Wiles' proof of FLT and AC
Edwin Clark
Gene W. Smith
(Edwin Clark) writes:
>Does Professor Wiles' proof require or use the axiom of choice?
Here's a challenge to the logicians--show that enough algebraic
geometry can be gotten out of the Peano postulates to give everything
you need. Then peddle the result as the "elementary proof" that
people say is lacking.
--
Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University
gsm...@kalliope.iwr.uni-heidelberg.de
Frederick W. Umminger
In article <20gdrq$k...@suntan.eng.usf.edu> you write:
>Does Professor Wiles' proof require or use the axiom of choice?
>
It clearly does not ultimately require it. FLT is
true unless there exists a counterexample, and any counterexample
can be proven to be such w/o using the axiom of choice. By the
consistency of AC, a proof using AC shows that no such counterexamples
exist, and hence proves FLT even in models of set theory in which AC
is false.
I have no idea whether or not Wiles actually invokes AC or not.
-Frederick Umminger
Matthew P Wiener
>>Does Professor Wiles' proof require or use the axiom of choice?
As another poster mentioned, Wiles' proof, if it uses AC, does not
need AC itself to say anything regarding FLT. I presume that AC is
not even needed for S-T-W.
>Here's a challenge to the logicians--show that enough algebraic
>geometry can be gotten out of the Peano postulates to give everything
>you need.
Actually, number theorists could probably do this themselves.
One special case of this appears in Takeuti TWO APPLICATIONS OF LOGIC
TO MATHEMATICS. He shows that analytic number theory is conservative
over PA. That is, he spells out in detail how to transfer any normal
looking proof using one variable complex analysis into a finitistic
induction style Peano arithmetic proof. In particular, Hardy's negative
prediction regarding elementary proofs of PNT was not just incorrect,
but flawed to its core.
The other possibility is that FLT is not provable in PA. Several
negative results regarding ways to prove FLT have been mentioned.
Perhaps the methodic counterexamples can be cobbled together into a
non-standard model of PA and not-FLT. Even if they don't model PA,
but only a large fragment, that would be of independent interest.
--
-Matthew P Wiener (wee...@sagi.wistar.upenn.edu)
Tal Kubo
>
Of course it does. From the proof sketches already posted, we can see that
it uses homological algebra and algebraic geometry, probably including
sheaves and the etale topology. So just track down all the references in
Wiles' forthcoming paper, the references to their references, and so on.
With probability 1 the calculation of some homology group will reference a
theorem whose proof uses free implies projective or some such thing. QED.
So it does indeed "use" AC, which is thus seen to be a de facto standard in
number theory.
Matthew P Wiener
>In article <1993Jun26.1...@sun0.urz.uni-heidelberg.de>, gsmith@lauren (Gene W. Smith) writes:
>>Here's a challenge to the logicians--show that enough algebraic
>>geometry can be gotten out of the Peano postulates to give everything
>>you need.
>Actually, number theorists could probably do this themselves. One
>special case of this appears in Takeuti TWO APPLICATIONS OF LOGIC TO
>MATHEMATICS.
Looking over Takeuti's book, I found two further comments of possible
interest.
> He shows that analytic number theory is conservative
>over PA. That is, he spells out in detail how to transfer any normal
>looking proof using one variable complex analysis into a finitistic
>induction style Peano arithmetic proof.
"Any normal looking proof" excludes the axiom of choice. This rules
out the Arzela-Ascoli theorem and the Riemann mapping theorem, although
particular instances could be analyzed on their own. Also, topological
arguments of the Riemann surface sort aren't covered.
> In particular, Hardy's negative
>prediction regarding elementary proofs of PNT was not just incorrect,
>but flawed to its core.
OK, so it's deeply flawed, but not to the core.
>The other possibility is that FLT is not provable in PA. Several
>negative results regarding ways to prove FLT have been mentioned.
>Perhaps the methodic counterexamples can be cobbled together into a
>non-standard model of PA and not-FLT. Even if they don't model PA,
>but only a large fragment, that would be of independent interest.
Takeuti refers, when conjecturing that FLT is independent of a certain
fragment of PA, to J C Shepherdson "Non-standard models for fragments
of number theory" THE THEORY OF MODELS (North-Holland, 1970) pp342-358.
He says Shepherdson proves something weaker, but whether that refers
to FLT in an even weaker fragment, or the independence of something
more general than FLT, is unclear, and our library doesn't have it.
Matthew P Wiener
>>Does Professor Wiles' proof require or use the axiom of choice?
>Of course it does. From the proof sketches already posted, [...]
Sarcasm aside, sidestepping such buried uses would be needed to verify
whether FLT is provable in PA.
Jan Johannsen
>In article <133...@netnews.upenn.edu>, weemba@sagi (Matthew P Wiener) writes:
>>In article <1993Jun26.1...@sun0.urz.uni-heidelberg.de>, gsmith@lauren (Gene W. Smith) writes:
>>>Here's a challenge to the logicians--show that enough algebraic
>>>geometry can be gotten out of the Peano postulates to give everything
>>>you need.
[some stuff deleted]
>>The other possibility is that FLT is not provable in PA. Several
>>negative results regarding ways to prove FLT have been mentioned.
>>Perhaps the methodic counterexamples can be cobbled together into a
>>non-standard model of PA and not-FLT. Even if they don't model PA,
>>but only a large fragment, that would be of independent interest.
>Takeuti refers, when conjecturing that FLT is independent of a certain
>fragment of PA, to J C Shepherdson "Non-standard models for fragments
>of number theory" THE THEORY OF MODELS (North-Holland, 1970) pp342-358.
>He says Shepherdson proves something weaker, but whether that refers
>to FLT in an even weaker fragment, or the independence of something
>more general than FLT, is unclear, and our library doesn't have it.
Shepherdson proves that there is a non-standard model of the fragment of PA
which has only induction for quantifier-free formulae in which the equations
x^3 + y^3 = z^3 and x^2 = 2y^2
have non-trivial solutions. Hence even very simple cases of FLT, as well as
rather trivial things like the irrationality of the squareroot of 2
cannot be proved in that fragment.
BTW, I have heard rumours that the Wiles' Proof of FLT has been withdrawn.
Is there something about this ?
Jan
---
--------------------------------------------------------------------------------
"... there ain't no devil, that's just God when he's drunk."
(Tom Waits "Heartattack and Vine")
--------------------------------------------------------------------------------
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Martensstrasse 3, D-8520 Erlangen, Germany
Tel.: 49-9131-857688
Email: joha...@informatik.uni-erlangen.de
--------------------------------------------------------------------------------
Note: from July 1 on the number 8520 has to be replaced by 91058 !
Torsten Ekedahl
Actually, from a formal point of view it is even worse. The only
published complete presentation of etale cohomology, SGA*, uses the
existence of universa (sets which are models for set theory) and even
hierarchies of universa. It is fairly clear, I would say, that for any
concrete application (at the level of Wiles' proof, say) certainly would
not need the full strength of that assumption, apart from the fact
that it is mainly introduced to get nicer statements (so much for a
few no-so average mathematician's for set theory).
The other direction, that of getting away even from AC is more
interesting. It would seem that AC is inextricably linked in, even the
existence of enough injectives in a category of sheaves probably needs
AC (may even imply it?). On the other hand one may use hyper-Cech
coverings (and even Cech coverings in most cases by a result of Artin)
to compute etale cohomology. I know that Deligne spent some time
proving that etale cohomology with finite coefficients is
constructible (probably not even using AC to prove it) which is much
stronger. When trying to prove l-adic cohomology constructible he got
stuck on bounding constructively the exponent on torsion but we can
get existence without AC even if not constructively.
That, of course, will not banish AC from Wiles' proof. The published
proof of Deligne for the function field Riemann hypothesis (which may
get used though perhaps the 1-dimensional case of Weil is enough or
even Hasses for genus 1) uses another of the recently hotly discussed
consequences of AC -- an embedding of the field of p-adic integers
into the complex numbers, this time, however, Deligne discusses this
in a note and sketches how to avoid it (incidentally he doesn't believe
in the existence of such an embedding). This still does not guarantee
that other authors have been as careful.
All in all I would say that it is not clear that AC would really be
needed for something like Wiles' proof (this is evidently a guess as I
haven't seen it) but I don't see anyone going through all the
references trying to remove all uses. Maybe, when someone builds a
usable proof checker a "Remove uses of AC" option could be provided...
--
Torsten Ekedahl
te...@matematik.su.se
Charles Yeomans[x2317]
>In article <1993Jun28.0...@husc14.harvard.edu> ku...@zariski.harvard.edu (Tal Kubo) writes:
>
>> In article <20gdrq$k...@suntan.eng.usf.edu> ecl...@gauss.math.usf.edu. (Edwin Clark) writes:
>> >Does Professor Wiles' proof require or use the axiom of choice? >
>>
>> Of course it does. From the proof sketches already posted, we can
>> see that it uses homological algebra and algebraic geometry,
>> probably including sheaves and the etale topology. So just track
>> down all the references in Wiles' forthcoming paper, the references
>> to their references, and so on. With probability 1 the calculation
>> of some homology group will reference a theorem whose proof uses
>> free implies projective or some such thing. QED. So it does indeed
>> "use" AC, which is thus seen to be a de facto standard in number
>> theory.
>
>Actually, from a formal point of view it is even worse. The only
>published complete presentation of etale cohomology, SGA*, uses the
>existence of universa (sets which are models for set theory) and even
Charles Yeomans
.
Centro Studi Univ California
>In article <20gdrq$k...@suntan.eng.usf.edu> you write:
>>Does Professor Wiles' proof require or use the axiom of choice?
> It clearly does not ultimately require it. FLT is
>true unless there exists a counterexample, and any counterexample
>can be proven to be such w/o using the axiom of choice. By the
>consistency of AC, a proof using AC shows that no such counterexamples
>exist, and hence proves FLT even in models of set theory in which AC
>is false.
Note that this requires going a little beyond the consistency strength
of the original theory. For example, suppose that Wiles' proof is
done in ZFC, and you want to show using this argument that it can
be done without AC, because Con(ZF) --> Con(ZFC). The argument above
works -- but not in ZF, since you need the hypothesis Con(ZF), which
is not a theorem of ZF. So you don't get a proof from ZF, but from
the slightly stronger theory ZF+Con(ZF).
On the other hand any statement of number theory (*not* just Pi^0_1 sentences
like FLT, true just in case they have no computable counterexamples) that
can be proved in ZFC, can also be proved in ZF. This is because ZF
proves that L is a (proper class) model of ZFC, so just relativize the
ZFC-proof to L. Of course the final theorem will also be relativized
to L, but since the final theorem talks only about numbers, and numbers
are all in L, you're OK.
What this basically means is that AC is never fundamentally necessary
for number theory in ZFC, but you're free to use it without risk whenever
it makes the proof more convenient.
This argument works for ZFC+large cardinals up to weakly compact or
a little beyond. Beyond that you can probably substitute other inner
models for L up to the superstrongs. Beyond that I don't know (off the
top of my head) any proof that AC can't affect number theory.
--
La superstizione porta sfortuna -- Umberto Eco
Mike Oliver
Torsten Ekedahl
>published complete presentation of etale cohomology, SGA*, uses the
>existence of universa (sets which are models for set theory) and even
How is Milne's book on etale cohomology not complete?
Well, if may memory serves me right (which it may not, if a disputed
once more I will have to actually go back to Milne) he does refer to SGA4
at some crucial places - the theorem on cohomological dimension of
fields is one candidate, of course this is due Tate but I am not sure
if he ever published it, another one is that one may use hyper-Cech
coverings to compute cohomology. Also Milne often assumes that one is
working over a field which in the case of Wiles' work is a no-no. For
instance, I don't think that constructibility of higher direct images
for schemes of finite type over a Dedekind ring (of SGA4 1/2) is
covered in Milne.
On the other hand, I agree that Milne covers most of the things one
needs for one's daily purposes and it would be a good starting point
for trying to get rid of strong set theory hypotheses in SGA*.
--
Torsten Ekedahl
te...@matematik.su.se
loa...@vax.oxford.ac.uk
Of course, by working inside the constructible universe L, any theorem of
number theory (to be precise, all quantifiers ranging over the natural numbers)
provable in ZF+V=L (and thus ZF+AC+GCH) is provable in ZF. Anyone but a
logician would probably regard that as cheating.
Tal Kubo
>published complete presentation of etale cohomology, SGA*, uses the
Milne's book?
>[...] The published
>proof of Deligne for the function field Riemann hypothesis (which may
>get used though perhaps the 1-dimensional case of Weil is enough or
>even Hasses for genus 1) uses another of the recently hotly discussed
>consequences of AC -- an embedding of the field of p-adic integers
>into the complex numbers, this time, however, Deligne discusses this
>in a note and sketches how to avoid it (incidentally he doesn't believe
>in the existence of such an embedding). [...]
Supposedly, Grothendieck also had constructivist tendencies. Just two
more sad examples of mathematicians whose output was curtailed by
unnecessary misgivings about AC.
Mikhail S. Verbitsky
>Actually, from a formal point of view it is even worse. The only
>published complete presentation of etale cohomology, SGA*, uses the
>existence of universa (sets which are models for set theory) and even
>hierarchies of universa. It is fairly clear, I would say, that for any
>concrete application (at the level of Wiles' proof, say) certainly would
>not need the full strength of that assumption, apart from the fact
>that it is mainly introduced to get nicer statements (so much for a
>few no-so average mathematician's for set theory).
Actually, there are at least three practically independent
books with closed presentation of etale cohomology:
SGA4, SGA4.5 and Milne's Etale Cohomology. The last two
(besides being readable - something that SGA4 does
not have) don't use set theory.
>The other direction, that of getting away even from AC is more
>interesting. It would seem that AC is inextricably linked in, even the
>existence of enough injectives in a category of sheaves probably needs
>AC (may even imply it?).
It does not need AC (only CC) at least if your topology
has a countable base. Proof: you can take a product
of a set of skyscraper sheaves located in points of your base variety
and require of stalks of these sheaves to be injective as modules.
This sheaf is injective. Now, for a sheaf $F$ you need to take
an open covering $U_i$ of you base variety and a set $A$
of sections of $F$ restricted to various $U_i$ such that any section
of $F$ over a small open set $U$ can be expressed as a linear
combination of restrictions of some sections of $A$ to $U$.
If your topology has a countable base, the set of $U_i$
can be chosen countable. Now, you take a point
$x_i$ in each $U_i$, embed $F|U_i$ in an injective module $G_i$
and take a product of skyscrapers with the stalk
$G_i$ in $x_i$. This is an injective sheaf if you
assume CC, and $F$ is embeddable in this sheaf.
Correct me if this is wrong somehow, but I
seem to repeat the proof used in Godement book.
>That, of course, will not banish AC from Wiles' proof. The published
>proof of Deligne for the function field Riemann hypothesis (which may
>get used though perhaps the 1-dimensional case of Weil is enough or
>even Hasses for genus 1) uses another of the recently hotly discussed
>consequences of AC -- an embedding of the field of p-adic integers
>into the complex numbers, this time, however, Deligne discusses this
>in a note and sketches how to avoid it (incidentally he doesn't believe
>in the existence of such an embedding). This still does not guarantee
>that other authors have been as careful.
This is indeed important case of using of AC, but
I am not sure if Wiles' proof substantially uses
the embedding of Z_p in C.
Another case when AC plays intrinsic role
in algebraic geometry is the algebraic K-theory,
when the cohomology of $GL(C)$ as a discrete
group are often used. For example, in Bloch, Zagier
and Goncharov calculations connected with
polylogarithm theory they often consider this
group and other huge spaces related to it. It
seems that this important theory is very difficult
exactly because of the non-constructive (=AC-dependent)
content. I am sure that Wiles' proof does not
use polylogarithms, though.
Misha.
Andrew Mullhaupt
>Here's a challenge to the logicians--show that enough algebraic
>geometry can be gotten out of the Peano postulates to give everything
>you need. Then peddle the result as the "elementary proof" that
>people say is lacking.
No. Elementary means not using Cauchy's theorem, (or in this case
higher cohomology) and I'd rather not tack on another asterisk on that.
How about choice-free or choiceless?
The motivation for these qualifications is the same - fear of infinity,
mistrust of point set topology, etc., but there is in my mind quite a
difference between things dependent on choice and things dependent on
Cauchy Theory.
Later,
Andrew Mullhaupt
Mikhail S. Verbitsky
>>Actually, from a formal point of view it is even worse. The only
>>published complete presentation of etale cohomology, SGA*, uses the
>>existence of universa (sets which are models for set theory) and even
>How is Milne's book on etale cohomology not complete?
>
>Well, if may memory serves me right (which it may not, if a disputed
>once more I will have to actually go back to Milne) he does refer to SGA4
>at some crucial places - the theorem on cohomological dimension of
>fields is one candidate, of course this is due Tate but I am not sure
>if he ever published it,
Another reference for this theorem is Serre's
_Cohomologie galoisienne_
> another one is that one may use hyper-Cech
>coverings to compute cohomology.
This is due to Labkin (sp.?), I guess, and one can find the
proof (for etale coverings) in Sullivan's Geometric Topology.
Besides, one commonly uses hyper-Cech coverings without
referring to SGA4 whatsoever, because this topic is covered
in many other papers.
Misha.
Philippe Gaucher
ver...@brauer.harvard.edu (Mikhail S. Verbitsky) wrote:
> Another case when AC plays intrinsic role
> in algebraic geometry is the algebraic K-theory,
> when the cohomology of $GL(C)$ as a discrete
> group are often used. For example, in Bloch, Zagier
> and Goncharov calculations connected with
> polylogarithm theory they often consider this
> group and other huge spaces related to it. It
> seems that this important theory is very difficult
> exactly because of the non-constructive (=AC-dependent)
> content. I am sure that Wiles' proof does not
> use polylogarithms, though.
I think, you are wrong. This theory is difficult because of its
non-constructive content, yes. But this non-constructive content
does not come from AC. It comes rather from the fact that it is
very difficult to take explicitly an element in any homotopy
group. More generally, suppose we have two topological spaces
X and Y. It is very difficult to take a precise continuous
function from X to Y.
In algebraic K-theory (at least in Quillen K-theory), this is
the main problem. It is the reason why the study of the Chern
character (a linear map from the Quillen K-theory of a ring
to the cyclic homology of the same ring) is so important. The
main problem in K-theory is the 'detection' of some element.
Thru cyclic homology, that becomes possible because cyclic
homology (like it is defined by Quillen and Loday in the
paper 'Lie algebra homology and cyclic homology') is like a
combinatoric object.
> Misha.
pg
**********************************************
Principe premier de l'entendement humain :
1) Je ne peux pas donc tu ne dois pas.
2) Je peux mais tu ne dois pas.
Principe second de l'entendement humain :
1) Il n'y a pas d'autre principe.
**********************************************
Torsten Ekedahl
>
> Actually, there are at least three practically independent
> books with closed presentation of etale cohomology:
> SGA4, SGA4.5 and Milne's Etale Cohomology. The last two
> (besides being readable - something that SGA4 does
> not have) don't use set theory.
theory hypotheses are no doubt not necessary but I still think that
both SGA4.5 and Milne do depend on SGA4. Milne makes references to it
at crucial points (I agree that two of my examples, Tate's therorem
and hyper coverings were badly chosen) and also almost everywhere
assumes that one is working over a base field which is not enough for
arithmetic applications. The basic etale cohomology in SGA4.5 is
really just an overview and makes constant references to SGA4. Pleas
do note that I said "complete presentation" :-). When it
comes to readability I am not arguing...
> >The other direction, that of getting away even from AC is more
> >interesting. It would seem that AC is inextricably linked in, even the
> >existence of enough injectives in a category of sheaves probably needs
> >AC (may even imply it?).
>
> It does not need AC (only CC) at least if your topology
> has a countable base. Proof:
Actually, I was thinking about the existence of injectives even in the
point case:
Does the assumption that any module over any ring can be embedded in
an injective module need AC? Does it imply it? (By one of the standard
proofs for the exisnce of injectives on may assume that the ring is
the Z. Still the proof of the fact that Q is injective as Z-module
seems to use AC.)
(In our situation one can probably get by with Z/n as the basic ring
but I dont't think that helps.)
> >That, of course, will not banish AC from Wiles' proof. The published
> >proof of Deligne for the function field Riemann hypothesis (which may
> >get used though perhaps the 1-dimensional case of Weil is enough or
> >even Hasses for genus 1) uses another of the recently hotly discussed
> >consequences of AC -- an embedding of the field of p-adic integers
> >into the complex numbers, this time, however, Deligne discusses this
> >in a note and sketches how to avoid it (incidentally he doesn't believe
> >in the existence of such an embedding). This still does not guarantee
> >that other authors have been as careful.
>
> This is indeed important case of using of AC, but
> I am not sure if Wiles' proof substantially uses
> the embedding of Z_p in C.
In the sketch of his proof that has appeared on the net there was a
surface to which he wanted to apply the Hodge-Tate decomposition.
Assuming that he also needs to know something about the absolute
values of eigenvalues of Frobenius then he implicitly uses that
embedding as he then uses Delinge's theorem. I don't know if that
assumption holds or not. Then again Deligne's sketch of how to avoid
that embedding can clearly be turned into a proof. It is no doubt the
case that Deligne chose between a more readable exposition with AC
(even though he dosen't seem to beleive in it) and a less readable one
without it and picked readability before his philosophical
convictions. (I hope this last sentence will not put me in the middle of
the AC war zone :-)
> Misha.
--
Torsten Ekedahl
te...@matematik.su.se
Gene W. Smith
Mullhaupt) writes:
>In article <1993Jun26.1...@sun0.urz.uni-heidelberg.de>
gsm...@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:
>>Here's a challenge to the logicians--show that enough algebraic
>>geometry can be gotten out of the Peano postulates to give everything
>>you need. Then peddle the result as the "elementary proof" that
>>people say is lacking.
>No. Elementary means not using Cauchy's theorem, (or in this case
>higher cohomology) and I'd rather not tack on another asterisk on that.
Elementary in the case of the prime number theorem meant avoiding the
use of complex analysis, but this is not what people are saying when
they remark on the lack of an "elementary" proof of FLT. If you go to
a number theory conference and attend a session on "Elementary Number
Theory", you will not find any complex analysis, but you won't find
any real analysis either.
What people seem to mean by "elementary number theory" in most
contexts, roughly, is not stepping outside of the circle of ideas
familiar to Fermat. Since the Peano postulates nicely describe the
sort of moves Fermat would understand, one can claim that translating
Kummer's work into Peano postulate terms (which could be done) would
in principle give a proof Fermat was prepared to follow, if he had
enough patience. The same would be true, also in principle, if the
FSRW proof was translated into moves of the Peano game.
In other words, it was a joke.
>How about choice-free or choiceless?
This is even weaker in its assumptions.
Centro Studi Univ California
>Of course, by working inside the constructible universe L, any theorem of
>number theory (to be precise, all quantifiers ranging over the natural numbers)
>provable in ZF+V=L (and thus ZF+AC+GCH) is provable in ZF. Anyone but a
>logician would probably regard that as cheating.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Why?
Matthew P Wiener
>>this time, however, Deligne discusses this in a note and sketches
>>how to avoid it (incidentally he doesn't believe in the existence of
>>such an embedding). [...]
>Supposedly, Grothendieck also had constructivist tendencies.
"Constructivist"? Future complaints from you regarding this term will
be ignored, or failing that, laughed at.
>Just two more sad examples of mathematicians whose output was
>curtailed by unnecessary misgivings about AC.
Your sarcasm is getting less and less accurate as time goes by. G&D
had no qualms about using AC when necessary. Their work on coherent
topoi, for example, uses AC. See Johnstone or Makkai&Reyes for more
readable expositions--their work includes topos-theoretic versions of
the Lowenheim-Skolem and Goedel completeness theorems.
All in all, the Bourbaki approach to logic and set theory was pathetic.
See the fairly recent Mathias? survey article in the INTELLIGENCER.
Mikhail S. Verbitsky
>an injective module need AC? Does it imply it? (By one of the standard
>proofs for the exisnce of injectives on may assume that the ring is
>the Z. Still the proof of the fact that Q is injective as Z-module
>seems to use AC.)
I think that if your ring in Noetherian and your module M is
countably generated the standard proof (from Hartshorne)
that M is embeddable in injective module works. This
proof does not use choice, but the proof that the
(non-countable) product of injective modules is injective does
use the (non-countable) choice.
You need Noether property because the submodule
of the countably generated module could be possibly
non-countably generated for general ring, so the countably
generated modules does not comprise abelian category
in non-Noetherian case. I an curious about that
last property (that the submodule of the
countably generated module is countably generated):
does it imply Noether property? If not, what
is it? Does/did someone study this?
Misha.
Matthew P Wiener
>In article <1993Jun29.1...@vax.oxford.ac.uk> loa...@vax.oxford.ac.uk writes:
>>Of course, by working inside the constructible universe L, any
>>theorem of number theory (to be precise, all quantifiers ranging
>>over the natural numbers) provable in ZF+V=L (and thus ZF+AC+GCH) is
>>provable in ZF. Anyone but a logician would probably regard that as
>>cheating.
>Why?
No one's answered that, so here's one stab. There's something about
metamathematical proofs that bother most people. They have a rabbit
out of the hat quality that just can't be eliminated. The way I think
of it is that proofs by various transference principles are a kind of
high level diagram chasing, except the object being punted around is
not some measly little typical item having its nose yanked in the
proper direction until it lands home, but it's "truth" itself being
kicked from place to place until it finally lands on the right target
and scores the big one.
In other words, one's natural expectation of a proof is that it will
allow you to dissect and reassemble the pieces of some fascinating
puzzle. Proofs by metamathematics don't do that. They dissect and
reassemble "truth" to fit the puzzle, and so the inner workings are
as mysterious as ever. They are the ultimate in pure existence proofs.
So instead of the crowd going "yay!" it goes "huh?"
Matthew P Wiener
> It does not need AC (only CC) at least if your topology
> has a countable base.
I'm curious why people consider CC (or DC) more tolerable than full AC?
If there's some working image of countable being just beyond the finite,
then surely there's at least the working image of the continuum being
just beyond the countable (ie, separability).
As an example, consider the Hewitt-Marczewski-Pondiczery theorem, which
states that a continuum-sized product of separable spaces is separable.
The proof relies on the fact that the continuum itself is separable, and
allows a more sophisticated version of the obvious countable product of
separable spaces is separable proof to go through (and _that_ proof is
itself just a more sophisticated version of the completely trivial finite
product of separable spaces is separable proof).
(I believe I cited this a month ago with the wrong Polish M-guy. To
confuse matters worse, Marczewski used to be named Szpilrajn.)
From the point of view of Wiles' proof, there is little difference
between AC and CC. Both are unnecessary for FLT, and probably the same
for STW, with or without semistability. What would be of separate
interest is not how much choice Wiles used--talk about the worm's eye
view!--but whether FLT is provable in PA. And for this, uses of AC
and CC are non-conservative over PA.
Of all the possible crackpot e-mails he could get, I couldn't imagine
anything more inane than a "did you use choice, huh, well then, it's
not a real proof, so there". I would be extremely surprised if his
paper even mentions choice explicitly, or even the metamathematical
principle that any of his implicit uses can be eliminated from the
final results.
Tal Kubo
>>>this time, however, Deligne discusses this in a note and sketches
>>>how to avoid it (incidentally he doesn't believe in the existence of
>>>such an embedding). [...]
>
>>Supposedly, Grothendieck also had constructivist tendencies.
>
>"Constructivist"? Future complaints from you regarding this term will
>be ignored, or failing that, laughed at.
"Had constructivist tendencies" != "was a constructivist". Both the
description and the inequality apply to a number of other well-known
mathematicians including Poincare, Weyl, and Manin. Among number-theorists
the number of people with such views is probably rather high. Note also
the qualifier "supposedly".
If you are interested in the details of what was intended and what
evidence there is for it, as opposed to retroactively justifying your and
Baez' earlier use of tendentious labels, email me. Since you mention it,
I'm still waiting for an explanation of the relevance of your earlier
remarks on "constructivizing" (and "unsurprising" and "took so long").
>>Just two more sad examples of mathematicians whose output was
>>curtailed by unnecessary misgivings about AC.
>
>Your sarcasm
This from sci.math's Flammophile of the Year.
>is getting less and less accurate as time goes by.
It is quite accurate. We can discuss the evidence by email if you like;
I'm disinclined to broadcast it over sci.math. As for "less and less", the
previous posting which you called sarcastic was a near-verbatim echo of
your own arguments for pervasiveness of AC. What do you find inaccurate there?
> G&D
>had no qualms about using AC when necessary.
Use, and how one understands the meaning of such use (theorem?
consistency result? extra hypotheses? beats honest work? removable but
improves the exposition?) are separate issues. See Prof. Ekedahl's
postings on Deligne's use of an AC-dependent embedding; what he describes
sounds like a fairly large qualm. You have yet to explain how Deligne's
apparent disbelief in AC did not induce the curtailment you predicted in
homological algebra, sheaves, etale cohomology, etc.
>Their work on coherent
>topoi, for example, uses AC. See Johnstone or Makkai&Reyes for more
>readable expositions--their work includes topos-theoretic versions of
>the Lowenheim-Skolem and Goedel completeness theorems.
I'm in no position to comment on the AC-dependence here, nor on the
differences between G&D and the secondary sources. I do know that
McLarty's book on categories and toposes has an AC-free (and even
intuitionistic) development of elementary topos theory.
>All in all, the Bourbaki approach to logic and set theory was pathetic.
>See the fairly recent Mathias? survey article in the INTELLIGENCER.
Huh?? G&D != Bourbaki. In particular their use of category theory
is very much non-Bourbaki.
Andrew Mullhaupt
>In article <18...@kepler1.rentec.com> and...@rentec.com (Andrew
>Mullhaupt) writes:
>>In article <1993Jun26.1...@sun0.urz.uni-heidelberg.de>
>gsm...@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:
>
>>No. Elementary means not using Cauchy's theorem, (or in this case
>>higher cohomology) and I'd rather not tack on another asterisk on that.
>
>Elementary in the case of the prime number theorem meant avoiding the
>use of complex analysis, but this is not what people are saying when
>they remark on the lack of an "elementary" proof of FLT.
I am telling people what to mean when they use the word "elementary"
in the context of number theory. It really means _not using Cauchy
theory_ especially when you consider what it means when you use
it in the sense _elementary functions_, a usage familiar to the
people who worried about an 'elementary proof' of the prime number
theorem.
>If you go to
>a number theory conference and attend a session on "Elementary Number
>Theory", you will not find any complex analysis, but you won't find
>any real analysis either.
I would include most simple Calculus in the context of elementary,
and I think number theorists have for a long time, which is how we
came by the rule of thumb 'up to but not including Cauchy theory'
for 'elementary'. So no, I would not expect to see much Lebesgue
integration, but I would not be surprised to see stuff like taylor
series.
My two papers relating to number theory, [R. M. Grassl and A. P. Mullhaupt,
"Hook and Shifted Hook Numbers" _Discrete Math_ v. 79 pp. 153-167
(1989/1990) and J. Douthett, R. Entringer and A. P. Mullhaupt, "Musical
Scale Construction: the Continued Fraction Compromise", _Utilitas
Mathematica_, v. 42, pp. 97-113 (1992)] both make essential use of
Calculus but not "higher" Calculus, and so I think of them as qualifying
as 'elementary'. Note that the Hook Number paper will be regarded
as _nonelementary_ by lots of combinatorialists, since the technique
there is to jettison the idea of thinking about tableaux immediately
and work out the analytic properties of rational and polynomial solutions
to appropriate functional equations which generate the various discrete
objects. But this pernicious opinion, (i.e. fear of generating functions)
which seems to be waning in combinatorialists, is in my mind one of the
major reasons why people tend to think that elementary is supposed to
mean 'without any infinities or limiting processes'. It has never meant
that in number theory, nor _can it ever mean that_
To see what I mean, suppose we took elementary to mean completely
devoid of limits or other devices of analysis. Then you cannot even
have an elementary _statement_ of the prime number theorem, no less
a proof. (E.g. where do you get a substitute for the log which is
both completely finitistic _and_ provides a sharp theorem?)
>What people seem to mean by "elementary number theory" in most
>contexts, roughly, is not stepping outside of the circle of ideas
>familiar to Fermat. Since the Peano postulates nicely describe the
>sort of moves Fermat would understand,
Oh yeah? Solve x^2 - 3 = 0 in Peano arithmetic, or (and I'd like
to see this), explain how Fermat would not have understood this 'move'.
> one can claim that translating
>Kummer's work into Peano postulate terms (which could be done) would
>in principle give a proof Fermat was prepared to follow, if he had
>enough patience. The same would be true, also in principle, if the
>FSRW proof was translated into moves of the Peano game.
They need to come up with a different word. 'Elementary' is taken.
>>How about choice-free or choiceless?
>This is even weaker in its assumptions.
Yes, but this is what I thought people were concerned about first.
Later,
Andrew Mullhaupt
Gene W. Smith
>I am telling people what to mean when they use the word "elementary"
>in the context of number theory.
The question is, *why* are you telling people "what to mean", given
that they don't always (in fact, I would say don't usually) actually
mean this?
>It really means _not using Cauchy theory_ especially when you consider
>what it means when you use it in the sense _elementary functions_, a
>usage familiar to the people who worried about an 'elementary proof'
>of the prime number theorem.
It really means this some of the time, when that is what is means.
>>If you go to a number theory conference and attend a session on
>>"Elementary Number Theory", you will not find any complex analysis,
>>but you won't find any real analysis either.
>I would include most simple Calculus in the context of elementary,
>and I think number theorists have for a long time, which is how we
>came by the rule of thumb 'up to but not including Cauchy theory'
>for 'elementary'. So no, I would not expect to see much Lebesgue
>integration, but I would not be surprised to see stuff like taylor
>series.
I've never seen analysis used in something called "session on elementary
number theory"; it always seems to be in "analytic number theory",
"modular forms", etc .etc. So in practice the Hardy-Erdos-Selberg
sense of this does not seem to be as canonical as you are claiming.
>My two papers relating to number theory, [R. M. Grassl and A. P. Mullhaupt,
>"Hook and Shifted Hook Numbers" _Discrete Math_ v. 79 pp. 153-167
>(1989/1990) and J. Douthett, R. Entringer and A. P. Mullhaupt, "Musical
>Scale Construction: the Continued Fraction Compromise", _Utilitas
>Mathematica_, v. 42, pp. 97-113 (1992)] both make essential use of
>Calculus but not "higher" Calculus, and so I think of them as qualifying
>as 'elementary'.
Maybe so. Thanks for mentioning the music paper; I shall look it up.
As a sort of amusing side-note to this, I've done stuff on musical
scales which *does* use complex analysis.
>But this pernicious opinion, (i.e. fear of generating functions) which
>seems to be waning in combinatorialists, is in my mind one of the
>major reasons why people tend to think that elementary is supposed to
>mean 'without any infinities or limiting processes'. It has never
>meant that in number theory, nor _can it ever mean that_
"It never can"?? This is an utterly bizarre claim; clearly a word
*can* come to mean anything at all.
>To see what I mean, suppose we took elementary to mean completely
>devoid of limits or other devices of analysis. Then you cannot even
>have an elementary _statement_ of the prime number theorem, no less
>a proof. (E.g. where do you get a substitute for the log which is
>both completely finitistic _and_ provides a sharp theorem?)
This is easy. Variants on the theme of 1+1/2+1/3+1/4+ ... + 1/n will
work just fine. We then formulate a statment in terms of
upper and lower bounds which avoids limit terminology.
>>What people seem to mean by "elementary number theory" in most
>>contexts, roughly, is not stepping outside of the circle of ideas
>>familiar to Fermat. Since the Peano postulates nicely describe the
>>sort of moves Fermat would understand,
>Oh yeah? Solve x^2 - 3 = 0 in Peano arithmetic, or (and I'd like
>to see this), explain how Fermat would not have understood this 'move'.
This is pretty easy, isn't it? We consider pairs of integers a, b;
and pairs of functions such as a1a2 + 3b1b2, a1b1 + b1a2 which relate
pairs such as a1, b1 and a2, b2. Since we can also "create" negative
integers in Peano arithmetic, we can easily deal with Z[sqrt(3)] in
Peano arithmetic, and reduce statments about it to statements about
integers.
There is more that one way to skin a cat; Gauss introduced class numbers
of quadratic fields in the context of binary forms.
As for Fermat not understanding the move, I am not going to explain
*that*, since no doubt he would understand it quite well.
Centro Studi Univ California
>>Why? [would anyone "regard as cheating" the elimination of AC from
>> the proof of a statement of number theory, said elimination
>> obtained by working in L.]
In article <134...@netnews.upenn.edu> wee...@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>No one's answered that, so here's one stab. There's something about
>metamathematical proofs that bother most people.
But really there is nothing at all metamathematical about the proof
obtained in this way. For each axiom of ZFC that you need in your
proof, you show that its relativization to L holds; these relativizations
have a mathematical meaning. Then you prove the theorem relativized
to L following the original proof, and observe that since all numbers
are in L, the relativization is equivalent to the original theorem.
You never need to use semantics at all.
There is a metamathematical quality, certainly, to the *method* of
transforming the proof, but the final proof obtained is not metamathematical.
>In other words, one's natural expectation of a proof is that it will
>allow you to dissect and reassemble the pieces of some fascinating
>puzzle. Proofs by metamathematics don't do that. They dissect and
>reassemble "truth" to fit the puzzle, and so the inner workings are
>as mysterious as ever. They are the ultimate in pure existence proofs.
This is of course nonsense when applied to FLT, which does not assert the
existence of anything, but rather the non-existence of a quadruple
of integers satisfying a certain decidable relation.
Matthew P Wiener
>>>Supposedly, Grothendieck also had constructivist tendencies.
>>"Constructivist"? Future complaints from you regarding this term will
>>be ignored, or failing that, laughed at.
>"Had constructivist tendencies" != "was a constructivist".
So?
>Note also the qualifier "supposedly".
So?
>If you are interested in the details of what was intended and what
>evidence there is for it, as opposed to retroactively justifying your and
>Baez' earlier use of tendentious labels, email me.
We did not consider them tendentious labels. You did. I then offered
a separate term to use for the stance you have been defending. You
did not like it either, and meanwhile, have offered no labels on your
own, just complaints.
Considering that you used "constructivist" in a context where you meant
having qualms about using AC, all future complaints from you regarding
this term will be ignored, or failing that, laughed at. All further
comments about what you really meant or whatever don't really matter on
this point--you did use the term.
>>Your sarcasm is getting less and less accurate as time goes by.
>It is quite accurate. We can discuss the evidence by email if you
>like; I'm disinclined to broadcast it over sci.math.
Considering that Grothendieck was willing to use inaccessibles or other
such egregious devices to do his K-theory in--unlike AC, one cannot prove
their relative consistency--the point of your sarcasm is getting less and
less accurate.
> As for "less and
>less", the previous posting which you called sarcastic was a
>near-verbatim echo of your own arguments for pervasiveness of AC.
>What do you find inaccurate there?
The _point_ of your sarcasm.
>You have yet to explain how Deligne's apparent disbelief in AC did
>not induce the curtailment you predicted in homological algebra,
>sheaves, etale cohomology, etc.
I did: he used it when necessary. I've cited particular examples.
Tell us: how does one do homological algebra without enough injectives
or projectives. All that glorious choice-free machinery isn't going
anywhere on its own.
>I'm in no position to comment on the AC-dependence here, nor on the
>differences between G&D and the secondary sources. I do know that
>McLarty's book on categories and toposes has an AC-free (and even
>intuitionistic) development of elementary topos theory.
So what? You don't need choice for elementary topos theory.
>>All in all, the Bourbaki approach to logic and set theory was pathetic.
>>See the fairly recent Mathias? survey article in the INTELLIGENCER.
>Huh?? G&D != Bourbaki. In particular their use of category theory
>is very much non-Bourbaki.
Of course. But the Bourbaki approach has cast a long pall on the general
French understanding of set theory and logic.
Matthew P Wiener
>But really there is nothing at all metamathematical about the proof
>obtained in this way. For each axiom of ZFC that you need in your
>proof, you show that its relativization to L holds; these relativizations
L is metamathematical.
>>Proofs by metamathematics ... are the ultimate in pure existence proofs.
>This is of course nonsense when applied to FLT, which does not assert
>the existence of anything, but rather the non-existence of a
>quadruple of integers satisfying a certain decidable relation.
I was referring to the existence of a logic-free proof of "ZF |- FLT".
Tal Kubo
>>devoid of limits or other devices of analysis. Then you cannot even
>>have an elementary _statement_ of the prime number theorem, no less
>>a proof. (E.g. where do you get a substitute for the log which is
>>both completely finitistic _and_ provides a sharp theorem?)
>
>This is easy. Variants on the theme of 1+1/2+1/3+1/4+ ... + 1/n will
>work just fine. We then formulate a statment in terms of
>upper and lower bounds which avoids limit terminology.
Rosser and Schoenfield proved upper and lower bounds for pi(n) of the form
n/(log(n) + const). Clearly log can be replaced by partial sums of the
harmonic series here. Their paper is based on making all sorts of careful
estimates and keeping track of remainder terms, in a pretty finitistic way.
For example, their bounds hold not only asymptotically, but for all n>53.
Benjamin J. Tilly
>
> To see what I mean, suppose we took elementary to mean completely
> devoid of limits or other devices of analysis. Then you cannot even
> have an elementary _statement_ of the prime number theorem, no less
> a proof. (E.g. where do you get a substitute for the log which is
> both completely finitistic _and_ provides a sharp theorem?)
>
What about f(n)=1/1+1/2+...+1/n? I believe that someone did come up
with an elementary proof of the prime number theorem that used this and
did not use any transcendental functions at all.
Ben Tilly
Tal Kubo
>a separate term to use
Haar Haar. What you offered as a separate term was "sigma-selectivist", a
private derisive term which you first used in the context of dismissing a
certain position as "too bizarre to credit". A red herring position
[hypothetical dechoicability implies non-use] which I had strenuously
disclaimed on multiple occasions (never mind that the very same
too-bizarre-to-credit reasoning showed up later as an explanation of how
nonstandard analysis arises in fluid mechanics).
After an unsuccessful request that you decode your terminology, I found out
from one of your later posts that the intended meaning was "bigot against
uncountable choice". Pretty non-tendentious. John Baez offered
"nonexistent constructivist mathematical physicist". Very non-tendentious.
More recently you've graciously offered such winners as "crackpot", "rabid
finitist", and "egregious". Not in the least tendentious.
> for the stance you have been defending.
What stance? All along I've been asking about evidence for various
assessments which have been made by you and others.
> You
>did not like it either, and meanwhile, have offered no labels on your
>own, just complaints.
As the earlier discussion of the Wiener Lemma paper showed, the labels
don't do anything to advance the discussion, and can create a lot of
misconceptions. Remember "constructivizing"? I'd prefer to have a
discussion based on the mathematics, not labels for positions.
> Considering that you used "constructivist" in
>a context where you meant having qualms about using AC,
The (con)text included "misgivings about AC", not "qualms about using AC".
>all future complaints from you regarding
>this term will be ignored, or failing that, laughed at.
Go ahead. The net provides all sorts of user-friendly ways to engage
in boneheaded displays to your heart's content.
> All further
>comments about what you really meant or whatever don't really matter on
>this point--you did use the term.
What irony: you and Dan Bernstein do subscribe to the same school of
textual interpretation, after all! Unlike Dan, you seem to have trouble
with reading comprehension, not to mention simple textual accuracy. The
C&S acknowledgements... Goedelizing... Loday/Rieffel... "highly relevant
here"... magic N-cubes and anthills... and on, and on.
>Considering that Grothendieck was willing to use inaccessibles or other
>such egregious devices to do his K-theory in--unlike AC, one cannot prove
>their relative consistency--the point of your sarcasm is getting less and
>less accurate.
I'd be interested to see where/how inaccessibles were used. All I know
for sure is that he introduced K_0-theory to get Chern classes
algebraically (lambda rings), and I don't recall any inaccessibles
there. As for "the point", in no way does G's use of inaccessibles
contradict it. See the remark about Poincare et al.
>> As for "less and
>>less", the previous posting which you called sarcastic was a
>>near-verbatim echo of your own arguments for pervasiveness of AC.
>>What do you find inaccurate there?
>
>The _point_ of your sarcasm.
The FLT post was right on target, sarcasm or not. What _point_ do you
find inaccurate?
>>You have yet to explain how Deligne's apparent disbelief in AC did
>>not induce the curtailment you predicted in homological algebra,
>>sheaves, etale cohomology, etc.
>
>I did: he used it when necessary. I've cited particular examples.
I see that the preceding portion of my message, addressing exactly this
point, has been elided. The point was not in regard to use, but attitude
toward such use. Assuming you meant what you said about eschewing formalism
as a philosophy, this makes quite a difference. Regarding "necessary", see
Prof. Ekedahl's comment, that AC was used mainly to get nicer statements;
this fits with what I've been saying all along about generality.
>Tell us: how does one do homological algebra without enough injectives
>or projectives. All that glorious choice-free machinery isn't going
>anywhere on its own.
Misha Verbitsky has kindly explained how to get injectives without too
much heavy lifting. Presumably the same holds for projectives.
>>Huh?? G&D != Bourbaki. In particular their use of category theory
>>is very much non-Bourbaki.
>
>Of course. But the Bourbaki approach has cast a long pall on the general
>French understanding of set theory and logic.
Maybe. But people who are tall enough might not be covered by the shadow.
Mikhail S. Verbitsky
>
>In article <134...@netnews.upenn.edu>
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>>Considering that Grothendieck was willing to use inaccessibles or other
>>such egregious devices to do his K-theory in--unlike AC, one cannot prove
>>their relative consistency--the point of your sarcasm is getting less and
>>less accurate.
>
>I'd be interested to see where/how inaccessibles were used. All I know
>for sure is that he introduced K_0-theory to get Chern classes
>algebraically (lambda rings), and I don't recall any inaccessibles
>there. As for "the point", in no way does G's use of inaccessibles
>contradict it. See the remark about Poincare et al.
Grotendieck's position on the use of set theory
was quite uneven. His first mathematic experience
was in the functional analysis connected with
functions of one real variable (very fascionable
10-20 years before that). His teachers
(Bourbaki elders) were functional analytists who (unlike
Kolmogorov/Gelfand school) used formal/non-constructive
agenda. This is why his books like SGA4 are
filled with (completely useless, IMHO) set theory.
I believe that Deligne rewrote SGA4 to SGA4.5
to get away from universum and topoi in algebraic
geometry. In 70-ies-80-ies, Grothendieck apparently
became a complete constructivist (cf his program).
Finally, his school (Deligne, most importantly)
seem trying to avoid unconstructive methods
as much as possible.
Misha.
Centro Studi Univ California
>L is metamathematical.
Eh? What in the world does this statement mean?
(You are presumably aware that L can be defined in purely combinatorial
terms, repeatedly closing under elementary operations on sets, without
any reference whatsoever to definability or semantics.)
MPW>Proofs by metamathematics ... are the ultimate in pure existence proofs.
MO>This is of course nonsense when applied to FLT, which does not assert
MO>the existence of anything, but rather the non-existence of a
MO>quadruple of integers satisfying a certain decidable relation.
MPW>I was referring to the existence of a logic-free proof of "ZF |- FLT".
But the proof of FLT that you get by working in L is even better than
that; it's a proof of FLT in ZF, not just a proof of "ZF |- FLT."
The latter statement asserts the existence of a proof of FLT in ZF,
so to prove it, just take the proof of FLT in ZF that you get by
the method described.
To sum up: Given any proof of a number-theoretic statement in ZFC, the
method of working in L allows you to transform it into a proof of the
same statement in ZF. This method is entirely effective; in fact, the
time it takes and the length of the final proof should be linear (?) in the
length of the original proof. (I admit to some doubt about the linearity;
anyone want to check me on this?)
AND the final proof is guaranteed 100% logic-free.
Centro Studi Univ California
letter. Try Kunen, _Set Theory_ or Jech, _Set Theory_; for a detailed
exposition of the combinatorial approach, I think there's an Omega book
called _Constructibility_.
In article <C9p0I...@dei.unipd.it> I wrote:
>To sum up: Given any proof of a number-theoretic statement in ZFC, the
>method of working in L allows you to transform it into a proof of the
>same statement in ZF. This method is entirely effective; in fact, the
>time it takes and the length of the final proof should be linear (?) in the
>length of the original proof. (I admit to some doubt about the linearity;
>anyone want to check me on this?)
PLEASE anyone want to check me on this?
(Actually, I no longer believe the linearity part; I suspect that proving
the L-relativization of an arbitrary instance of Aussonderung or of Replace-
ment might take longer. But maybe it's still true in an environment in
which the instances you need are limited to what can occur in geometry?)
>AND the final proof is guaranteed 100% logic-free.
(Whatever that means. I admit to not having had a terribly specific
meaning in mind; just that whatever might have been the semantic inspirations
of the manipulations we performed on the proof, we haven't introduced
references to semantics in the final proof.)
Matthew P Wiener
>>L is metamathematical.
>Eh? What in the world does this statement mean?
L serves as a useful view of mathematics as a whole, among other things.
>(You are presumably aware that L can be defined in purely combinatorial
>terms, repeatedly closing under elementary operations on sets, without
>any reference whatsoever to definability or semantics.)
Yes. So what? Doing logic without the usual accoutrements is still logic.
>MPW>I was referring to the existence of a logic-free proof of "ZF |- FLT".
>But the proof of FLT that you get by working in L is even better than
>that; it's a proof of FLT in ZF, not just a proof of "ZF |- FLT."
Yes, but it's not logic-free, which is what the non-meta people want.
Meanwhile, I really don't understand what in the world you're arguing
about. You asked why people consider the metamathematical elimination
of AC from a derivation of ZFC |- FLT to be "cheating" (as opposed to
the more laborious line-by-line effectivization) and I explained it to
you.
Centro Studi Univ California
>Meanwhile, I really don't understand what in the world you're arguing
>about. You asked why people consider the metamathematical elimination
>of AC from a derivation of ZFC |- FLT to be "cheating" (as opposed to
>the more laborious line-by-line effectivization) and I explained it to
>you.
OK, let's take this point first. You explained a possible reason, fine.
But I got the impression, possibly false, that you didn't think that
people who felt that way were unjustified, even if they felt that way
after they understood the argument.
In addition you seem to use the words "metamathematical" and "logical"
in a way that robs them of definable content, if not the rather
ad-hominem meaning of "something invented by logicians."
But I agree that the thread is getting silly, and I'm willing to stop it.
I just want to make clear my position that working in L to eliminate the
necessity of AC in proofs of number-theoretic statements is just as
mathematical as anything else. I don't see any reason why anyone who
is willing to accept all the machinery of elliptic curves to prove FLT
should balk at using "constructible elliptic curves" (that is, those in L)
to do the same work, especially after seeing the combinatorial definition
of L (which, I repeat, is logic-free, even if inspired by logic).
Matthew P Wiener
>>Meanwhile, I really don't understand what in the world you're arguing
>>about. You asked why people consider the metamathematical elimination
>>of AC from a derivation of ZFC |- FLT to be "cheating" (as opposed to
>>the more laborious line-by-line effectivization) and I explained it to
>>you.
>OK, let's take this point first. You explained a possible reason, fine.
>But I got the impression, possibly false, that you didn't think that
>people who felt that way were unjustified, even if they felt that way
>after they understood the argument.
A proof is a proof, of course, but somehow some proofs leave many people
unsatisfied. Why is a psychological issue, not a mathematical one. I
have no opinion about people's preferences, but I think I identified the
core issues regarding what makes proofs by metamathematics unsatisfying
to many people.
>In addition you seem to use the words "metamathematical" and "logical"
>in a way that robs them of definable content, if not the rather
>ad-hominem meaning of "something invented by logicians."
What else could it mean? In the end, logic is just so much mathematics,
so drawing the line is a taxonomic convenience.
>But I agree that the thread is getting silly, and I'm willing to stop it.
>I just want to make clear my position that working in L to eliminate the
>necessity of AC in proofs of number-theoretic statements is just as
>mathematical as anything else.
Of course.
> I don't see any reason why anyone who
>is willing to accept all the machinery of elliptic curves to prove FLT
>should balk at using "constructible elliptic curves" (that is, those in L)
>to do the same work,
If some fantastic combinatorial argument involving morasses had been
worked out to prove FLT, you can be pretty certain that the reaction
would be to develop a constructible-free notion of `baby morass' that
is provable to exist in ZF and suffices for this hypothetical proof.
Why? Because FLT has nothing to do with L: it's a theorem! The very
argument that says FLT proofs in L transfer to V also tells us that such
a proof, while a proof, is perforce irrelevant to FLT. NSA suffers from
the same fate. A minority like it, but the majority has merely stopped
disparaging Leibniz.
> especially after seeing the combinatorial definition
>of L (which, I repeat, is logic-free, even if inspired by logic).
It's logic-free only by change of notation. See above about taxonomy.