basic to the world?

I don't say that there aren't other approaches. But Josh's position

is very mainstream. Josh just states it a lot more clearly than is

usual. He's a character in a novel. You'd expect that.

In academia, math and philosophy are separated. Gödel's

Incompleteness Theorem is discussed as an exercise with formalisms

in the math literature. The relationship of logic to ontology is

dealt with in the philosophical literature. But if either mean

anything in reality (and in a novel, things must be meaningful

to the characters) the twain must meet.

Even in SF, writers are often just plain indifferent to accuracy.

I took great trouble to make _God Proof_ not just a book

that stirs the imagination, but one that is as accurate as a book

which avoids equations and technical language can be.

It's easy to check out how well I've succeeded: _The God Proof_ is

available as a free download (http://www.lulu.com/content/933192).

Those who want a print edition can order one from either Lulu or

Amazon.

I don't claim _The God Proof_ is inerrant. (In another of his

interpretations of Gödel's work, Josh claims that the

Second Incompleteness Theorem in fact

shows that inerrancy is possible only if inerrancy is not claimed.)

Readers might find statements which are not just informal, but plain

ol' incorrect. I'm grateful to have those pointed out.

thanks,

Jeffrey Kegler

Aug 19, 2008, 3:13:16 AM8/19/08

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If arithetic is consistent, then it cannot prove its own consistency,

as you say. This is what Godel proved.

The premise here, that arithmetic is consistent, Godel did not

prove, meta or otherwise.

This supposed meta-proof is flawed in that it has not been shown

that there is something arithmetic cannot prove. It has only

been shown that arithmetic cannot prove its consistency if it

is consistent.

Your assertion "Godel noticed that arithmetic cannot prove its

own consistency" is wrong. What is correct is "Godel noticed

that arithmetic, if consistent, cannot prove its own consistency."

It is not unusual for people to leave out the conditional clause.

Aug 19, 2008, 9:00:26 AM8/19/08

to

Joe McCarthy was nothing more than a barbaric sadist in that movie.

Sure, in every movie villains must be always ultraevil and without no

soul.

Aug 19, 2008, 9:11:44 AM8/19/08

to

On 18 elo, 17:06, LauLuna <laureanol...@yahoo.es> wrote:

>

> Does anyone knows of any decent fiction work about Gödel, Cantor,

> etc. ? I enjoy that stuff whenever no gross inaccuracies make me

> angry.

There is a book about Gauss, which I haven`t read:

http://www.amazon.com/Measuring-World-Novel-Daniel-Kehlmann/dp/0375424466

Aug 19, 2008, 12:11:10 PM8/19/08

to

Nice suggestion. Also translogi's and MoeBlee's.

Thanks.

Aug 19, 2008, 12:38:49 PM8/19/08

to

Of course McCarthy was meant to be quite unlikable in certain definite

ways. I'd have to see the movie again, though, to assess whether the

character was drawn as one-dimensionally as you describe.

MoeBlee

Aug 19, 2008, 12:45:47 PM8/19/08

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On Aug 18, 8:50 pm, jeffreykeg...@gmail.com wrote:

> Gödel noticed that arithmetic cannot prove its own consistency.

> This (the fact that there is something arithmetic cannot prove) is

> a meta-proof that arithmetic is consistent.

No, we have: IF PA (or whatever system rich enough for a certain

amount of arithmetic) is consistent THEN PA does not prove its own

consistency. That in itself is not a proof of the consistency of PA,

since in itself it does not reveal that there is a formula that PA

does not prove but only that IF PA is consistent THEN there is a

formula that PA does not prove, which is known already anyway. So a

proof of the consistency of PA must come from some other means.

MoeBlee

Aug 19, 2008, 12:55:23 PM8/19/08

to

On Aug 19, 9:11 am, LauLuna <laureanol...@yahoo.es> wrote:

I recommend Neal Stephenson (_Cryptonomicon_, _The Baroque Cycle_) and

Rudy Rucker. Rucker is a retired math professor and SF enthusiast.

He knows math and loves it.

I recommend Neal Stephenson (_Cryptonomicon_, _The Baroque Cycle_) and

Rudy Rucker. Rucker is a retired math professor and SF enthusiast.

He knows math and loves it.

Stephenson is an extremely strong writer. When he gets technical or

shows equations, he often either gets it wrong or makes a joke of it,

I can't tell which. But he's excellent at reaching the imagination.

After reading Stephenson, you feel you know what it would have been

like to hang out with Isaac Newton or Alan Turing. It's nice piece of

craftmanship when a novel nails the technical stuff (I work hard at

it), but that's not what a novel is for. If a novel fills you with

enthusiasm for the topic, and fires up the intuition, it's done it's

job. Both Rucker and Stephenson are good at this.

Rudy Rucker has a website with a very active and interesting blog:

http://www.cs.sjsu.edu/faculty/rucker/ .

hope that helps, jeffrey

Aug 19, 2008, 12:58:45 PM8/19/08

to

I just remember him punching Marilyn Monroe to the stomach with a

tight glove on his fist and because of that Marilyn Monroe had an

miscarriage.

I thought that was quite nonrealistic. Whatkind of a man would punch

Marilyn Monroe?

Aug 19, 2008, 1:13:16 PM8/19/08

to

A fictionalized Joe McCarthy.

MoeBlee

Aug 19, 2008, 2:23:06 PM8/19/08

to

MoeBlee wrote:

> jeffreykeg...@gmail.com wrote:

>

> > Gödel noticed that arithmetic cannot prove its own consistency.

> > This (the fact that there is something arithmetic cannot prove) is

> > a meta-proof that arithmetic is consistent.

>

> No, we have: IF PA (or whatever system rich enough for a certain

> amount of arithmetic) is consistent THEN PA does not prove its own

> consistency.

This is indeed the content of Godel's second incompleteness theorem,

as I said in another post. But isn't some business at the end of his

paper where he writes something like "Thus, the formula is decided by meta-mathematical

means"?

> That in itself is not a proof of the consistency of PA,

> since in itself it does not reveal that there is a formula that PA

> does not prove but only that IF PA is consistent THEN there is a

> formula that PA does not prove, which is known already anyway. So a

> proof of the consistency of PA must come from some other means.

--

hz

Aug 19, 2008, 1:27:23 PM8/19/08

to

On Tue, 19 Aug 2008 09:55:23 -0700 (PDT), jeffre...@gmail.com

<jeffre...@gmail.com> said:

> On Aug 19, 9:11 am, LauLuna <laureanol...@yahoo.es> wrote:

> I recommend Neal Stephenson (_Cryptonomicon_,

<jeffre...@gmail.com> said:

> On Aug 19, 9:11 am, LauLuna <laureanol...@yahoo.es> wrote:

> I recommend Neal Stephenson (_Cryptonomicon_,

My favorite book of the last five years by a long shot.

Aug 19, 2008, 4:51:42 PM8/19/08

to

On Aug 19, 10:27 am, Chris Menzel <cmen...@remove-this.tamu.edu>

wrote:

> On Tue, 19 Aug 2008 09:55:23 -0700 (PDT), jeffreykeg...@gmail.com

> <jeffreykeg...@gmail.com> said:

>

> > On Aug 19, 9:11 am, LauLuna <laureanol...@yahoo.es> wrote:

> > I recommend Neal Stephenson (_Cryptonomicon_,

>

> My favorite book of the last five years by a long shot.

wrote:

> On Tue, 19 Aug 2008 09:55:23 -0700 (PDT), jeffreykeg...@gmail.com

> <jeffreykeg...@gmail.com> said:

>

> > On Aug 19, 9:11 am, LauLuna <laureanol...@yahoo.es> wrote:

> > I recommend Neal Stephenson (_Cryptonomicon_,

>

> My favorite book of the last five years by a long shot.

Does it do a good job of involving some actual mathematics in the

story? If so, I think I'll read it, since I've heard good things about

it from other people too.

MoeBlee

Aug 19, 2008, 5:12:41 PM8/19/08

to

Isn't it from 1999?

How time flies when you get older...

--

Cheers,

Herman Jurjus

Aug 19, 2008, 6:24:32 PM8/19/08

to

On Tue, 19 Aug 2008 23:12:41 +0200, Herman Jurjus <hju...@hetnet.nl> said:

> Chris Menzel wrote:

>> On Tue, 19 Aug 2008 09:55:23 -0700 (PDT), jeffre...@gmail.com

>> <jeffre...@gmail.com> said:

>>> On Aug 19, 9:11 am, LauLuna <laureanol...@yahoo.es> wrote:

>>> I recommend Neal Stephenson (_Cryptonomicon_,

>>

>> My favorite book of the last five years by a long shot.

>

> Isn't it from 1999?

> Chris Menzel wrote:

>> On Tue, 19 Aug 2008 09:55:23 -0700 (PDT), jeffre...@gmail.com

>> <jeffre...@gmail.com> said:

>>> On Aug 19, 9:11 am, LauLuna <laureanol...@yahoo.es> wrote:

>>> I recommend Neal Stephenson (_Cryptonomicon_,

>>

>> My favorite book of the last five years by a long shot.

>

> Isn't it from 1999?

What I meant was: it is my favorite of the books I've *read* in the last

five years.

> How time flies when you get older...

Alas.

Aug 19, 2008, 6:29:14 PM8/19/08

to

On Tue, 19 Aug 2008 13:51:42 -0700 (PDT), MoeBlee <jazz...@hotmail.com>

said:

> On Aug 19, 10:27 am, Chris Menzel <cmen...@remove-this.tamu.edu>

> wrote:

>> On Tue, 19 Aug 2008 09:55:23 -0700 (PDT), jeffreykeg...@gmail.com

>> <jeffreykeg...@gmail.com> said:

>>

>> > On Aug 19, 9:11 am, LauLuna <laureanol...@yahoo.es> wrote:

>> > I recommend Neal Stephenson (_Cryptonomicon_,

>>

>> My favorite book of the last five years by a long shot.

>

> Does it do a good job of involving some actual mathematics in the

> story?

said:

> On Aug 19, 10:27 am, Chris Menzel <cmen...@remove-this.tamu.edu>

> wrote:

>> On Tue, 19 Aug 2008 09:55:23 -0700 (PDT), jeffreykeg...@gmail.com

>> <jeffreykeg...@gmail.com> said:

>>

>> > On Aug 19, 9:11 am, LauLuna <laureanol...@yahoo.es> wrote:

>> > I recommend Neal Stephenson (_Cryptonomicon_,

>>

>> My favorite book of the last five years by a long shot.

>

> Does it do a good job of involving some actual mathematics in the

> story?

More computer science than math proper. To the best of my recollection

he got things largely right.

> If so, I think I'll read it, since I've heard good things about

> it from other people too.

It's a fantastic read, regardless of the math/CS content. Some of the

funniest and truest bits have to do with the geek mindset. Look for the

part about the proper technique for eating Cap'n Crunch cereal. The

story (well, stories, really) is *way* broader than the geek stuff,

however.

Aug 20, 2008, 9:57:11 AM8/20/08

to

>

> - Show quoted text -

Thanks a lot.

Let me add that I am enjoying your novel; though I may sometimes get a

little angry about some passages, I am delighted to have found it

available online. It's certainly a pleasant read.

And I'm learning a lot of slang too ;)

Aug 20, 2008, 3:44:58 PM8/20/08

to

On Aug 19, 3:29 pm, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:

> It's a fantastic read, regardless of the math/CS content.

Okay, thanks, Chris. I'll grab a copy of it.

MoeBlee

Aug 22, 2008, 6:49:40 AM8/22/08

to

herbzet <her...@gmail.com> writes:

> This is indeed the content of Godel's second incompleteness theorem,

> as I said in another post. But isn't some business at the end of

> his paper where he writes something like "Thus, the formula is

> decided by meta-mathematical means"?

Yes, the truth of the Gödel sentence of P is established by the first

incompleteness theorem, given that we know P is consistent. The

consistency of P is of course not in any way derived from the

incompleteness theorem.

--

Aatu Koskensilta (aatu.kos...@uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"

- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Aug 23, 2008, 9:05:28 AM8/23/08

to

Aatu Koskensilta wrote:

> herbzet writes:

>

> > This is indeed the content of Godel's second incompleteness theorem,

> > as I said in another post. But isn't [there] some business at the end

> > of his paper where he writes something like "Thus, the formula is

> > decided by meta-mathematical means"?

>

> Yes, the truth of the Gödel sentence of P is established by the first

> incompleteness theorem, given that we know P is consistent. The

> consistency of P is of course not in any way derived from the

> incompleteness theorem.

Right, but let me ask, whatever it is that "we" may know about

the consistency of P, what grounds did Godel give in the paper

for this knowledge? What I have read (in translation) seems

quite obscure.

--

hz

Aug 23, 2008, 4:29:42 PM8/23/08

to

>

> - Show quoted text -

Browsed "The god proof" online,

Would like to have the printed version.

Liked the stories about the Vienna Circle, Einstein Spinoza Leibniz

ed. (And they do confirm to what i have read elsewhere)

Not so sure about all the Tibetan Buddhism in it, (Skipped most of it)

Nice read for when i have a lay in,

Aug 23, 2008, 7:48:22 PM8/23/08

to

On Aug 23, 2:05 pm, herbzet <herb...@gmail.com> wrote:

> Right, but let me ask, whatever it is that "we" may know about

> the consistency of P, what grounds did Godel give in the paper

> for this knowledge? What I have read (in translation) seems

> quite obscure.

Remember, Gödel officially proved a conditional claim: IF an

axiomatized system containing enough arithmetic is (omega-)consistent,

THEN it is negation incomplete.

He didn't have to argue that familiar systems like PM or Zermelo set

theory *are* (omega)-consistent (though they doubtless are). The point

of the theorem is that those working in Hilbert's program in effect

wanted complete formal theories that contain enough arithmetic and are

(omega)-consistent. Gödel first theorem shows that Hilbertians can't

get they want. Which is why Hilbert was pissed off when he heard about

the result! :-)

Aug 23, 2008, 11:11:10 PM8/23/08

to

Gödel's papers are not the best source for this background. Gödel was

basically a philosopher, but he had a fear (to the point of pathology)

of controversy. Gödel always assumed his readers know the current

philosophical background cold, and tried to let the math speak for

itself. Where Gödel thought his philosophical point of view would not

be well received (which was often), he kept silent or restricted

himself to making his point indirectly.

On this, I've currently looking at Wilfred Sieg's "Hilbert's Programs

1917-1922" from the Bulletin of Symbolic Logic, and Sieg works

seemsvery informative. According to Sieg and the Stanford

Encyclopedia, Poincaré had argued in 1906 that proving the consistency

of arithmetic would just be using induction to prove induction (see

Sieg, p. 7). Hilbert's finitist point of view (according to Sieg) was

a long delayed, and problematic attempt to get around the circularity

(forgive the pun). An introduction to this discussion is in the

online Stanford Encyclopedia article on "Hilbert's Program".

I'm clearly going to have to reread _God Proof_ to see what exactly

what I say, and what I need to revise. While apparently the

formalists were willing to the risk the accusation of circularity in

carrying out the formalist agenda, my belief (expressed earlier in

this thread) that the same kind of circularity would be seen as

orthodox when it came to proofs of consistency that require

metamathematical steps, is clearly wrong. The nearest I can find is

Smullyan's "No, the fact that PA, if consistent, cannot prove its own

consistency -- this fact does not constitute the slightest rational

grounds for doubting the consistency of PA." This is not quite the

same as a claim that lack of a consistency proof within the formality

is metamathematical evidence of consistency.

Which brings up the question, why do we think PA is consistent? The

consensus on sci.logic? Gentzen's proof? But I've a bad habit of

writing long posts, so I'll leave it for another time.

jeffrey kegler

Aug 23, 2008, 11:20:16 PM8/23/08

to

Peter_Smith wrote:

> herbzet wrote:

>

> > Right, but let me ask, whatever it is that "we" may know about

> > the consistency of P, what grounds did Godel give in the paper

> > for this knowledge? What I have read (in translation) seems

> > quite obscure.

>

> Remember, Gödel officially proved a conditional claim: IF an

> axiomatized system containing enough arithmetic is (omega-)consistent,

> THEN it is negation incomplete.

Yes, I've mentioned that previously in this thread.

> He didn't have to argue that familiar systems like PM or Zermelo set

> theory *are* (omega)-consistent (though they doubtless are). The point

> of the theorem is that those working in Hilbert's program in effect

> wanted complete formal theories that contain enough arithmetic and are

> (omega)-consistent. Gödel first theorem shows that Hilbertians can't

> get they want. Which is why Hilbert was pissed off when he heard about

> the result! :-)

What you say is undoubtedly so; I'm just asking what Godel was

saying in that obscure passage towards the end where he says something

like "Thus, the proposition is decided by metamathematical means". I'm

sure I'm not the only one who finds that passage obscure.

What is at issue here is the claim earlier in the thread that Godel

gave a meta-proof of the consistency of arithmetic (whatever "meta-proof"

might be interpreted to mean here). So I'm asking what Godel was

saying in his paper of 1931 towards the end there.

--

hz

Aug 24, 2008, 12:06:34 AM8/24/08

to

The lack of consistency proof within the formality _would_ be

a metaproof of the consistency of the formality, if you could

establish that the formalism lacks such a proof.

> Which brings up the question, why do we think PA is consistent? The

> consensus on sci.logic? Gentzen's proof? But I've a bad habit of

> writing long posts, so I'll leave it for another time.

>

> jeffrey kegler

Well personally, and I'm not an expert, I have a clear picture,

a model, of the naturals in my head. That's pretty convincing

to me.

I could say too that the set omega is also clearly a model of

of PA. But that just shifts the burden to another foot --

is ZF set theory consistent?

I guess you have to just get used to the fact that you're not going

to get a final demonstration of consistency of these systems -- one

can only definitively demonstrate inconsistency. PA and ZFC have

the weight of experience behind them -- it's hardly imaginable that

PA could be inconsistent. If a contradiction turned up in ZFC (happened

once already in naive set theory, pretty shocking) we'd just patch it up

again and go on.

--

hz

Aug 24, 2008, 12:18:23 AM8/24/08

to

Are you referring to this passage, "Thus the proposition that is

undecidable IN THE SYSTEM PM still was decided by metamathematical

considerations. The precise analysis of this curious situation leads

to surprising results concerning consistency proofs for formal

systems, ..."? [ Note: where I have caps, Gödel has italics ] This

is at the end of section 1 of his 1931 paper.

Alas, this does not rescue me from having falsely claimed an

orthodoxly-accepted meta-proof of consistency. (And it wasn't the

source of my error.) "The proposition" referred to is an example of

another undecidable proposition in PM (PM is the system of Whitehead &

Russell's _Principia Mathematica_). Gödel says that even though his

proposition is not decidable in PM -- that is, cannot be proved in PM

to be either or true or false, it still can be decided

"metamathematically", that is, by arguments which are not formalized.

(I'll return to this business of meta-mathematics.)

This undecidability in PM, which Gödel has just announced, he says in

the next sentence will have "surprising results concerning consistency

proofs for formal systems". He does NOT say he has a consistency

proof or meta-proof, but only "surprising results" about consistency

proofs, and as usual he's choosing his words carefully. So no claimed

meta-proof of consistency.

The meta-proof Gödel does claim is that his formally undecidable

proposition is in fact, in meta-mathematical terms, true. Gödel has

shown a certain formula is not provable in PM. This formula, when

interpreted meta-mathematically, says that it is not provable in PM.

So meta-mathematically (translation: by our intuitively obvious

notions of what truth is), the proposition is clearly true, though we

can't prove it in PM. In fact, it's true, because we can't prove it

in PM, and because we can prove we can't prove it. So we have a meta-

proof, but no proof.

Note that "meta-mathematical" here means "appealing to notions of

truth outside of the formalism". Some schools of philosophy of

mathematics have claimed there is no such thing as informal

mathematical truth. Gödel disagreed, though here he's not coming out

and saying so directly.

jeffrey kegler

Aug 24, 2008, 1:07:38 AM8/24/08

to

jeffre...@gmail.com wrote:

That's probably what I'm recalling.

> Alas, this does not rescue me from having falsely claimed an

> orthodoxly-accepted meta-proof of consistency. (And it wasn't the

> source of my error.) "The proposition" referred to is an example of

> another undecidable proposition in PM (PM is the system of Whitehead &

> Russell's _Principia Mathematica_). Gödel says that even though his

> proposition is not decidable in PM -- that is, cannot be proved in PM

> to be either or true or false, it still can be decided

> "metamathematically", that is, by arguments which are not formalized.

> (I'll return to this business of meta-mathematics.)

>

> This undecidability in PM, which Gödel has just announced, he says in

> the next sentence will have "surprising results concerning consistency

> proofs for formal systems". He does NOT say he has a consistency

> proof or meta-proof, but only "surprising results" about consistency

> proofs, and as usual he's choosing his words carefully. So no claimed

> meta-proof of consistency.

>

> The meta-proof Gödel does claim is that his formally undecidable

> proposition is in fact, in meta-mathematical terms, true. Gödel has

> shown a certain formula is not provable in PM.

I doubt it. That would prove "metamathematically" that PM is

consistent. He probably shows this proposition is not provable

in PM on the condition that PM is consistent.

But please continue:

> This formula, when

> interpreted meta-mathematically, says that it is not provable in PM.

> So meta-mathematically (translation: by our intuitively obvious

> notions of what truth is), the proposition is clearly true, though we

> can't prove it in PM. In fact, it's true, because we can't prove it

> in PM, and because we can prove we can't prove it. So we have a meta-

> proof, but no proof.

>

> Note that "meta-mathematical" here means "appealing to notions of

> truth outside of the formalism". Some schools of philosophy of

> mathematics have claimed there is no such thing as informal

> mathematical truth. Gödel disagreed, though here he's not coming out

> and saying so directly.

>

> jeffrey kegler

Good post, despite the fact that I doubt one point.

A formula G that can be interpreted as saying "G has no proof in

theory T" will be true just if G has no proof in theory T.

It will be true just if it has no proof. That's the beauty part.

--

hz

Aug 24, 2008, 3:19:00 PM8/24/08

to

On Aug 24, 6:07 am, herbzet <herb...@gmail.com> wrote:> hz- Hide quoted text -

>

> - Show quoted text -- Hide quoted text -

>

> - Show quoted text -

>

> - Show quoted text -- Hide quoted text -

>

> - Show quoted text -

You (and Jeffrey) have me wondering now.

As far as i do recall Godel used a system P (what to be honnest i do

not know the axiomatisation from but i guess i can find it)

Later proofs use PA (Peano arithmetic) or Q ( Robinson arithmetic, Pa

without the axiom scheme of induction)

But here is suddenly a referal to PM

PA (Peano arithmetic) and PM (sytem of the Principa Mathematica) are

different things.

and I am not aware if and howfar they are identical.

(It is just that i want clarity here)

I think around the 1930 there was an allergy for model theory only

Hilbert style proofs were deemed good enough. (I sound negative here

but i do in general agree with it)

I think that the constructive/ intuitionistic school (Brouwer, Heyting

cs.) and Modal Logic (Lewis book was just comming out) were the main

reason for this allergy.

Also the Principa Mathematica was only using this kind of proof.

Truthtables (which i think were the first part of model theory) only

came in use later (I still like to think that Post was before

Wittgenstein, but the race is close and neither used the form that

they are now tought)

Gentzens article on natural deduction and Sequent calculi was not even

published.

I am also unsure how to prove (in)completenes in another than Hilbert

style proof is possible,

(Is there a proofstyle that is not reducable to a Hilbert style

proof?, maybe some Modal logics defined by accesibility relations that

are not replaceble by axioms)

Still a lot to think about, and that is what a good book does.

Greetings

Aug 24, 2008, 2:26:13 PM8/24/08

to

On Sat, 23 Aug 2008 20:11:10 -0700 (PDT), jeffre...@gmail.com

<jeffre...@gmail.com> said:

>

> Gödel's papers are not the best source for this background. Gödel was

> basically a philosopher,

<jeffre...@gmail.com> said:

>

> Gödel's papers are not the best source for this background. Gödel was

> basically a philosopher,

I'm not sure what you think it takes to be "basically a philosopher",

but me, I'd sure say that Gödel was basically a logician.

Aug 25, 2008, 1:48:30 AM8/25/08

to

translogi wrote:

> You (and Jeffrey) have me wondering now.

>

> As far as i do recall Godel used a system P (what to be honnest i do

> not know the axiomatisation from but i guess i can find it)

>

> Later proofs use PA (Peano arithmetic) or Q ( Robinson arithmetic, Pa

> without the axiom scheme of induction)

>

> But here is suddenly a referal to PM

>

> PA (Peano arithmetic) and PM (sytem of the Principa Mathematica) are

> different things.

> and I am not aware if and howfar they are identical.

>

> (It is just that i want clarity here)

Godel's paper is on the web as a .pdf file at

http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf

transated to english. It contains parts 1 and 2 (of 4).

In part 1 Godel sketches the proof as if it were in the theory

PM, which was familiar to his readers. It's just a quick sketch

and it assumes without mention the consistency of PM, and indeed

the truth of the theorems of PM, as his readers would.

After indicating the construction of the self-referencing proposition

R_q(q) which states its own unprovability, he shows that it must

be undecidable in PM. He remarks that from its undecidability

its truth follows immediately, since it is unprovable.

In part 2 he begins the actual proof:

"We will now exactly implement the proof sketched above, and

will first give an exact description of the formal system P,

for which we want to show the existence of undecidable theorems.

By and large, P is the system that you get by building the logic

of PM on top the Peano axioms (numbers as individuals, successor-

relation as undefined basic concept)."

Basically he's just getting rid of the parts of PM not needed

for his number theoretical proof; the natural numbers will not

be constructed out of classes -- their existence is just assumed

via the Peano axioms.

--

hz

Aug 26, 2008, 5:59:24 AM8/26/08

to

herbzet wrote:

> Right, but let me ask, whatever it is that "we" may know about

> the consistency of P, what grounds did Godel give in the paper

> for this knowledge?

> Right, but let me ask, whatever it is that "we" may know about

> the consistency of P, what grounds did Godel give in the paper

> for this knowledge?

Gödel didn't give any grounds in the paper, rather, he used the

consistency of P which is in any case obvious: P is true in V_omega

+omega.

Aug 26, 2008, 6:29:34 PM8/26/08

to

On Aug 24, 6:18 am, jeffreykeg...@gmail.com wrote:

> Note that "meta-mathematical" here means "appealing to notions of

> truth outside of the formalism". Some schools of philosophy of

> mathematics have claimed there is no such thing as informal

> mathematical truth. Gödel disagreed, though here he's not coming out

> and saying so directly.

Consider that the meta-math of a particular system can be formalized

in another system. Nevertheless, it's true that Gödel believed human

reason can eventually prove any mathematical truth, a feat no correct

formal system can achieve.

Regards

Aug 26, 2008, 11:19:50 PM8/26/08

to

LauLuna wrote:

> On Aug 24, 6:18 am, jeffreykeg...@gmail.com wrote:

>

>> Note that "meta-mathematical" here means "appealing to notions of

>> truth outside of the formalism". Some schools of philosophy of

>> mathematics have claimed there is no such thing as informal

>> mathematical truth. Gödel disagreed, though here he's not coming out

>> and saying so directly.

>

> Consider that the meta-math of a particular system can be formalized

> in another system.

> On Aug 24, 6:18 am, jeffreykeg...@gmail.com wrote:

>

>> Note that "meta-mathematical" here means "appealing to notions of

>> truth outside of the formalism". Some schools of philosophy of

>> mathematics have claimed there is no such thing as informal

>> mathematical truth. Gödel disagreed, though here he's not coming out

>> and saying so directly.

>

> Consider that the meta-math of a particular system can be formalized

> in another system.

The meta-math of which could be formalized yet in another system,

the meta-math of which could be formalized yet in another system,

...

to infinity. It seems to be the case.

> Nevertheless, it's true that Gödel believed human

> reason can eventually prove any mathematical truth, a feat no correct

> formal system can achieve.

How could a finite mortal being as human "prove _any_ mathematical truth"?

For example, could a human being prove the truth of GC, or of cGC, or the

truth of there's neither?

--

"To discover the proper approach to mathematical logic,

we must therefore examine the methods of the mathematician."

(Shoenfield, "Mathematical Logic")

Sep 5, 2008, 2:45:35 PM9/5/08

to

What you ALL ought to be doing is going back to reexamine the terms of

the original set theory debate. Godel was a very sloppy investigator

of these issues, which are only now coming to light in the renaissance

in the historiography of set theory which we are currently enjoying.

Godel comes off very badly--even the editors in the Feferman edition

make fun of him. Quite the intellectual slob and cafe intellectual.

You should CERTAINLY read A. Garciadiego, BERTRAND RUSSELL AND THE

ORIGINS OF THE SET-THEORETIC 'PARADOXES.'

And by the way, when you find the constructivist intervention in the

Pythagorean theorem, let me know first thing!

SSRN-Paradox, Natural Mathematics, Relativity and Twentieth ...Apr 18,

2006 ... Ryskamp, John Henry, "Paradox, Natural Mathematics,

Relativity and Twentieth-Century Ideas" . Available at SSRN:

http://ssrn.com/abstract= ...

papers.ssrn.com/sol3/papers.cfm?abstract_id=897085 - Similar pages -

Note this

by J RYSKAMP - All 3 versions

Sep 5, 2008, 2:53:02 PM9/5/08

to

On Sep 5, 11:45 am, jrysk...@gmail.com wrote:

> What you ALL ought to be doing is going back to reexamine the terms of

> the original set theory debate. Godel was a very sloppy investigator

> of these issues, which are only now coming to light in the renaissance

> in the historiography of set theory which we are currently enjoying.

> Godel comes off very badly--even the editors in the Feferman edition

> make fun of him. Quite the intellectual slob and cafe intellectual.

Why "OUGHT" [emphasis added] such claims be of concern to me? I mean,

how do the bear upon the correctness of any particular mathematical

arguments?

MoeBlee

Sep 6, 2008, 2:39:43 AM9/6/08

to

On Aug 23, 7:48 pm, Peter_Smith <ps...@cam.ac.uk> wrote:

> Remember, Gödel officially proved a conditional claim: IF an

> axiomatized system containing enough arithmetic is (omega-)consistent,

> THEN it is negation incomplete.

>

> He didn't have to argue that familiar systems like PM or Zermelo set

> theory *are* (omega)-consistent (though they doubtless are).

> Remember, Gödel officially proved a conditional claim: IF an

> axiomatized system containing enough arithmetic is (omega-)consistent,

> THEN it is negation incomplete.

>

> He didn't have to argue that familiar systems like PM or Zermelo set

> theory *are* (omega)-consistent (though they doubtless are).

Indeed, he didn't, and more to the point, NEITHER DOES ANYONE ELSE.

But plenty of people here, historically, most notoriously Torkel

Franzen and

Aatu Koskensilta, HAVE felt the NEED to argue this very thing.

It is one thing to say (as you did) that it is doubtless that PA is

consistent.

It is ENTIRELY ANOTHER thing to say the formalized Con(PA) "is true",

since

there are models of PA in which that sentence is false, and since that

very thing

is basically what G2 proves.

Sep 6, 2008, 2:42:11 AM9/6/08

to

On Aug 26, 5:59 am, aatu.koskensi...@xortec.fi wrote:

> Gödel didn't give any grounds in the paper, rather, he used the

> consistency of P which is in any case obvious: P is true in V_omega

> +omega.

> Gödel didn't give any grounds in the paper, rather, he used the

> consistency of P which is in any case obvious: P is true in V_omega

> +omega.

And how was THAT proved?

And why was anybody supposed to believe that THAT system was

consistent?

More to the point, if anything is true, then you don't NEED to say "in

V_omega+omega" OR in ANYwhere ELSE! If you DO have to say that it is

true "in X", for ANY X, then the only reason WHY you have to say this

is that there ALSO exists a Y with the property that it is NOT true

"in Y"! And if there is a Y wherein it is NOT true, then saying that

it IS true is just WRONG.

Sep 5, 2008, 9:21:16 PM9/5/08

to

On Fri, 5 Sep 2008 11:45:35 -0700 (PDT), jrys...@gmail.com

<jrys...@gmail.com> said:

> On Aug 20, 12:44 pm, MoeBlee <jazzm...@hotmail.com> wrote:

>> On Aug 19, 3:29 pm, Chris Menzel <cmen...@remove-this.tamu.edu>

>> wrote:

>>

>> > It's a fantastic read, regardless of the math/CS content.

>>

>> Okay, thanks, Chris. I'll grab a copy of it.

>>

>> MoeBlee

>

> What you ALL ought to be doing is going back to reexamine the terms of

> the original set theory debate.

<jrys...@gmail.com> said:

> On Aug 20, 12:44 pm, MoeBlee <jazzm...@hotmail.com> wrote:

>> On Aug 19, 3:29 pm, Chris Menzel <cmen...@remove-this.tamu.edu>

>> wrote:

>>

>> > It's a fantastic read, regardless of the math/CS content.

>>

>> Okay, thanks, Chris. I'll grab a copy of it.

>>

>> MoeBlee

>

> What you ALL ought to be doing is going back to reexamine the terms of

> the original set theory debate.

You seem to be implying that one can't both enjoy fine fiction and keep

up with recent historical scholarship.

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