Meyer's Argument against Gödel's Theorem
LauLuna
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?
Peter_Smith
> 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).
george
> > 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.
Jesse F. Hughes
> 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
LauLuna
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
translogi
> 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)
MoeBlee
> 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
jeffre...@gmail.com
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
herbzet
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.
Gc
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.
Gc
>
> 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
LauLuna
Nice suggestion. Also translogi's and MoeBlee's.
Thanks.
MoeBlee
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
MoeBlee
> 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
jeffre...@gmail.com
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
Gc
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?
MoeBlee
A fictionalized Joe McCarthy.
MoeBlee
herbzet
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
Chris Menzel
<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.
MoeBlee
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
Herman Jurjus
Isn't it from 1999?
How time flies when you get older...
--
Cheers,
Herman Jurjus
Chris Menzel
> 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.
Chris Menzel
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.
LauLuna
>
> - 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 ;)
MoeBlee
> It's a fantastic read, regardless of the math/CS content.
Okay, thanks, Chris. I'll grab a copy of it.
MoeBlee
Aatu Koskensilta
> 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
herbzet
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
translogi
>
> - 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,
Peter_Smith
> 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! :-)
jeffre...@gmail.com
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
herbzet
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
herbzet
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
jeffre...@gmail.com
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
herbzet
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
translogi
>
> - 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
Chris Menzel
<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.
herbzet
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
aatu.kos...@xortec.fi
> 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.
LauLuna
> 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
Nam Nguyen
> 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")
jrys...@gmail.com
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
MoeBlee
> 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
george
> 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.
george
> 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.
Chris Menzel
<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.