Cantor's argument is erroneous
LudovicoVan
mathematics.
The basic idea in the argument is that there is no bijection between
the set of counting numbers and the set of infinite binary strings.
But such a bijection exists, it can be expressed in terms of limit
points, or by transfinite induction; informally, it can be defined as
the correspondence between the paths and the leaf (i.e. limit) nodes
in the infinite binary tree. This invalidates all results relating to
Cantor's transfinite.
In particular, it is invalid to state that the set of infinite binary
strings is uncountable. It is countable, being in bijection with a
subset of a countable set, the set of nodes in the infinite binary
tree. The other option is that one drops the countability of infinite
sets completely, but I can see no advantage in banning the
transfinite.
The problem is possibly much broader, and deeper, because it is the
very soundness of archimedean arithmetic that seems at stake here. I
say "soundness" because from the archimedean framework a tension
results, between computability and tractability, that manifests a
deeper tension between sound and unsound mathematics, and then even
logic.
-LV
Dirk Van de moortel
249a94e0-bd9e-4a1d...@37g2000yqp.googlegroups.com
Don Stockbauer
Just 2 (count 'em) infinities:
1. Potential
2. Actual
LudovicoVan
> Cantor's argument is erroneous and its adoption leads to unsound
> mathematics.
> The problem is possibly much broader, and deeper, because it is the
> very soundness of archimedean arithmetic that seems at stake here. I
> say "soundness" because from the archimedean framework a tension
> results, between computability and tractability, that manifests a
> deeper tension between sound and unsound mathematics, and then even
> logic.
The actual world is non-archimedean as it is non-euclidean?
-LV
Brian Chandler
Excellent! You win ten points, Sir. Though I wonder if perhaps your
critics might (variously) say:
1. There are but two infinities, actual and potential.
2. 0.9 repeating is below one by the infinitely nonzero small.
... And you have, I fancy, no answer to these other powerful points.
So you lose your ten points, and you owe me a drink.
Brian Chandler
LudovicoVan
Powerful points? Have a look at the p-adics. The standard reals embed
a contradiction.
> So you lose your ten points, and you owe me a drink.
Maybe the other way round? A Guinness please.
-LV
Brian Chandler
> On 29 Mar, 15:23, Brian Chandler <imaginator...@despammed.com> wrote:
> > LudovicoVan wrote:
> > > On 29 Mar, 13:11, LudovicoVan <ju...@diegidio.name> wrote:
> > > > Cantor's argument is erroneous and its adoption leads to unsound
> > > > mathematics.
> > > [snip]
> > > > The problem is possibly much broader, and deeper, because it is the
> > > > very soundness of archimedean arithmetic that seems at stake here. I
> > > > say "soundness" because from the archimedean framework a tension
> > > > results, between computability and tractability, that manifests a
> > > > deeper tension between sound and unsound mathematics, and then even
> > > > logic.
> >
> > > The actual world is non-archimedean as it is non-euclidean?
> >
> > Excellent! You win ten points, Sir. Though I wonder if perhaps your
> > critics might (variously) say:
> >
> > 1. There are but two infinities, actual and potential.
> >
> > 2. 0.9 repeating is below one by the infinitely nonzero small.
> >
> > ... And you have, I fancy, no answer to these other powerful points.
>
> Powerful points? Have a look at the p-adics. The standard reals embed
> a contradiction.
I'm not sure about this... you're not Archie's sock-puppet are you?
Anyway, I could only suggest you read this essay: http://www.elsewhere.org/pomo/
Brian Chandler
Newberry
> Cantor's argument is erroneous and its adoption leads to unsound
> mathematics.
>
> The basic idea in the argument is that there is no bijection between
> the set of counting numbers and the set of infinite binary strings.
> But such a bijection exists, it can be expressed in terms of limit
> points, or by transfinite induction; informally, it can be defined as
> the correspondence between the paths and the leaf (i.e. limit) nodes
> in the infinite binary tree. This invalidates all results relating to
> Cantor's transfinite.
>
> In particular, it is invalid to state that the set of infinite binary
> strings is uncountable. It is countable, being in bijection with a
> subset of a countable set, the set of nodes in the infinite binary
> tree. The other option is that one drops the countability of infinite
> sets completely, but I can see no advantage in banning the
> transfinite.
Everybody knows this. The point is that they managed to create a
consistent axiomatic system, where Cantor's argument goes through. So
you have to come up with an axiomatic system of set theory where
Cantor's argument does not go through.
Such systems alredy exist e.g. Myhill's constructive set theory or
Markov's theory. I do not know why these are not more widely known or
what the problems with them are. But if you think that Cantor's
argument is wrong this is the question you may want to answer.
LudovicoVan
> On Mar 29, 5:11 am, LudovicoVan <ju...@diegidio.name> wrote:
>
> > Cantor's argument is erroneous and its adoption leads to unsound
> > mathematics.
>
> > The basic idea in the argument is that there is no bijection between
> > the set of counting numbers and the set of infinite binary strings.
> > But such a bijection exists, it can be expressed in terms of limit
> > points, or by transfinite induction; informally, it can be defined as
> > the correspondence between the paths and the leaf (i.e. limit) nodes
> > in the infinite binary tree. This invalidates all results relating to
> > Cantor's transfinite.
>
> > In particular, it is invalid to state that the set of infinite binary
> > strings is uncountable. It is countable, being in bijection with a
> > subset of a countable set, the set of nodes in the infinite binary
> > tree. The other option is that one drops the countability of infinite
> > sets completely, but I can see no advantage in banning the
> > transfinite.
>
> Everybody knows this. The point is that they managed to create a
> consistent axiomatic system, where Cantor's argument goes through.
What goes through is an invalid argument, based on the false
assumption that such a bijection cannot be formulated, and coming to
the self-contradictory conclusion of the existence of uncountable
sets. In fact, ex falso quod libet, and as I have suggested in the OP,
you can produce such bijection, so the opposite of Cantor's
assumption, within the same language: so even formal consistency I
suppose is lost.
> So
> you have to come up with an axiomatic system of set theory where
> Cantor's argument does not go through.
>
> Such systems alredy exist e.g. Myhill's constructive set theory or
> Markov's theory. I do not know why these are not more widely known or
> what the problems with them are. But if you think that Cantor's
> argument is wrong this is the question you may want to answer.
I can only guess why Myhill or Markiv are not widely adopted, and this
risks to be quite beyond the scope of the present topic.
-LV
leland....@gmail.com
That's quite a claim given that there are full formal machine
checkable (and checked!) proofs browsable on the web:
http://us.metamath.org/mpegif/ruc.html . To claim you don't like the
axioms used is one thing, but to claim that the result doesn't
actually follow from the axioms is something else again. Until your
proof that a bijection exists is made as formal and explicit and is as
well checked as the one linked above (which proves in ruclem39 that no
such bijection exists) I suggest you don't have a leg to stand on
here.
lwa...@lausd.net
> On Mar 29, 5:11 am, LudovicoVan <ju...@diegidio.name> wrote:
> > Cantor's argument is erroneous and its adoption leads to unsound
> > mathematics.
> > strings is uncountable. It is countable, being in bijection with a
> > subset of a countable set, the set of nodes in the infinite binary
> > tree. The other option is that one drops the countability of infinite
> > sets completely, but I can see no advantage in banning the
> > transfinite.
> Everybody knows this.
I already know that LV has been labeled "crank" before. He is
clearly another opponent of standard set theory. Notice that
LV's argument against ZFC main piggy-backs off of WM's
objection to the infinite binary tree.
> [Y]ou have to come up with an axiomatic system of set theory where
> Cantor's argument does not go through.
That I've already tried, several times.
> Such systems alredy exist e.g. Myhill's constructive set theory or
> Markov's theory. I do not know why these are not more widely known or
> what the problems with them are. But if you think that Cantor's
> argument is wrong this is the question you may want to answer.
There's also NFU, where non-Cantorian sets exist. Yet, despite
the existence of NFU, Markov, and Myhill, LV is still considered
to be a so-called "crank."
Perhaps LV wouldn't be considered a "crank" anymore, if he were
to fully adopt NFU, Markov, or Myhill. Then he'd be an adherent
of a rigorous theory, so that he wouldn't be a "crank" anymore. I
have no access to the Markov or Myhill theories though, so I
can't tell which one LV is most likely to adopt, if any.
LudovicoVan
> axioms used is one thing, but to claim that the result doesn't
> actually follow from the axioms is something else again.
That's quite the opposite of what I said, and we had already agreed
"the proof goes through": the problem is in the assumptions, then
embedded as axioms. It's not by axiom that you state the truth of a
statement, it's the other way round. A pure "formal machine" can't
tell anything but its own consistency.
-LV
Virgil
<249a94e0-bd9e-4a1d...@37g2000yqp.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
> Cantor's argument is erroneous and its adoption leads to unsound
> mathematics.
>
> The basic idea in the argument is that there is no bijection between
> the set of counting numbers and the set of infinite binary strings.
> But such a bijection exists, it can be expressed in terms of limit
> points, or by transfinite induction; informally, it can be defined as
> the correspondence between the paths and the leaf (i.e. limit) nodes
> in the infinite binary tree.
The whole point of a complete INFINITE binary tree is that it has
absolutely NO leaf nodes at all. Every node has two child nodes.
So that any such argument which relies on the existence of what is by
definition prohibited is invalid.
> This invalidates all results relating to
> Cantor's transfinite.
Not according any standard form of logic, it doesn't.
>
> In particular, it is invalid to state that the set of infinite binary
> strings is uncountable. It is countable, being in bijection with a
> subset of a countable set, the set of nodes in the infinite binary
> tree.
Suppose that S is the set of all binary strings, functions from N to
{0,1} and f:N -> S is any function from N to that set S,
then for any n in N, f(n) is a function from N to {0,1}, whose value, 0
or 1, for any m in N may be denoted by f(n)(m).
But the consider the function g:N -> {0,1} defined by g(n) = 1 - f(n)(n)
If every such function, g, is listed, then there must exist an n for
which g = f(n) and then, for that n, one must have g(n) = f(n)(n).
But then, for that n, 1 - f(n)(n) = f(n)(n) or f(n)(n) = 1/2.
Which is impossible.
So in LudovicoVan's world either 0 = 1/2 or 1/2 = 1, either of which
makes his world of little interest to mathematicians.
Virgil
<4b6730c3-6402-422e...@k2g2000yql.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
But the ideal world of mathematics is less handicapped.
Virgil
<47213c5b-6234-411c...@g19g2000yql.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
> What goes through is an invalid argument, based on the false
> assumption that such a bijection cannot be formulated, and coming to
> the self-contradictory conclusion of the existence of uncountable
> sets. In fact, ex falso quod libet, and as I have suggested in the OP,
> you can produce such bijection, so the opposite of Cantor's
> assumption, within the same language: so even formal consistency I
> suppose is lost.
Cantor has shown, quite simply and unambiguously that any injection from
the naturals to the set of ALL binary sequences must fail to be
surjective.
Your handwaving in no way invalidates that proof.
LudovicoVan
> In article
> <249a94e0-bd9e-4a1d-b1cf-51c11c82f...@37g2000yqp.googlegroups.com>,
>
> LudovicoVan <ju...@diegidio.name> wrote:
> > Cantor's argument is erroneous and its adoption leads to unsound
> > mathematics.
>
> > The basic idea in the argument is that there is no bijection between
> > the set of counting numbers and the set of infinite binary strings.
> > But such a bijection exists, it can be expressed in terms of limit
> > points, or by transfinite induction; informally, it can be defined as
> > the correspondence between the paths and the leaf (i.e. limit) nodes
> > in the infinite binary tree.
>
> The whole point of a complete INFINITE binary tree is that it has
> absolutely NO leaf nodes at all. Every node has two child nodes.
I have already pointed out in another thread that the tree you allude
to is not _complete_: your arguments just do not hold for the
transfinite.
-LV
Jesse F. Hughes
>> In particular, it is invalid to state that the set of infinite binary
>> strings is uncountable. It is countable, being in bijection with a
>> subset of a countable set, the set of nodes in the infinite binary
>> tree. The other option is that one drops the countability of infinite
>> sets completely, but I can see no advantage in banning the
>> transfinite.
>
> Everybody knows this.
It's not clear to me what you think everyone knows.
Surely, you don't mean that everyone knows that the set of infinite
binary strings is in bijection with the set of nodes in the infinite
binary tree, because that just ain't so.
So what did you mean?
--
Jesse F. Hughes
"I get to make things move just by saying a few things. When I post
now the math world has to tremble, even if it does so quietly, hoping
that no one else notices." -- James S. Harris has the power.
Virgil
<47213c5b-6234-411c...@g19g2000yql.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
> > > In particular, it is invalid to state that the set of infinite binary
> > > strings is uncountable. It is countable, being in bijection with a
> > > subset of a countable set, the set of nodes in the infinite binary
> > > tree. The other option is that one drops the countability of infinite
> > > sets completely, but I can see no advantage in banning the
> > > transfinite.
No such bijection exists between the set of nodes and the set of paths
nor between N and the set of all binary sequences.
The Cantor proof of this fact has never been satisfactorily challenged.
Dave Seaman
> On 29 Mar, 16:48, Newberry <newberr...@gmail.com> wrote:
Evidently you don't understand the difference between an assumption and a
conclusion. The nonexistence of a bijection between the naturals and the reals
is a conclusion of Cantor's argument, not an assumption. You have not
identified a false assumption.
By the way, I am curious to know what it would mean for an argument to "go
through" if the argument is invalid. Cantor's argument is not invalid. You
have not identified a flaw in the argument.
--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
Virgil
<f66e5d65-31c9-409a...@o36g2000yqh.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
The set of all infinite binary sequences is complete in the only
relevant sense, it contains all infinite binary sequences.
And every node (using, say, Ullrich's definition of node) appears as a
finite initial segment of uncountably many of those infinite binary
sequences, so that the countability of nodes is irrelevant to the
countability of paths.
leland....@gmail.com
Forgive me for confusing what you were trying to claim, but your talk
of "coming to the self-contradictory conclusion" and "In fact, ex
falso quod libet, and as I have suggested in the OP, you can produce
such bijection, so the opposite of Cantor's assumption, within the
same language: so even formal consistency I suppose is lost." rather
seemed to imply you were talking about self-contradiction and formal
inconsistency which is evidently not the case for the axiom system
given.
Perhaps instead you can point out which of the axioms you feel are
wrong and introducing the problems -- the page I linked kindly lists
all the 29 axioms used in the proof: which ones are the faulty ones?
Tim Little
> In particular, it is invalid to state that the set of infinite
> binary strings is uncountable. It is countable, being in bijection
> with a subset of a countable set, the set of nodes in the infinite
> binary tree.
What bijection would that be? E.g. to which node does the infinite
binary string 01-repeating correspond?
- Tim
Daryl McCullough
>> Such systems alredy exist e.g. Myhill's constructive set theory or
>> Markov's theory. I do not know why these are not more widely known or
>> what the problems with them are. But if you think that Cantor's
>> argument is wrong this is the question you may want to answer.
>
>There's also NFU, where non-Cantorian sets exist. Yet, despite
>the existence of NFU, Markov, and Myhill, LV is still considered
>to be a so-called "crank."
Why can't you get this into your head: the reason LV and
WM and various denizens of sci.logic are considered
crackpots is not because they use unorthodox mathematical
foundations. You can use whatever foundations you like;
that doesn't make you a crackpot. What makes you a
crackpot is being incapable of coherent mathematical
arguments.
If you can't give a rigorous argument for what you
are claiming, then what you are doing is not mathematics.
It is "wanking".
>Perhaps LV wouldn't be considered a "crank" anymore, if he were
>to fully adopt NFU, Markov, or Myhill.
If he learned how to do mathematical reasoning, then he
wouldn't be considered a crackpot, regardless of what
system he adopted.
You think that the causality is: Certain people adopt
unorthodox mathematical beliefs, and because of this,
they are considered crackpots.
The causality is actually the other way around: Because
these people are crackpots (unable to do mathematical
reasoning), they are drawn to unorthodox mathematical
beliefs, where they can hide their incompetence.
Non-crackpots sometimes use unorthodox mathematical
systems.
--
Daryl McCullough
Ithaca, NY
Jesse F. Hughes
> There's also NFU, where non-Cantorian sets exist. Yet, despite
> the existence of NFU, Markov, and Myhill, LV is still considered
> to be a so-called "crank."
Look, which of the following do you think that LV is trying to show
us?
(1) In some unspecified theory, Cantor's theorem is false.
(2) Cantor's theorem is false in ZFC.
(3) Cantor's theorem is "really" false, no matter what ZFC says.
If he's really trying to show (1), then he might not be a crank, but
he's remarkably bad at basic communication. But you don't *really*
think that's his goal, do you?
If he's trying to show (2), then he's failing. If he's trying to show
(3), then I can't imagine what he means because I don't understand
(3). Nonetheless, I sometimes get the impression that our local
cranks think in terms of something like (3) rather than (1) or (2).
Note: if he's arguing for *anything but* (1), then NFU, etc., is
irrelevant.
--
Jesse F. Hughes
"But regardless of my goofs, my reality of journals is different from
ANY of yours, as they just treat me in a special way."
-- James S. Harris
David C. Ullrich
<ju...@diegidio.name> wrote:
>Cantor's argument is erroneous and its adoption leads to unsound
>mathematics.
>
>The basic idea in the argument is that there is no bijection between
>the set of counting numbers and the set of infinite binary strings.
>But such a bijection exists, it can be expressed in terms of limit
>points, or by transfinite induction; informally, it can be defined as
>the correspondence between the paths and the leaf (i.e. limit) nodes
>in the infinite binary tree. This invalidates all results relating to
>Cantor's transfinite.
This is nonsense. Being charitable about the notion of "limit node":
Yes, there's a bijection between the limit nodes and the paths.
This does not give a bijection between the poitive integers
and the paths.
>In particular, it is invalid to state that the set of infinite binary
>strings is uncountable. It is countable, being in bijection with a
>subset of a countable set, the set of nodes in the infinite binary
>tree. The other option is that one drops the countability of infinite
>sets completely, but I can see no advantage in banning the
>transfinite.
>
>The problem is possibly much broader, and deeper, because it is the
>very soundness of archimedean arithmetic that seems at stake here. I
>say "soundness" because from the archimedean framework a tension
>results, between computability and tractability, that manifests a
>deeper tension between sound and unsound mathematics, and then even
>logic.
>
>-LV
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Brian Chandler
> On Sun, 29 Mar 2009 05:11:57 -0700 (PDT), LudovicoVan
> <ju...@diegidio.name> wrote:
>
> >Cantor's argument is erroneous and its adoption leads to unsound
> >mathematics.
> >
> >The basic idea in the argument is that there is no bijection between
> >the set of counting numbers and the set of infinite binary strings.
> >But such a bijection exists, it can be expressed in terms of limit
> >points, or by transfinite induction; informally, it can be defined as
> >the correspondence between the paths and the leaf (i.e. limit) nodes
> >in the infinite binary tree. This invalidates all results relating to
> >Cantor's transfinite.
>
> This is nonsense. Being charitable about the notion of "limit node":
> Yes, there's a bijection between the limit nodes and the paths.
Do tell -- in the charitable interpretation, what _is_ a "limit node"?
(Just asking: I don't know if there is some sort of usage along these
lines, but it looks to me as though "limit node" could only really
mean "path", in which case the putative bijection seems rather
degenerate.)
Brian Chandler
jhon...@usa.com
> Cantor's argument is erroneous and its adoption leads to unsound
> mathematics.
>
> The basic idea in the argument is that there is no bijection between
> the set of counting numbers and the set of infinite binary strings.
> But such a bijection exists, it can be expressed in terms of limit
> points, or by transfinite induction; informally, it can be defined as
> the correspondence between the paths and the leaf (i.e. limit) nodes
> in the infinite binary tree. This invalidates all results relating to
> Cantor's transfinite.
>
> strings is uncountable. It is countable, being in bijection with a
> subset of a countable set, the set of nodes in the infinite binary
> tree. The other option is that one drops the countability of infinite
> sets completely, but I can see no advantage in banning the
> transfinite.
>
> The problem is possibly much broader, and deeper, because it is the
> very soundness of archimedean arithmetic that seems at stake here. I
> say "soundness" because from the archimedean framework a tension
> results, between computability and tractability, that manifests a
> deeper tension between sound and unsound mathematics, and then even
> logic.
>
> -LV
I also think the argument is invalid because it is arriving to the
conclusion of the existence of uncountable sets which is contradictory
by itself.
Jhonas
Brian Chandler
> On Mar 29, 2:11 pm, LudovicoVan <ju...@diegidio.name> wrote:
> > Cantor's argument is erroneous and its adoption leads to unsound
> > mathematics.
>
> I also think the argument is invalid because it is arriving to the
> conclusion of the existence of uncountable sets which is contradictory
> by itself.
It is? ("contradictory by itself", I mean) Could you share with us
what _you_ mean by an "uncountable set", and possible give some
indication of why this is "contradictory"?
Brian Chandler
Jesse F. Hughes
> I also think [Cantor's] argument is invalid because it is arriving
> to the conclusion of the existence of uncountable sets which is
> contradictory by itself.
I hesitate to ask, but how is that contradictory by itself?
--
Jesse F. Hughes
"Society's going to hell in a handbasket. I blame the media-blamers."
-- Dale Dribble, /King of the Hill/
Mariano Suárez-Alvarez
Pray tell us: in what way is that existence contradictory
by itself?
-- m
Dave L. Renfro
> Cantor's argument is erroneous and its adoption leads
> to unsound mathematics.
Well, at least this is a step up from some of the
Cantor threads, since you're only saying the argument
is incorrect, not the result itself.
Dave L. Renfro
MoeBlee
> Yet, despite
> the existence of NFU, Markov, and Myhill, LV is still considered
> to be a so-called "crank."
Because he's not just pointing to some theory in which a certain
statement is not a theorem; but rather he's STUBBORNLY, IGNORANTLY,
and IRRATIONALLY PERSISTING (in face of all explanation) to make
INcorrect statements about the LOGIC of a certain proof in a certain
system.
Please tell me you understand this distinction that has been pointed
out to you a thousand times already.
MoeBlee
ross.fi...@gmail.com
That infinite sets are equivalent, in terms of standard, finitary
analysis based on the least upper bound property of the standard real
numbers, doesn't affect the finitary analysis in the countable
additivity that is about symmetries and measure. Basically saying
that the infinite is Cantor's Absolute infinite, beyond all the
Aleph's, leaves standard analysis alone.
The point is to make it meaningful to also have all the other
infinities of mathematics than "one big infinity", including, for
example, Cantorian and post-Cantorian ZFC and as well nonstandard
analysis with the small axiomatics.
Systems in nature are better explained with concrete mathematics and
the synthesis for integration of the normalizing bounds. Combinatoric
expansion in space obviously has a complete role in the mathematical
framework, for that, set theory is admittedly perfect in terms of its
formalization of set theoretic operations in the finite and small like
the finite spaces where all the functions between the integer spaces
are integer spaces, in configuration.
If you're going to say that Cantor's argument is flawed, keep in mind
Cantor has more than one.
Regards,
Ross F.
MoeBlee
> That infinite sets are equivalent, in terms of standard, finitary
> analysis based on the least upper bound property of the standard real
> numbers, doesn't affect the finitary analysis in the countable
> additivity that is about symmetries and measure. Basically saying
> that the infinite is Cantor's Absolute infinite, beyond all the
> Aleph's, leaves standard analysis alone.
>
> The point is to make it meaningful to also have all the other
> infinities of mathematics than "one big infinity", including, for
> example, Cantorian and post-Cantorian ZFC and as well nonstandard
> analysis with the small axiomatics.
>
> Systems in nature are better explained with concrete mathematics and
> the synthesis for integration of the normalizing bounds. Combinatoric
> expansion in space obviously has a complete role in the mathematical
> framework, for that, set theory is admittedly perfect in terms of its
> formalization of set theoretic operations in the finite and small like
> the finite spaces where all the functions between the integer spaces
> are integer spaces, in configuration.
>
> If you're going to say that Cantor's argument is flawed, keep in mind
> Cantor has more than one.
Surely you won't mind that I decline to spend my lunch hour mulling
over your gibberish or even mulling over the question of what
motivates you to go around mumbling gibberish.
MoeBlee
Brian Chandler
> On Mar 30, 9:51 am, MoeBlee <jazzm...@hotmail.com> wrote:
> > On Mar 29, 11:57 am, lwal...@lausd.net wrote:
> >
> > > Yet, despite
> >
>
Yes, Ross, brilliant. But did you know:
1. There are but two infinities, actual and potential.
2. 0.9 repeating is below one by the infinitely nonzero small.
And have you read:
http://www.elsewhere.org/pomo/
HTH
Brian Chandler
Virgil
<e0ddfaf6-c04b-49a6...@i28g2000prd.googlegroups.com>,
Brian Chandler <imagin...@despammed.com> wrote:
If one identifies a non-limit node in a tree as the minimal chain (set)
of edges linking it with the root of the tree, then presumably a limit
node, at least in an infinite tree, would be an endless chain of such
linked edges.
That would biject limit nodes with paths in a complete infinite tree.
At least that is the only interpretation I can see right now that is not
nonsensical.
But with this definition, one can easily show, a la Cantor, that there
are "more" limit nodes than non-limit nodes (more infinite rooted
chains of edges that finite rooted chains of edges).
Virgil
<6408d357-313b-4ec4...@r18g2000vbi.googlegroups.com>,
jhon...@usa.com wrote:
Contradictory of what? A set,S, being countable merely means that there
is a surjection from the set of naturals, N, to S, a mapping which "uses
up" every member of S.
What natural rule requires the existence of such a mapping?
Particularly when it has been shown false.
Virgil
<6bc00ce1-558d-4d7c...@o11g2000yql.googlegroups.com>,
But fails to identify wherein the error lies.
Ralf Bader
The usual definition of a tree in set theory (elsewhere there are other
definiions) is: A tree is a partially ordered set (A,<), such that for any
a e A the set P_a = {b e A| b<a} is well-ordered by <. (e.g. Jech, Set
Theory, 3rd ed., Def. 9.10)
The elements of A are the nodes, in the usual parlance of this thread. An
edge would be given by two nodes, a and b, such that b<a and there
is no c e A such that b<c<a. This edge is starting in a and ending in b.
There is at most one edge ending in a, for any a e A; but there need not be
an edge ending in a. The nodes which don't have an edge ending in them
would be aptly called limit nodes.
For the "full binary tree", one can take for A the set of all sequences of
finite length of the symbols 0 and 1; and s < t iff s is an initial segment
of t. Another binary tree (in the set-theoretical sense) can be obtained by
taking for A the set of all sequences of length <=omega and < defined as
above. Then all sequences of length omega are limit nodes; and they
correspond to the infinite paths.
Dave L. Renfro
>> Well, at least this is a step up from some of the
>> Cantor threads, since you're only saying the argument
>> is incorrect, not the result itself.
Virgil wrote:
> But fails to identify wherein the error lies.
Or what to do with the huge edifice of useful and
seemingly self-consistent results that would crumble
if the real numbers were not uncountable -- most of modern
real analysis, most of modern complex analysis, most of
modern functional analysis, much of general topology
and metric space theory (concept of second countability,
concept of separability, etc.), and many many other areas.
It's like someone saying there's a problem with spoken
language, and yet everyone is still talking to each other,
there are movies and TV, etc.
Dave L. Renfro
lwa...@lausd.net
> lwal...@lausd.net writes:
> > There's also NFU, where non-Cantorian sets exist. Yet, despite
> > the existence of NFU, Markov, and Myhill, LV is still considered
> > to be a so-called "crank."
> Look, which of the following do you think that LV is trying to show
> us?
> (1) In some unspecified theory, Cantor's theorem is false.
> (2) Cantor's theorem is false in ZFC.
> (3) Cantor's theorem is "really" false, no matter what ZFC says.
> If he's really trying to show (1), then he might not be a crank, but
> he's remarkably bad at basic communication. But you don't *really*
> think that's his goal, do you?
Whether a particular so-called "crank" is arguing in favor of
(1), (2), or (3) depends on the "crank."
For many "cranks," especially MR, for example, I believe that
they are arguing in favor of (1). When someone makes comments
such as ".9 repeating. Is below one by the infinitely nonzero
small," since standard analysis doesn't have infinitely small
nonzero infinitesimals, it's natural for me to assume that
they are discussing a theory in which there _are_ infinitely
small infinitesimals.
If someone posts "the flag is red, white, and green," would
it make more sense to say, "You 'crank'! The flag is actually
red, white, and blue, idiot!" or assume that the poster is
from another country, such as Italy or Mexico, whose flag
really is red, white, and green? Yet the standard analysts do
the equivalent of the former all the time.
> If he's trying to show (2), then he's failing.
In the case of WM, for example, we can say that case (2) is
the most likely, since WM does believe that he has a proof
that ZFC is inconsistent. This is the equivalent of saying
that "the colors of the U.S. flag are red, white, and green,"
which I don't believe. Similarly, I don't believe that WM
really has a proof that ZFC is inconsistent.
But of course, this thread isn't about MR or WM -- we're
actually discussing LV here. One could make a reasonable
argument that LV is in case (2), since his argument about the
binary tree is similar to that of WM, who is already known to
believe that ZFC is inconsistent. But I'm still not sure
whether _LV_ finds ZFC inconsistent or not.
> If he's trying to show
> (3), then I can't imagine what he means because I don't understand
> (3).
Consider Goedel, the mathematician who first proved ZFC+CH to
be consistent (if ZFC is itself consistent). Now I know that
many posters don't trust Wikipedia and so I don't like having
to keep going back to Wikipedia for quotes, but here's a
relevant quote I can't find anywhere else:
"Gödel believed that CH is false and that his proof that CH
is consistent only shows that the Zermelo-Frankel axioms are
defective. Gödel was a platonist and therefore had no
problems with asserting the truth and falsehood of statements
independent of their provability."
I believe that case (3) entails replacing Goedel with LV and
CH with Cantor's theorem, to obtain:
"[LV] believed that [Cantor's Theorem] is false and that
[the] proof [of Cantor's Theorem] only shows that the
Zermelo-Frankel axioms are defective. [LV] was a platonist
and therefore had no problems with asserting the truth and
falsehood of statements independent of their provability."
So case (3) can entail a type of Platonism. If Goedel is
free to believe that CH is "really" false, no matter what ZFC
says, then LV is free to believe that Cantor's Theorem is
false, no matter what ZFC says.
Which case is most likely, (1), (2), or (3)? I'd say (2) only
because of LV's argument being similar to WM's. But even if
WM and LV object to ZFC and try to find proofs, however
invalid, that ZFC is inconsistent, then I at least want to
find theories which they won't want to prove inconsistent. So
even in case (2), I want to find an alternate theory which
matches WM's and LV's Platonic intuitions, in the hopes that
WM and LV will accept it rather than attempt to prove the
theory inconsistent.
After all, if LV doesn't want us to use ZFC, there must be
_some_ theory that LV would have us use instead of ZFC. And
I'm trying to find what such a theory might be.
George Greene
> But such a bijection exists, it can be expressed in terms of limit
> points, or by transfinite induction;
That is not even the issue, dumbass.
The issue IS that AFTER it exists, IT HAS AN ANTI-DIAGONAL.
So it couldn't exist after all, since the bijection canNOT biject
ANY natnum with the anti-diagonal OF THE BIJECTION.
> informally, it can be defined as
> the correspondence between the paths and the leaf (i.e. limit) nodes
> in the infinite binary tree.
The infinite binary tree DOES NOT HAVE ANY limit nodes or leaves!
George Greene
In the first place, IT ISN'T an "argument":
IT'S A PROOF.
In the second place, BECAUSE it's a proof, YOU CAN'T just SAY
"it's erroneous"; you have to PROVE THAT, TOO.
ALL erroneous proofs ARE PROVABLY erroneous (at 1st-order anyway).
WE HAVE a proof.
YOU DON'T.
So you lose.
The easiest way to refute a proof is to derive a contradiction from
the axioms from which the conclusion was derived (if a contradiction
follows, then EVERYthing, INCLUDING BOTH the conclusion AND its
denial, ALSO follows, so the fact that the conclusion follows ceases
to
be a reason to believe that it is true). YOU CAN'T derive a
contradiction
from any proof of Cantor's theorem, YET WE CAN construct an anti-
diagonal, and THEREBY derive a contradiction, from any bijection
YOU supply.
So, again,
YOU LOSE.
IT DOESN'T MATTER how carefully you construct your bijection:
IT STILL MUST HAVE an anti-diagonal.
IT STILL MUST HAVE A UNIQUE "real" or "subset" of the base set,
that is NOT in its range.
George Greene
> strings is uncountable. It is countable, being in bijection with a
> tree.
The set of infinite binary strings IS NOT in bijection with the
set of nodes in the tree, DUMBASS!
If B is your bijection, then its anti-diagonal is NOT mapped TO ANY
node!
This requires a little notation; if the nodes are numbered with all
the natnums
starting with 0, then the paths also have EDGES that can be numbered
in order,
starting from 0 at the root, and the 0th (or first) edge of any path
p (beginning
from the root) is p_0, and this p_x= 0 or 1 depending on whether that
edge
goes left or right.
Then the anti-diagonal is defined by a_i = 1- b(i)_i
where b(i) is the path that B maps node i to.
For any node i, obviously, that node is NOT the one mapped to this
path
because IT GOES THE OTHER WAY at the ith step.
So this path IS NOT mapped to any node.
IF you have a bijection.
If you began by assuming that your bijection covered everything,
THEN you now have
A CONTRADICTION (though of course that assumption was never used).
Tim Little
> After all, if LV doesn't want us to use ZFC, there must be _some_
> theory that LV would have us use instead of ZFC. And I'm trying to
> find what such a theory might be.
Why must there be such a theory any more rigorous than "if LV says it
is so, then it is so?"
Do you also believe that there is some theory in which JSH *has*
solved the factoring problem, and in which his proof of doing so is
both valid and sound?
- Tim
MoeBlee
> If someone posts "the flag is red, white, and green," would
> it make more sense to say, "You 'crank'! The flag is actually
> red, white, and blue, idiot!" or assume that the poster is
> from another country, such as Italy or Mexico, whose flag
> really is red, white, and green? Yet the standard analysts do
> the equivalent of the former all the time.
No, they don't. The analogy is this:
Crank: The flag is red, white, and green.
Reasonable person: Are you talking about the U.S. flag?
Crank. I'm talking about the real flag. There is only one real flag
and it is red, white, and green.
Reasonable person: No, there are flags for each country and even more
kinds of flags. In the U.S., when one says "the flag" it is ordinarily
assumed that the reference is to the U.S. flag. In other places, "the
flag" would be assumed to refer to the flag of that place. Now, if you
are saying that some country other than the U.S. has a red, white, and
green flag, then we have no argument. But if you claim that the U.S.
flag is red, white, and green, then you are wrong; otherwise what is
your proof.
Crank: I said I'm talking about the real flag. If the U.S. flag is
different from the real flag, then the U.S. flag is wrong. Anyone can
see this.
> In the case of WM, for example, we can say that case (2) is
> the most likely, since WM does believe that he has a proof
> that ZFC is inconsistent. This is the equivalent of saying
> that "the colors of the U.S. flag are red, white, and green,"
> which I don't believe. Similarly, I don't believe that WM
> really has a proof that ZFC is inconsistent.
And not only does he not have a good proof, he's shown to himself to
be a thoroughgoing crank.
> But of course, this thread isn't about MR or WM -- we're
> actually discussing LV here. One could make a reasonable
> argument that LV is in case (2), since his argument about the
> binary tree is similar to that of WM, who is already known to
> believe that ZFC is inconsistent. But I'm still not sure
> whether _LV_ finds ZFC inconsistent or not.
What makes you think he even has a view that is as COHERENT as
claiming ZFC is inconsistent? What makes you think he even understands
what 'inconsistent' means?
> "Gödel believed that CH is false and that his proof that CH
> is consistent only shows that the Zermelo-Frankel axioms are
> defective. Gödel was a platonist and therefore had no
> problems with asserting the truth and falsehood of statements
> independent of their provability."
>
> I believe that case (3) entails replacing Goedel with LV and
> CH with Cantor's theorem, to obtain:
>
> "[LV] believed that [Cantor's Theorem] is false and that
> [the] proof [of Cantor's Theorem] only shows that the
> Zermelo-Frankel axioms are defective. [LV] was a platonist
> and therefore had no problems with asserting the truth and
> falsehood of statements independent of their provability."
That would be fine, except that is NOT what LV has said. Rather, LV's
remarks show that he doesn't even understand the notion of proof.
Where Godel might believe a certain theorem of a certain theory to be
false, Godel would not contest that that theory does actually prove
that theorem. Godel would not dispute the finitistic fact of the
existence of a certain sequence of formulas.
Moreover, Godel wouldn't say such ill-informed and confused things as
found in the first post of this thread.
So, nope, your comparison of LV with Godel fails.
MoeBlee
Dik T. Winter
...
> > On Mar 29, 5:11=A0am, LudovicoVan <ju...@diegidio.name> wrote:
> > > Cantor's argument is erroneous and its adoption leads to unsound
> > > mathematics.
> Perhaps LV wouldn't be considered a "crank" anymore, if he were
> of a rigorous theory, so that he wouldn't be a "crank" anymore. I
> have no access to the Markov or Myhill theories though, so I
> can't tell which one LV is most likely to adopt, if any.
He would not be considered a "crank" if he did not state "Cantor's
argument is erroneous". In that case he has to show where and why the
argument is erroneous, which he does not do. Note: Cantor's argument
is within something akin to ZFC, so to state it is erroneous means
that it is erroneous in the realm of ZFC and it does not matter what
other theory LV would like.
Are you really unable to recognise unsound arguments?
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Jesse F. Hughes
> On Mar 29, 7:28 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> lwal...@lausd.net writes:
>> > There's also NFU, where non-Cantorian sets exist. Yet, despite
>> > the existence of NFU, Markov, and Myhill, LV is still considered
>> > to be a so-called "crank."
>> Look, which of the following do you think that LV is trying to show
>> us?
>> (1) In some unspecified theory, Cantor's theorem is false.
>> (2) Cantor's theorem is false in ZFC.
>> (3) Cantor's theorem is "really" false, no matter what ZFC says.
>> If he's really trying to show (1), then he might not be a crank, but
>> he's remarkably bad at basic communication. But you don't *really*
>> think that's his goal, do you?
>
> Whether a particular so-called "crank" is arguing in favor of
> (1), (2), or (3) depends on the "crank."
My question was specifically about LV.
> For many "cranks," especially MR, for example, I believe that
> they are arguing in favor of (1). When someone makes comments
> such as ".9 repeating. Is below one by the infinitely nonzero
> small," since standard analysis doesn't have infinitely small
> nonzero infinitesimals, it's natural for me to assume that
> they are discussing a theory in which there _are_ infinitely
> small infinitesimals.
What a truly bizarre interpretation. The fact is that MR has betrayed
absolutely no hint that he understands that his claims are false in
R. Indeed, when explicitly asked, MR was incapable of saying whether
he means (1), (2) or (3).
Your sense of charity is boundless.
[...]
Ah, but this analogy is very weak. Cantor's theorem is provable in
ZF, while CH is not provable. Goedel presumably thought that ZFC was
sound but did not prove enough. LV believes that ZF is unsound (or
inconsistent).
Since I don't really have any intuitions about sets *aside* from ZF,
I'm still puzzled by someone who would declare that ZF is
unsound. Where does their concept of sets come from?
> So case (3) can entail a type of Platonism. If Goedel is
> free to believe that CH is "really" false, no matter what ZFC
> says, then LV is free to believe that Cantor's Theorem is
> false, no matter what ZFC says.
Of course he's free to believe it, but it's still a pretty vague
belief to me.
> Which case is most likely, (1), (2), or (3)? I'd say (2) only
> because of LV's argument being similar to WM's. But even if
> WM and LV object to ZFC and try to find proofs, however
> invalid, that ZFC is inconsistent, then I at least want to
> find theories which they won't want to prove inconsistent. So
> even in case (2), I want to find an alternate theory which
> matches WM's and LV's Platonic intuitions, in the hopes that
> WM and LV will accept it rather than attempt to prove the
> theory inconsistent.
>
> After all, if LV doesn't want us to use ZFC, there must be
> _some_ theory that LV would have us use instead of ZFC. And
> I'm trying to find what such a theory might be.
Why must there be some theory that satisfies his desires? That's a
rather presumptuous and optimistic assumption.
--
Jesse F. Hughes
"And I will dream that I live underground and people will say, 'How
did you get there?'
"'I just live there,' I will tell them." -- Quincy P. Hughes, Age 4
Jesse F. Hughes
> In article <8433581e-4b20-4a3a...@v37g2000vbb.googlegroups.com> lwa...@lausd.net writes:
> ...
> > > On Mar 29, 5:11=A0am, LudovicoVan <ju...@diegidio.name> wrote:
> > > > Cantor's argument is erroneous and its adoption leads to unsound
> > > > mathematics.
> ...
> > Perhaps LV wouldn't be considered a "crank" anymore, if he were
> > to fully adopt NFU, Markov, or Myhill. Then he'd be an adherent
> > of a rigorous theory, so that he wouldn't be a "crank" anymore. I
> > have no access to the Markov or Myhill theories though, so I
> > can't tell which one LV is most likely to adopt, if any.
>
> He would not be considered a "crank" if he did not state "Cantor's
> argument is erroneous". In that case he has to show where and why the
> argument is erroneous, which he does not do. Note: Cantor's argument
> is within something akin to ZFC, so to state it is erroneous means
> that it is erroneous in the realm of ZFC and it does not matter what
> other theory LV would like.
>
> Are you really unable to recognise unsound arguments?
Every argument is sound. We just have to find the right theory (and
perhaps the right logic) to make it so.
It is a thankless task, but LWalker is a man with a mission.
--
"Basically I see myself as a hero in a great drama, and that is part of
how I motivate myself through failures and a lot of negativity, like
from people like you. So the Hammer is part of my own personal story,
my personal myth." -- James S. Harris, a legend in his own mind
Dik T. Winter
> David C. Ullrich wrote:
...
> > This is nonsense. Being charitable about the notion of "limit node":
> > Yes, there's a bijection between the limit nodes and the paths.
>
> Do tell -- in the charitable interpretation, what _is_ a "limit node"?
If I remember right, in the surreals they indeed just are the nodes created
at time omega (with a big bang). But in them a lot more is created of
course.
> (Just asking: I don't know if there is some sort of usage along these
> lines, but it looks to me as though "limit node" could only really
> mean "path", in which case the putative bijection seems rather
> degenerate.)
As I have not yet seen a coherent definition by the proposers of the
infinite tree of either path or node I wonder.
Dik T. Winter
...
> (3) Cantor's theorem is "really" false, no matter what ZFC says.
>
> If he's really trying to show (1), then he might not be a crank, but
> he's remarkably bad at basic communication. But you don't *really*
> think that's his goal, do you?
>
> (3), then I can't imagine what he means because I don't understand
> (3).
It is what WM calls "absolute mathematics", although I have no idea what
that means. And each time when I try to pin it down he evades and goes
on with something else. For some reason WM is of the opinion that when
a particular word is used in mathematics it can have only one meaning
that is universal and does not require definition.
lwa...@lausd.net
> On 2009-03-30, lwal...@lausd.net <lwal...@lausd.net> wrote:
> > After all, if LV doesn't want us to use ZFC, there must be _some_
> > theory that LV would have us use instead of ZFC. And I'm trying to
> > find what such a theory might be.
> Why must there be such a theory any more rigorous than "if LV says it
> is so, then it is so?"
The point I was trying to make is, there are all these posters
trying to tell us how bad ZFC is. Suppose we all listened to
them and abandoned ZFC. I want to know what theory they would
want us to start using once we'd have abandoned ZFC.
In other words, LV believes that the theory ZFC is unsound, so
now I want to know what theory he believes to be _sound_.
So I want to fill in the blank:
LV believes that ZFC is unsound.
LV believes that ___ is sound.
Don't tell us what theory we should _not_ use, tell us what
theory we _should_ use (in LV's opinion).
> Do you also believe that there is some theory in which JSH *has*
> solved the factoring problem, and in which his proof of doing so is
> both valid and sound?
Notice that I don't post in the JSH threads, because I don't
believe that there's a theory in which JSH is correct.
Normally, I say that I don't support theories that go against
the well-known arithmetic of small finite numbers. But some
of the numbers mentioned in the JSH threads (such as RSA
numbers, for example) _aren't_ small anymore. The largest RSA
number is greater than the number of particles in the known
universe, and so WM, for example, wouldn't even accept the
largest RSA number as a natural number! Neither would AP, who
has declared sufficiently large standard natural numbers to
be infinite!
Notice how an ultrafinitist could claim to have solved the
factoring problem by providing a factoring method that is
efficient only for sufficiently small natural numbers, then
state that the natural numbers that are too large to be
effectively factored by the method don't really exist.
Of course, it's mentioned in the JSH threads that his methods
aren't efficient even for much smaller numbers, numbers that
are so small that even ultrafinitists accept them as actually
being natural numbers.
Therefore, JSH really does contradict the mathematics of
small finite numbers. Thus I don't give JSH the time of day.
lwa...@lausd.net
> In article <e0ddfaf6-c04b-49a6-873c-fb0b398b2...@i28g2000prd.googlegroups.com> Brian Chandler <imaginator...@despammed.com> writes:
> > Do tell -- in the charitable interpretation, what _is_ a "limit node"?
> If I remember right, in the surreals they indeed just are the nodes created
> at time omega (with a big bang). But in them a lot more is created of
> course.
I've noticed how both Winter in this thread, and Bader in the current
WM thread, have mentioned the similarity between the infinite tree of
surreals and WM's tree.
If we consider the tree of surreals whose birthday is at most omega,
then this binary tree really does have exactly as many nodes as it
have paths, since each path has a leaf node, a surreal whose birthday
is exactly omega.
Thus, for this tree, WM is right that # of nodes = # of paths.
But of course, unfortunately for WM, this common cardinality of the
nodes and paths is uncountable. This is because every standard real
is considered to have been born by birthday omega, and there are
uncountably many standard reals.
At one time I wondered whether these surreals could represent a
theory in which WM would be right, but it most likely won't work.
Tim Little
> So I want to fill in the blank:
>
> LV believes that ZFC is unsound.
> LV believes that ___ is sound.
Fair enough. If you can manage that, and not have them change their
mind as soon as it entails a result they don't like, you would have
accomplished something that I would find very impressive indeed.
- Tim
Brian Chandler
> On Mar 30, 5:05 pm, Tim Little <t...@little-possums.net> wrote:
> > On 2009-03-30, lwal...@lausd.net <lwal...@lausd.net> wrote:
> The point I was trying to make is, there are all these posters
> trying to tell us how bad ZFC is. Suppose we all listened to
> them and abandoned ZFC. I want to know what theory they would
> want us to start using once we'd have abandoned ZFC.
For each crank P, the theory P would want us to use is "P's theory",
which typically is simply incoherent, and where not quite incoherent
is oracular. That is, it includes lots of provisions where in practice
the only way to determine the answer to a question is to ask P.
> In other words, LV believes that the theory ZFC is unsound,
Why do you assume this? Can't you read? LV says: "Cantor's argument is
erroneous". Note that he doesn't point out any flaw in Cantor's
argument, he merely informs us that he doesn't find the result of the
argument acceptable, which he regards as the same thing as the
argument being "erroneous".
> Don't tell us what theory we should _not_ use, tell us what
> theory we _should_ use (in LV's opinion).
Huh? Who are you asking to tell you what theory we[?] "should" use in
LV's opinion? Have you tried asking LV? Have you noticed anyone else
getting sensible answers from LV?
> Notice how an ultrafinitist could claim to have solved the
> factoring problem by providing a factoring method that is
> efficient only for sufficiently small natural numbers, then
> state that the natural numbers that are too large to be
> effectively factored by the method don't really exist.
Fascinating! So ultrafinitists can crack encryption by fiat...
Brian Chandler
herbzet
lwa...@lausd.net wrote:
> Consider Goedel, the mathematician who first proved ZFC+CH to
> be consistent (if ZFC is itself consistent). Now I know that
> many posters don't trust Wikipedia and so I don't like having
> to keep going back to Wikipedia for quotes, but here's a
> relevant quote I can't find anywhere else:
>
> "Gödel believed that CH is false and that his proof that CH
> is consistent only shows that the Zermelo-Frankel axioms are
> defective.
It shows nothing of the sort, unless you consider incompleteness
to be a defect.
> Gödel was a platonist and therefore had no
> problems with asserting the truth and falsehood of statements
> independent of their provability."
>
> I believe that case (3) entails replacing Goedel with LV and
> CH with Cantor's theorem, to obtain:
>
> "[LV] believed that [Cantor's Theorem] is false and that
> [the] proof [of Cantor's Theorem] only shows that the
> Zermelo-Frankel axioms are defective. [LV] was a platonist
> and therefore had no problems with asserting the truth and
> falsehood of statements independent of their provability."
>
> So case (3) can entail a type of Platonism. If Goedel is
> free to believe that CH is "really" false, no matter what ZFC
> says, then LV is free to believe that Cantor's Theorem is
> false, no matter what ZFC says.
That's nothin'!
*I* believe that there is a greatest prime number, no matter
what Peano arithmetic or Euclid say. They're just wrong.
But I'm not a crank, because Godel thought the cardinality
of the continuum was aleph_2!
--
hz
herbzet
"Jesse F. Hughes" wrote:
> "Dik T. Winter" writes:
>
> > Are you [lwalke3] really unable to recognise unsound arguments?
>
> Every argument is sound. We just have to find the right theory (and
> perhaps the right logic) to make it so.
Now you've got it.
--
hz
Virgil
wrote:
> In article <8763hrg...@phiwumbda.org> "Jesse F. Hughes"
> <je...@phiwumbda.org> writes:
> ...
> > (3) Cantor's theorem is "really" false, no matter what ZFC says.
> >
> > If he's really trying to show (1), then he might not be a crank, but
> > he's remarkably bad at basic communication. But you don't *really*
> > think that's his goal, do you?
> >
> > If he's trying to show (2), then he's failing. If he's trying to show
> > (3), then I can't imagine what he means because I don't understand
> > (3).
>
> It is what WM calls "absolute mathematics", although I have no idea what
> that means. And each time when I try to pin it down he evades and goes
> on with something else. For some reason WM is of the opinion that when
> a particular word is used in mathematics it can have only one meaning
> that is universal and does not require definition.
And that meaning for WM varies, like WM's potentially infinite sets,
over time.
Ralf Bader
> On Mar 30, 5:05 pm, Tim Little <t...@little-possums.net> wrote:
>> On 2009-03-30, lwal...@lausd.net <lwal...@lausd.net> wrote:
>> > After all, if LV doesn't want us to use ZFC, there must be _some_
>> > theory that LV would have us use instead of ZFC. And I'm trying to
>> > find what such a theory might be.
>> Why must there be such a theory any more rigorous than "if LV says it
>> is so, then it is so?"
>
> The point I was trying to make is, there are all these posters
> trying to tell us how bad ZFC is. Suppose we all listened to
> them and abandoned ZFC. I want to know what theory they would
> want us to start using once we'd have abandoned ZFC.
Who is us? And what do people use ZFC for? Here is a listing of mathematical
subjects:
http://www.ams.org/msc/
What do you think in how many of these subjects ZFC is explicitly used?
> In other words, LV believes that the theory ZFC is unsound, so
> now I want to know what theory he believes to be _sound_.
>
> So I want to fill in the blank:
>
> LV believes that ZFC is unsound.
> LV believes that ___ is sound.
>
> Don't tell us what theory we should _not_ use, tell us what
> theory we _should_ use (in LV's opinion).
And what do you think how many of the people working in the subjects of
mentioned classification care what LV thinks they should use?
Probably LV doesn't have any kind of "theory" at all. Concerning
Mueckenheim, it is a bit different. He has a kind of "theory" according to
which mathematics is a part of physics; you can read more about it in
various "papers" he has dumped on the arXiv. Numbers are physical objects
(maybe like elementary particles), and knowledge about them is obtained by
experiment and observation. No need to say more about that.
lwa...@lausd.net
wrote:
> ross.finlay...@gmail.com wrote:
> > That infinite sets are equivalent, in terms of standard, ...
> Yes, Ross, brilliant. But did you know:
> 1. There are but two infinities, actual and potential.
> 2. 0.9 repeating is below one by the infinitely nonzero small.
> And have you read:
> http://www.elsewhere.org/pomo/
I see that Chandler is starting to link to that Postmodernism
Generator website whenever someone makes a post in support of
an alternate theory. And I see that the PoMo Generator is a
random generator of big, impressive-sounding words in order
to make what sounds like a credible essay, but really isn't.
Evidently the point being made is that posters like RF and MR
(someone wondered whether MR was a real person earlier) make
arguments in favor of their alternate theories that sound as
if they're just meaningless words put together.
I want to see a post in support of an alternate theory that
isn't going to be compared to the PoMo generator. Is such a
post possible, or will the standard analysts keep comparing
any such attempt to PoMo?
Brian Chandler
> On Mar 30, 12:38 pm, Brian Chandler <imaginator...@despammed.com>
> wrote:
> > ross.finlay...@gmail.com wrote:
> > > That infinite sets are equivalent, in terms of standard, ...
> > Yes, Ross, brilliant. But did you know:
> > 1. There are but two infinities, actual and potential.
> > 2. 0.9 repeating is below one by the infinitely nonzero small.
> > And have you read:
> > http://www.elsewhere.org/pomo/
>
> I see that Chandler is starting to link to that Postmodernism
> Generator website whenever someone makes a post in support of
> an alternate theory.
Kindly name the "alternate theory" that Ross is posting in support of.
While you're at it, why not name the "alternate theory" that the PoMo
generator is advocating.
In particular, Ross has been saying "infinite sets are equivalent" for
years now, and has never managed to answer the obvious question: "What
do you mean by this?"
Brian Chandler
Of course infinite sets are equivalent, given the right equivalence
relation...
David C. Ullrich
<imagin...@despammed.com> wrote:
>David C. Ullrich wrote:
>> On Sun, 29 Mar 2009 05:11:57 -0700 (PDT), LudovicoVan
>> <ju...@diegidio.name> wrote:
>>
>> >Cantor's argument is erroneous and its adoption leads to unsound
>> >mathematics.
>> >
>> >The basic idea in the argument is that there is no bijection between
>> >the set of counting numbers and the set of infinite binary strings.
>> >But such a bijection exists, it can be expressed in terms of limit
>> >points, or by transfinite induction; informally, it can be defined as
>> >the correspondence between the paths and the leaf (i.e. limit) nodes
>> >in the infinite binary tree. This invalidates all results relating to
>> >Cantor's transfinite.
>>
>> This is nonsense. Being charitable about the notion of "limit node":
>> Yes, there's a bijection between the limit nodes and the paths.
>
>Do tell -- in the charitable interpretation, what _is_ a "limit node"?
>
>(Just asking: I don't know if there is some sort of usage along these
>lines, but it looks to me as though "limit node" could only really
>mean "path", in which case the putative bijection seems rather
>degenerate.)
Well yes, in any case the bijection in question is going to be
more or less obvious. But one can easily give a realization
of an infinite binary tree where there's a natural notion
of "limit" node which is not literally the same thing as
"path".
For example, say the 2^n nodes at level n are the 2^n
intervals, each of length 3^(-n), that come up in the
standard construction of the middle-thirds Cantor
set. Order the nodes by inclusion. Then a limit node
"is" just a point of the Cantor set...
(Yes, that's bending over backwards to be "charitable";
yes, in an abstract setting the only reasonable definition
of "limit node" is "path"...)
>
>Brian Chandler
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Gus Gassmann
[...]
> Normally, I say that I don't support theories that go against
> the well-known arithmetic of small finite numbers. But some
> of the numbers mentioned in the JSH threads (such as RSA
> numbers, for example) _aren't_ small anymore. The largest RSA
> number is greater than the number of particles in the known
> universe, and so WM, for example, wouldn't even accept the
> largest RSA number as a natural number!
I think that is misreading WM. He has no problem with the existence of
a googolplex, for instance. WM thinks that the natural numbers have
"holes" in them. The "average" 100-digit number does not exist and
will never exist, but a whole pile of them do. And here is one I just
created this morning:
820719687095959909098560295297219287625 \
094762986795865265845658429847446856623
I believe by this act of creation I have also created its successor
and perhaps a few other numbers, as well, but this does in no way fill
in all the holes. Peano was a wuss.
mikek...@googlemail.com
> On Mar 30, 5:05 pm, Tim Little <t...@little-possums.net> wrote:
> > On 2009-03-30, lwal...@lausd.net <lwal...@lausd.net> wrote:
> > > After all, if LV doesn't want us to use ZFC, there must be _some_
> > > theory that LV would have us use instead of ZFC. And I'm trying to
> > > find what such a theory might be.
> > Why must there be such a theory any more rigorous than "if LV says it
> > is so, then it is so?"
>
> The point I was trying to make is, there are all these posters
> trying to tell us how bad ZFC is. Suppose we all listened to
> them and abandoned ZFC. I want to know what theory they would
> want us to start using once we'd have abandoned ZFC.
>
> In other words, LV believes that the theory ZFC is unsound, so
> now I want to know what theory he believes to be _sound_.
>
> So I want to fill in the blank:
>
> LV believes that ZFC is unsound.
> LV believes that ___ is sound.
>
> Don't tell us what theory we should _not_ use, tell us what
> theory we _should_ use (in LV's opinion).
LV doesn't believe ZFC is unsound. LV doesn't have an alternate
theory. LV is a raving incompetent or a troll.
I simply cannot fathom why you are so eternally optimistic that idiots
are going to supply you with interesting non-standard set theories to
study. If you're interested in such theories then go read the
literature or make a thread requesting pointers on where to read up on
the subject. _Don't_ interlocute into threads started by trolls or
incompetents and try to defend them and pretend that their detractors
are being unreasonable. It's asinine and annoying and is never going
to get you the results you want. Surely you can see that, after a year
or more of pointless disingenuity?
Also, please stop _lying_ about what people post. It's dishonest and
very rude.
Daryl McCullough
>I see that Chandler is starting to link to that Postmodernism
>Generator website whenever someone makes a post in support of
>an alternate theory.
Why do you keep saying the same falsehoods? It isn't that
Ross (or WM or LV) are supporting some alternative theory,
it is that what they are doing is not rigorous mathematics.
If somebody adds fractions by writing 1/2 + 1/3 = 2/5, they
aren't working in alternative theory, they are being incompetent
at mathematics.
You are the one who keeps conflating "being incompetent" and
"being alternative". That's completely wrong. A competent
mathematician can work with alternative theories. People have
given the examples of (1) NFU, (2) Aczel's non-well-founded
set theory, (3) Robinson's nonstandard analysis, (4) Martin
Lof's type theory, (5) lambda calculus. Competent mathematicians
can work with all these theories and will not be called
crackpots for doing so. People are called crackpots for
doing bad (sloppy, nonrigorous) mathematics.
--
Daryl McCullough
Ithaca, NY
Jesse F. Hughes
> I want to see a post in support of an alternate theory that
> isn't going to be compared to the PoMo generator. Is such a
> post possible, or will the standard analysts keep comparing
> any such attempt to PoMo?
Why do you keep pretending that the issue is standard vs. nonstandard
theories rather than clear, sound mathematical reasoning
vs. incoherent proclamations?
Here, I'll make a post like you want. I like ZFA, the theory of
non-well-founded sets. It is not the standard set theory, but it's
fun and allows for some fairly natural constructions of certain
objects.
I support ZFA.
--
Jesse F. Hughes
"Hey, I look stupid all the time. That's not news."
-- James S. Harris
Dik T. Winter
...
> The largest RSA
> number is greater than the number of particles in the known
> universe, and so WM, for example, wouldn't even accept the
> largest RSA number as a natural number!
You are extremely bad at reading. WM himself has contradicted that. As
he has written, he accepts 10^(10^(10^10)) as a natural number. He only
states that not so many natural numbers can exist at the same time. So,
if 10^(10^(10^))) is a natural number at a certain moment there are numbers
smaller than that that do not exist. And when you fill in one of the holes,
some other number ceases to exist.
Gus Gassmann
> In article <75cf28c9-3888-46c6-944e-0ff8a9169...@l13g2000vba.googlegroups.com> lwal...@lausd.net writes:
> ...
> > The largest RSA
> > number is greater than the number of particles in the known
> > universe, and so WM, for example, wouldn't even accept the
> > largest RSA number as a natural number!
>
> You are extremely bad at reading. WM himself has contradicted that. As
> he has written, he accepts 10^(10^(10^10)) as a natural number. He only
> states that not so many natural numbers can exist at the same time. So,
> if 10^(10^(10^))) is a natural number at a certain moment there are numbers
> smaller than that that do not exist. And when you fill in one of the holes,
> some other number ceases to exist.
_May_ cease to exist. I am not sure we are at the point yet that
numbers are starting to disappear. But it will happen. No doubt about
it.
stephen.p.harris
> On 29 Mar, 16:48, Newberry <newberr...@gmail.com> wrote:
>> On Mar 29, 5:11 am, LudovicoVan <ju...@diegidio.name> wrote:
>>
>>> Cantor's argument is erroneous and its adoption leads to unsound
>>> mathematics.
>> Everybody knows this. The point is that they managed to create a
>> consistent axiomatic system, where Cantor's argument goes through.
>
> What goes through is an invalid argument, based on the false
> assumption that such a bijection cannot be formulated, and coming to
> the self-contradictory conclusion of the existence of uncountable
> sets. In fact, ex falso quod libet, and as I have suggested in the OP,
> you can produce such bijection, so the opposite of Cantor's
> assumption, within the same language: so even formal consistency I
> suppose is lost.
>
>> So
>> you have to come up with an axiomatic system of set theory where
>> Cantor's argument does not go through.
>>
>> Such systems alredy exist e.g. Myhill's constructive set theory or
>> Markov's theory. I do not know why these are not more widely known or
>> what the problems with them are. But if you think that Cantor's
>> argument is wrong this is the question you may want to answer.
>
> I can only guess why Myhill or Markiv are not widely adopted, and this
> risks to be quite beyond the scope of the present topic.
>
> -LV
>
>>> The problem is possibly much broader, and deeper, because it is the
>>> very soundness of archimedean arithmetic that seems at stake here. I
>>> say "soundness" because from the archimedean framework a tension
>>> results, between computability and tractability, that manifests a
>>> deeper tension between sound and unsound mathematics, and then even
>>> logic.
OP doesn't mean original post but refers to a person the original
poster. Mathematics is not established by physical reality. There
are no infinities in reality; Zeno's Paradox is purely an abstraction.
You wrote a bunch of gibberish. The Continuum Hypothesis has never
been proven. It assumes the Axiom of Choice which is logically
independent of the other axioms of Zermelo–Fraenkel set theory.
So it is speculation about an abstraction, infinity. There is no
fact of the matter. If you make assumptions called axioms then
Cantor's diagonal argument works like Godel's Incompleteness.
"In the philosophy of mathematics, constructivism asserts that it is
necessary to find (or "construct") a mathematical object to prove that
it exists. When one assumes that an object does not exist and derives a
contradiction from that assumption, one still has not found the object
and therefore not proved its existence, according to constructivists."
"Constructivists opposed those "proofs by contradiction" that were used
to establish the existence of infinite sets of irrational and
transcendental numbers."
"Nevertheless, the constructivists eventually lost the battle to the
formalists. Just as one cannot imagine mathematics giving up on the
number zero, so mathematicians refused to give up on the properties
of the real numbers that Cantor had demonstrated."
So the dispute between constructivists and formalists/Cantor has been
going on for over 100 years. Since there is no fact of the matter, just
like is the Buddha, or the Tao, supreme undifferentiated consciousness,
I think the word "erroneous" is misused. Like calling the best flavor of
ice cream chocolate correct, and thinking the strawberry flavor is an
error. This topic is no more than philosophical opinions disputing areas
where there is no fact of the matter, because the areas are symbolic
inventions of the mind which don't reflect a physical reality of truth.
No formalist or constructivist is ever going to be right just uneducated
and the law of the excluded middle is just another story in Dragnet=NYC,
Stephen
Jesse F. Hughes
> So I want to fill in the blank:
>
> LV believes that ZFC is unsound.
> LV believes that ___ is sound.
LV has never said the former statement, as far as I know. He says
that Cantor's proof is "erroneous". That sounds to me as if he's
claiming that the proof is not valid in ZFC.
--
"Yeah, I know, it's quantum [computing], and all kind of interesting physics
associated with what is to many a mystical word, but I have a B.Sc. in physics,
and I know that you're just talking about specialized mechanical devices when
you talk about quantum computing." -- James S. Harris
Aatu Koskensilta
> Also, please stop _lying_ about what people post. It's dishonest and
> very rude.
Lwalke is not lying, he's merely working in a very non-standard theory
of human behaviour.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
> People are called crackpots for doing bad (sloppy, nonrigorous)
> mathematics.
Usually they aren't. We've all produced oodles of bad, sloppy,
non-rigorous mathematics, in our student days in the very least, if
not at a later time, but I doubt many of us got called crakpots for
that.
Aatu Koskensilta
> I support ZFA.
Crank! Off with your non-standard head!
Peter Webb
"Brian Chandler" <imagin...@despammed.com> wrote in message
news:e0ddfaf6-c04b-49a6...@i28g2000prd.googlegroups.com...
> David C. Ullrich wrote:
>> On Sun, 29 Mar 2009 05:11:57 -0700 (PDT), LudovicoVan
>> <ju...@diegidio.name> wrote:
>>
>> >Cantor's argument is erroneous and its adoption leads to unsound
>> >mathematics.
>> >
>> >the set of counting numbers and the set of infinite binary strings.
>> >But such a bijection exists, it can be expressed in terms of limit
>> >points, or by transfinite induction; informally, it can be defined as
>> >the correspondence between the paths and the leaf (i.e. limit) nodes
>> >in the infinite binary tree. This invalidates all results relating to
>> >Cantor's transfinite.
>>
>> This is nonsense. Being charitable about the notion of "limit node":
>> Yes, there's a bijection between the limit nodes and the paths.
>
> Do tell -- in the charitable interpretation, what _is_ a "limit node"?
>
The equivalent of a Cauchy sequence using a binary tree metaphor.
That a Real can be expressed as the limit of a sequence of finite binary
trees - which is what I suspect a "limit node" is - should not come as a
surprise, as a Real is also the limit of a sequence of rationals (in fact
many such), and they are countable.
James Burns
> stevend...@yahoo.com (Daryl McCullough) writes:
>
>>People are called crackpots for doing bad (sloppy,
>>nonrigorous) mathematics.
>
> Usually they aren't. We've all produced oodles of bad,
> sloppy, non-rigorous mathematics, in our student days
> in the very least, if not at a later time, but I doubt
> many of us got called crakpots for that.
So, what would be a compact description of people that
do get called cranks or crackpots?
After a small amount of reflection, I offer my theory:
Crackpots, like everyone else, started with bad, sloppy,
non-rigorous mathematics, but then, instead of erasing
the results that came from poor arguments, they erased
the arguments that gave poor results. By "poor results"
I mean of course, results that they did not want, that
did not match their intuitions, that did not win them
world-wide adoration and boatloads of money, and so on.
What other theories of crankogenesis do people hold?
Jim Burns
Daryl McCullough
>
>stevend...@yahoo.com (Daryl McCullough) writes:
>
>> People are called crackpots for doing bad (sloppy, nonrigorous)
>> mathematics.
>
>Usually they aren't. We've all produced oodles of bad, sloppy,
>non-rigorous mathematics, in our student days in the very least, if
>not at a later time, but I doubt many of us got called crakpots for
>that.
I guess the difference is that the non-crackpot is correctable:
he can be convinced that he is wrong (or that that he doesn't
understand something as well as he thought he did). The crackpot
is incorrigible.
As I have said before, there is a sense in which we are
all crackpots, only on different topics. Lwalke is a crackpot
on the topic of crackpots.
Aatu Koskensilta
> I guess the difference is that the non-crackpot is correctable: he
> can be convinced that he is wrong (or that that he doesn't
> understand something as well as he thought he did). The crackpot is
> incorrigible.
This is probably just a matter of personal inclination but I would
include pretentiousness in the mix of required traits. I've met many
people who have to their satisfaction explained to themselves the
fundamental nature of mathematics, time, life in general, who have a
solution to the liar, the problem of free will, and what not, and
whose musings on these subjects are of little if any philosophical or
mathematical value in the traditional sense -- their musings are
simply something that gives them some pleasure. These people are
entirely harmless, and do not go about foaming at the mouth about evil
Cantorians, conspiracies to stop them posting on Usenet, and so on. It
would be highly obnoxious to try and ram technicalities down their
throats, and we're probably best served by reminding ourselves we too
inevitably foster all sorts of somewhat silly pet ideas in areas
outside our expertise, often without being at all aware of this.
The true crackpot, in contrast, combines his eccentric ideas with
tedious pretentiousness. Towards them I have no sympathy; or, as Paul
Feyerabend once put it
Now I have no objection to incompetence but I do object when
incompetence is accompanied by boredom and self-righteousness.
> As I have said before, there is a sense in which we are
> all crackpots, only on different topics.
Indeed.
> Lwalke is a crackpot on the topic of crackpots.
It is, I posit, a rather refreshing form of crackpottery.
Herbert Newman
> As I have said before, there is a sense in which we are
> all crackpots, only on different topics. Lwalke is a crackpot
> on the topic of crackpots.
Right. Hence I once called him a "meta-crank". :-)
Herb
Jesse F. Hughes
> "Jesse F. Hughes" <je...@phiwumbda.org> writes:
>
>> I support ZFA.
>
> Crank! Off with your non-standard head!
You'll have to find it first! Ha ha! (Hint: I keep it in my head.)
--
Jesse F. Hughes
"I'm ruler", said Yertle, "of all that I see.
But I don't see enough. That's the trouble with me."
-- Yertle the Turtle, by Dr. Suess
Virgil
"Peter Webb" <webbf...@DIESPAMDIEoptusnet.com.au> wrote:
What are you claiming countable?
The set of all sequences of rationals certainly isn't, and the set of
reals also isn't, at least for the standard definition of countability.
lwa...@lausd.net
> lwal...@lausd.net writes:
> > So I want to fill in the blank:
> > LV believes that ZFC is unsound.
> > LV believes that ___ is sound.
> LV has never said the former statement, as far as I know.
But Hughes himself did. Since no one will believe this claim
unless I provide a verbatim quote, such a verbatim quote is
in order here.
So here is the verbatim quote from Hughes, which he posted
last night, March 30th, a little after 8:30 PM, U.S. Eastern
Daylight time (after midnight the morning of 31st March,
Greenwich Mean Time):
> Goedel presumably thought that ZFC was
> sound but did not prove enough. LV believes that ZF is unsound (or
> inconsistent).
OK, so according to what Hughes wrote last night, LV believes
that ZF is unsound or inconsistent, and of course if ZF is
inconsistent, then so is ZFC. Thus, based on what Hughes said
last night, it is all right for me to write that LV believes
ZF(C) to be unsound and inconsistent.
> He says
> that Cantor's proof is "erroneous". That sounds to me as if he's
> claiming that the proof is not valid in ZFC.
Based on both the post from last night and this comment here,
we are in case (2) of the three cases Hughes presented
yesterday about LV's perspective. I'd already said that this
is the most likely of the three cases. Of course, we know
that whatever errors LV found in Cantor's proof are not
really sufficient to prove ZF(C) to be inconsistent.
What I was wondering in this post is, suppose someone were to
say, "LV, you're right! Cantor's proof really is erroneous,
and ZF(C) really is unsound and inconsistent. Therefore, I
want to use a new theory without Cantor's erroneous proof, a
theory that is really is sound. What theory should I use?" I
want to know what theory that would be, a theory that we
ought to use in lieu of the allegedly unsound ZF(C) with its
allegedly erroneous Cantor proof.
That's all I want to know in this post.
lwa...@lausd.net
> On Mar 31, 3:42 am, lwal...@lausd.net wrote:
> > Don't tell us what theory we should _not_ use, tell us what
> > theory we _should_ use (in LV's opinion).
> are going to supply you with interesting non-standard set theories to
> study. If you're interested in such theories then go read the
> literature or make a thread requesting pointers on where to read up on
> the subject.
Here we go again with reading books. Since I can't afford these
books and I'm not a professor with university access, the only
way for me to access these books would be to _steal_ them.
That's right, _steal_ books. I can't make the money for the
books appear out of thin air! Therefore, the only way possible
for me to access these books would be to _steal_ them.
> Also, please stop _lying_ about what people post.
If this refers to:
> > LV believes that ZFC is unsound.
> LV doesn't believe ZFC is unsound.
then, even if it does count as a lie about what LV has himself
written, it's not a lie about what Jesse Hughes has written. A
verbatim quote:
lwa...@lausd.net
wrote:> Why do you assume this? Can't you read?
Yes, in fact, I _did_ read -- a post from Jesse Hughes to be
exact, where he states:
"Goedel presumably thought that ZFC was
sound but did not prove enough. LV believes that ZF is unsound (or
inconsistent)."
> > Notice how an ultrafinitist could claim to have solved the
> > factoring problem by providing a factoring method that is
> > efficient only for sufficiently small natural numbers, then
> > state that the natural numbers that are too large to be
> > effectively factored by the method don't really exist.
> Fascinating! So ultrafinitists can crack encryption by fiat...
Well, such ultrafinitists wouldn't have exactly cracked
encryption, since they still wouldn't be able to hack into a
system encrypted by the factors of a number that they don't
believe doesn't exist. They'd merely be able to factor all the
numbers that they do believe exist.
Since I don't know enough about non-"crank" ultrafinitists (i.e.
ultrafinitists other than those who post at sci.math), I don't
know what a non-"crank" ultrafinitist would say about the
factoring of naturals that they don't believe to exist, or the
encryption based on the factors of "non-existent" naturals.
I'd like to know a non-"crank" ultrafinitist's opinion of the
factoring problem -- unless I'm required to read a book that I
can't afford, since the only way for me to access the book
would be for me to _steal_ it...
MoeBlee
> On Mar 31, 4:13 am, mikekell...@googlemail.com wrote:
>
> > On Mar 31, 3:42 am, lwal...@lausd.net wrote:
> > > Don't tell us what theory we should _not_ use, tell us what
> > > theory we _should_ use (in LV's opinion).
> > I simply cannot fathom why you are so eternally optimistic that idiots
> > are going to supply you with interesting non-standard set theories to
> > study. If you're interested in such theories then go read the
> > literature or make a thread requesting pointers on where to read up on
> > the subject.
>
> Here we go again with reading books.
I want to participate in horse riding, but I don't have a horse and I
don't have a pass to any stables.
I want to learn archery, but I don't have a bow or arrows and I don't
have access to an archery range.
I want to learn about the history of medieval Russia, but I don't have
any way to get books on the subject and I don't have a pass to any
museums or libraries.
So, instead I tell people they're not riding horses correctly, that
there is a better way to shoot arrows, and that standard research in
medieval Russian history is all horribly biased
This all seems to work for me. Don't you agree?
MoeBlee
lwa...@lausd.net
> lwal...@lausd.net wrote:
> > So case (3) can entail a type of Platonism. If Goedel is
> > free to believe that CH is "really" false, no matter what ZFC
> > says, then LV is free to believe that Cantor's Theorem is
> > false, no matter what ZFC says.
> That's nothin'!
> *I* believe that there is a greatest prime number, no matter
> what Peano arithmetic or Euclid say. They're just wrong.
An ultrafinitist, perhaps? Since ultrafinitists believe that
there is a largest natural number, it follows that they
would believe in a greatest prime number as well.
Once again, I can't be sure as to what ultrafinitists, even
the alleged non-"crank" ultrafinitists, have to say about the
existence of large prime numbers and factoring.
> But I'm not a crank, because Godel thought the cardinality
> of the continuum was aleph_2!
Of course there can be a rigorous theory in which there are
exactly aleph_2 reals, namely ZFC+"c=aleph_2."
lwa...@lausd.net
> lwal...@lausd.net wrote:
> > The point I was trying to make is, there are all these posters
> > trying to tell us how bad ZFC is. Suppose we all listened to
> > them and abandoned ZFC. I want to know what theory they would
> > want us to start using once we'd have abandoned ZFC.
> subjects:http://www.ams.org/msc/
> What do you think in how many of these subjects ZFC is explicitly used?
One so-called "crank" (not a sci.math poster) once estimated that
two-thirds of modern mathematics is dependent on set theory.
Sections 26-xx through 49-xx are related to the uncountability of
the real numbers, to which LV is opposed.
> > Don't tell us what theory we should _not_ use, tell us what
> > theory we _should_ use (in LV's opinion).
> And what do you think how many of the people working in the subjects of
> mentioned classification care what LV thinks they should use?
> Probably LV doesn't have any kind of "theory" at all.
Well, that's what I'm trying to find out. I want to know what
theory LV would have us use to replace 26-xx to 49-xx.
> Mueckenheim, it is a bit different. He has a kind of "theory" according to
> which mathematics is a part of physics; you can read more about it in
> various "papers" he has dumped on the arXiv. Numbers are physical objects
> (maybe like elementary particles), and knowledge about them is obtained by
> experiment and observation.
OK, so WM is more focused on sections 70-xx through 86-xx,
since these are related to physics. Apparently, anything
outside these sections are irrelevant to WM -- well, let's
include 97-xx, Mathematics Education, since WM is after
all a professor and instructor of mathematics.
Dave L. Renfro
> Here we go again with reading books. Since I can't
> afford these books and I'm not a professor with
> university access, the only way for me to access
> these books would be to _steal_ them.
Maybe you could give up internet access for a few
weeks to pay for a book? Or, if you're on a fixed
income, you can tutor high school students in your
spare time (what I've been doing most days of the
week, after I get off work). Or, you can try using
interlibrary loan with your town's public library.
Of course, the obvious thing would be to simply
look through university math department web pages
for notes and handouts posted by faculty for their
courses.
Dave L. Renfro
lwa...@lausd.net
wrote:
> On Mar 31, 12:42 am, lwal...@lausd.net wrote:
> > Normally, I say that I don't support theories that go against
> > the well-known arithmetic of small finite numbers. But some
> > of the numbers mentioned in the JSH threads (such as RSA
> > numbers, for example) _aren't_ small anymore. The largest RSA
> > number is greater than the number of particles in the known
> > universe, and so WM, for example, wouldn't even accept the
> > largest RSA number as a natural number!
> I think that is misreading WM. He has no problem with the existence of
> a googolplex, for instance. WM thinks that the natural numbers have
> "holes" in them. The "average" 100-digit number does not exist and
> will never exist, but a whole pile of them do. And here is one I just
> created this morning:
> 820719687095959909098560295297219287625 \
> 094762986795865265845658429847446856623
So in other words, WM really would accept the existence of all
the natural numbers involved in RSA and encryption, both the
P2's and their prime factors? OK.
As far as the number Gassmann mentions, it's about as large as
some of the smallest RSA numbers. I have no idea whether it's
prime, a P2, or has many small factors. I wonder what WM would
say about the existence of its factors.
Meanwhile, note that AP has declared any number larger than
googol^5 to be infinite. The number Gassmann mentions would
still be finite, but the largest of the RSA numbers would be
considered infinite. So I also wonder what AP would say about
the largest RSA number and its factorization.
mikek...@googlemail.com
> On Mar 31, 4:13 am, mikekell...@googlemail.com wrote:
>
> > On Mar 31, 3:42 am, lwal...@lausd.net wrote:
> > > Don't tell us what theory we should _not_ use, tell us what
> > > theory we _should_ use (in LV's opinion).
> > I simply cannot fathom why you are so eternally optimistic that idiots
> > are going to supply you with interesting non-standard set theories to
> > study. If you're interested in such theories then go read the
> > literature or make a thread requesting pointers on where to read up on
> > the subject.
>
> Here we go again with reading books. Since I can't afford these
> books and I'm not a professor with university access, the only
> way for me to access these books would be to _steal_ them.
>
> That's right, _steal_ books. I can't make the money for the
> books appear out of thin air! Therefore, the only way possible
> for me to access these books would be to _steal_ them.
You can't access the math textbooks you would like. So the logical
thing to do then is to rabidly defend mathematical incompetents on the
internet and smear anyone who dismisses them? You've been doing this
for over a year and a half. How is it working out for you so far?
Would it be unkind of me to point out that the huge amount of time
you've wasted defending the indefensible and attacking perfectly
reasonable people could'be been spent earning money somehow to buy
some actual math books?
> > Also, please stop _lying_ about what people post.
>
> If this refers to:
> <snip>
It doesn't.
It refers to when you say such-and-such is proposing an alternate
theory to ZFC when they're doing nothing of the sort and, in fact,
they're just terminally confused about something.
It refers to when you attack people for refusing to consider any
theory besides ZFC when they repeatedly and explicitly tell you that
they do not hold that position.
It's inane. Stop it.
mikek...@googlemail.com
>
> What I was wondering in this post is, suppose someone were to
> say, "LV, you're right! Cantor's proof really is erroneous,
> and ZF(C) really is unsound and inconsistent. Therefore, I
> want to use a new theory without Cantor's erroneous proof, a
> theory that is really is sound. What theory should I use?" I
> want to know what theory that would be, a theory that we
> ought to use in lieu of the allegedly unsound ZF(C) with its
> allegedly erroneous Cantor proof.
>
> That's all I want to know in this post.
And you're never going to get what you want out of people like LV. Can
you honestly not see that after a year and a half of trying?
Stop wasting your time.
lwa...@lausd.net
> On Mar 31, 12:42 pm, lwal...@lausd.net wrote:
> > Here we go again with reading books.
> I want to participate in horse riding, but I don't have a horse and I
> don't have a pass to any stables.
What's the analogy here? Obviously I correspond to the horse
rider without a horse, but what about the person telling
people they're not riding horses correctly? That sounds more
like what the standard analysts are doing. They tell people
(the so-called "cranks") they're not riding horses (or doing
math) correctly. Also, riding horses requires finding a physical
horse and riding it. It's not something that can be done on
Usenet, as opposed to math.
> I want to learn about the history of medieval Russia, but I don't have
> any way to get books on the subject and I don't have a pass to any
> museums or libraries.
> So, instead I tell people [...] that standard research in
> medieval Russian history is all horribly biased
A more apt analogy, I admit.
I am not a historian, but I've heard the old cliche that history
books are written by the winners. In that sense, _all_ history
can be said to be biased.
But there is one difference between history and math. In history
either an event occurred, or an event didn't occur. Thus, going
back to MoeBlee's example of Russian history, either Ivan the
Terrible was the tsar of Russia, or he wasn't. There is no room
for an alternate opinion.
But in math, there can be many different theories. In some
theories the existence of infinite sets is provable, and in
others it's not provable. In some theories, there can be
infinitesimals, while in other theories, the numbers are
Archimedean and so there are no infinitesimals. And so on.
Some posters at sci.math are labeled "cranks." Why? The
prevailing opinion appears to be that the reason is that their
arguments lack rigor. Then I want to have the opportunity to
make the "cranks'" arguments more rigorous, so that they won't
be called "cranks" anymore. There'll be no reason to call them
"cranks" anymore if their arguments were rigorous.
I'm not trying to rewrite Russian history. I'm trying to come
up with a theory so that "cranks" are no longer "cranks." But
I don't want to hear that the only way to accomplish this is
to spend money I don't have.
Jesse F. Hughes
> On Mar 31, 5:59 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> lwal...@lausd.net writes:
>> > So I want to fill in the blank:
>> > LV believes that ZFC is unsound.
>> > LV believes that ___ is sound.
>> LV has never said the former statement, as far as I know.
>
> But Hughes himself did. Since no one will believe this claim
> unless I provide a verbatim quote, such a verbatim quote is
> in order here.
Fair enough. I believe I misrepresented LV's views until I saw
someone else state more clearly what LV has said, namely, not that a
particular theory is bad, but that a particular (and clearly valid)
argument is "erroneous".
--
"So now you see a math person coming out to talk about *his* program
which is fast as he says it can count over 89 billions primes in less
than a second. How is that objective? It's childish."
-- James S. Harris, on objective facts.
Jesse F. Hughes
> On Mar 30, 9:13 pm, Brian Chandler <imaginator...@despammed.com>
> wrote:
>> lwal...@lausd.net wrote:
>> > In other words, LV believes that the theory ZFC is unsound,
>> Why do you assume this? Can't you read?
>
> Yes, in fact, I _did_ read -- a post from Jesse Hughes to be
> exact, where he states:
>
> "Goedel presumably thought that ZFC was
> sound but did not prove enough. LV believes that ZF is unsound (or
> inconsistent)."
Yes, I was in error. Brian's correct that LV has not said ZF is bad,
but a particular argument (which is clearly valid in ZF) is
"erroneous".
--
But in our enthusiasm, we could not resist a radical overhaul of the
system, in which all of its major weaknesses have been exposed,
analyzed, and replaced with new weaknesses.
-- Bruce Leverett (presumably with apologies to Ambrose Bierce)
Ralf Bader
> On Mar 31, 2:46 am, Gus Gassmann <horand.gassm...@googlemail.com>
> wrote:
>> On Mar 31, 12:42 am, lwal...@lausd.net wrote:
>> > Normally, I say that I don't support theories that go against
>> > the well-known arithmetic of small finite numbers. But some
>> > of the numbers mentioned in the JSH threads (such as RSA
>> > numbers, for example) _aren't_ small anymore. The largest RSA
>> > number is greater than the number of particles in the known
>> > universe, and so WM, for example, wouldn't even accept the
>> > largest RSA number as a natural number!
>> I think that is misreading WM. He has no problem with the existence of
>> a googolplex, for instance. WM thinks that the natural numbers have
>> "holes" in them. The "average" 100-digit number does not exist and
>> will never exist, but a whole pile of them do. And here is one I just
>> created this morning:
>> 820719687095959909098560295297219287625 \
>> 094762986795865265845658429847446856623
>
> So in other words, WM really would accept the existence of all
> the natural numbers involved in RSA and encryption, both the
> P2's and their prime factors? OK.
Try again to understand what has been written above.
> As far as the number Gassmann mentions, it's about as large as
> some of the smallest RSA numbers. I have no idea whether it's
> prime, a P2, or has many small factors. I wonder what WM would
> say about the existence of its factors.
He would say something like "blurgoffwhackidilipsuköafdslgbklkb"
> Meanwhile, note that AP has declared any number larger than
> googol^5 to be infinite.
I don't note what AP declares. I find it sufficient to note a certain part
of what Mueckenheim says. Mueckenheim displays the syndrome of crankism in
such a full-fledged way that it is superfluous to study other sources in
the field.
> The number Gassmann mentions would
> still be finite, but the largest of the RSA numbers would be
> considered infinite. So I also wonder what AP would say about
> the largest RSA number and its factorization.
Ultrafinitists which are not braindead leave it open what the largest
existing number actually is. There is a story about Harvey Friedman asking
Yessenin-Volpin about the existence of numbers. Friedman's first question
was whether 2 exists. Answer: Yes. Then he asked whether 4 exists.
Answer "uh, yes", and for each following power of 2 Yessenin's answer
needed about twice the time of the previous one.
Ralf
--
How lucky we are that Cantor introduced curly brackets! But it was no[t]
he who introduced the silly distinction between a and {a} that enables
so called mathematicians to build card houses on nothing.
(Prof. Dr. W. Mückenheim, mathematical mastermind of "Augsburg University of
Applied Science" , in sci.math)
MoeBlee
> On Mar 31, 12:54 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Mar 31, 12:42 pm, lwal...@lausd.net wrote:
> > > Here we go again with reading books.
> > I want to participate in horse riding, but I don't have a horse and I
> > don't have a pass to any stables.
> > So, instead I tell people they're not riding horses correctly,
>
> What's the analogy here? Obviously I correspond to the horse
> rider without a horse, but what about the person telling
> people they're not riding horses correctly? That sounds more
> like what the standard analysts are doing. They tell people
> (the so-called "cranks") they're not riding horses (or doing
> math) correctly.
The cranks are most often not doing math (riding horses) correctly. A
mathematician (horse rider) knows something about that. Your role is
to tell the mathematicians (horse riders) that they're not doing math
(riding horses) properly since they don't allow for just any gibberish
to qualify as math (horse riding). Lying in the mud and letting horses
trample over you is not a form of horse riding.
> Also, riding horses requires finding a physical
> horse and riding it. It's not something that can be done on
> Usenet, as opposed to math.
No, understanding math (at this level) requires access to writings
about the subjecvt. One needs books or classes or tutoring or
comprehensive lecture notes, or whatever it takes.
> > I want to learn about the history of medieval Russia, but I don't have
> > any way to get books on the subject and I don't have a pass to any
> > museums or libraries.
> > So, instead I tell people [...] that standard research in
> > medieval Russian history is all horribly biased
>
> A more apt analogy, I admit.
>
> I am not a historian, but I've heard the old cliche that history
> books are written by the winners. In that sense, _all_ history
> can be said to be biased.
Oh, criminy, one can find some point of difference with virtually ANY
such analogy.
> But there is one difference between history and math. In history
> either an event occurred, or an event didn't occur. Thus, going
> back to MoeBlee's example of Russian history, either Ivan the
> Terrible was the tsar of Russia, or he wasn't. There is no room
> for an alternate opinion.
There is no room for an alternate opinion as to whether a certain
sequence of formulas is or is not a proof in a system of recursive
axioms and recursive rules.
> But in math, there can be many different theories. In some
> theories the existence of infinite sets is provable, and in
> others it's not provable. In some theories, there can be
> infinitesimals, while in other theories, the numbers are
> Archimedean and so there are no infinitesimals. And so on.
And who has ever disagreed?
> Some posters at sci.math are labeled "cranks." Why?
We, and I, have TOLD you why, over and over and over!!!
I see below that you as much as undercut all of that by describing it
as merely "a lack of rigor".
> The
> prevailing opinion appears to be that the reason is that their
> arguments lack rigor.
Not JUST lack rigor, but are INCOHERENT. And OTHER attributes in
sufficient measure - spreading plain misinformation and confusion
about ZFC, stubborn illogic, circular repetition, uttely idiosyncratic
notation and terminology explained only in a circle of ad hoc
announcements, "theories" that no one can determine what is correct
for or not without personally consulting the crank himself for his
fiat on the matter (what another poster called the 'oracle-like'
nature of crank posting), evasion often of the form of such statements
as "Mine is the theory of reality; you just can't see it", various
straw men and red herrings, acting as if a poster's remarks are merely
personal insults when the poster gave BOTH mathematical argument and
explanation as well as giving DESERVED insults, continual shifting of
what point is at stake in the discussion, etc.
MoeBlee
FredJeffries
>
> I'd like to know a non-"crank" ultrafinitist's opinion of the
I have no idea whether any of the following [are/consider themselves
to be/would be considered by you to be] an ultrafinitist. But an
ultrafinitist might think along similar lines...
Brian Rotman's "Will the Digital Computer Transform Classical
Mathematics?"
http://www.wideopenwest.com/~brian_rotman/Phil.Trans.html
in the section "asymptotic growth" asks:
"What sense does it make sense for a science of computing to formulate
its fundamental concept of feasibility in terms of an infinitistic
criterion that necessarily appeals to numbers outside the reach of its
computational processes? Pragmatically, what is the relevance of the P
= NP question for categorizing (let alone understanding) the
possibilities and limits of computation? Why would its solution have
anything to say about real feasibility, about the possibility of
finding or not being able to find algorithms that worked within the
limits and freedoms of the real world rather than those of the
idealized, boundless universe of infinitistic mathematics?"
and quotes Arnold G. Reinhold's "P=?NP Doesn't Affect Cryptography"
http://theworld.com/%7Ereinhold/p=np.txt
Who says, "The still-unsolved problem of whether 'P=NP' has no
practical bearing on problems in cryptography"
Reinhold then quotes Steve Morgan:
"Even if factoring *is* polynomial, it isn't necessarily practical.
I'm sure we all remember from our "sorting and searching" classes just
how slow O(n**2) algorithms are. And only a few applications justify
an O(n**3) algorithm. A polynomial algorithm of, say O(n**20) is
essentially intractable even for small values of n."
Reinhold adds "Conversely, an algorithm that ran in O(exp(n**0.1))
would make factoring billion bit numbers easy. The equivalence
relation used to define the class P is not a meaningful one when real
computations are involved. Most polynomials represent a computation
cost far to high to bear."
Later:
"To say that functions which are usefully 'one-way' can't exist if
P=NP is nonsense. In reality, polynomial time functions can be too
hard to compute, while superpolynomial functions can be perfectly
computable for all practical purposes."
FredJeffries
> > still be finite, but the largest of the RSA numbers would be
> > considered infinite. So I also wonder what AP would say about
> > the largest RSA number and its factorization.
>
> Ultrafinitists which are not braindead leave it open what the largest
> existing number actually is. There is a story about Harvey Friedman asking
> Yessenin-Volpin about the existence of numbers. Friedman's first question
> was whether 2 exists. Answer: Yes. Then he asked whether 4 exists.
> Answer "uh, yes", and for each following power of 2 Yessenin's answer
> needed about twice the time of the previous one.
>
> Ralf
Hey! That's MY story:
http://groups.google.com/group/sci.math/msg/90c1f4ed1161a3a8?hl=en
which I stole from
http://dialinf.wordpress.com/2009/02/16/achilles-tortoise-and-yessenin-volpin/
who got it from
http://www.math.ohio-state.edu/~friedman/pdf/Princeton532.pdf
<quote>
I have seen some ultrafinitists go so far as to challenge the
existence of 2^100 as a natural number, in the sense of there being a
series of "points" of that length. There is the obvious "draw the
line" objection, asking where in 2^1, 2^2, 2^3, ... , 2^100 do we stop
having "Platonistic reality"? Here this ... is totally innocent, in
that it can be easily be replaced by 100 items (names) separated by
commas. I raised just this objection with the (extreme) ultrafinitist
Yessenin-Volpin during a lecture of his. He asked me to be more
specific. I then proceeded to start with 2^1 and asked him whether
this is "real" or something to that effect. He virtually immediately
said yes. Then I asked about 2^2, and he again said yes, but with a
perceptible delay. Then 2^3, and yes, but with more delay. This
continued for a couple of more times, till it was obvious how he was
handling this objection. Sure, he was prepared to always answer yes,
but he was going to take 2^100 times as long to answer yes to 2^100
then he would to answering 2^1. There is no way that I could get very
far with this.
</quote>
Virgil
<1c1263af-8a8d-4bcd...@b6g2000pre.googlegroups.com>,
lwa...@lausd.net wrote:
> On Mar 31, 12:54 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> > On Mar 31, 12:42 pm, lwal...@lausd.net wrote:
> > > Here we go again with reading books.
> > I want to participate in horse riding, but I don't have a horse and I
> > don't have a pass to any stables.
> > So, instead I tell people they're not riding horses correctly,
>
> What's the analogy here? Obviously I correspond to the horse
> rider without a horse, but what about the person telling
> people they're not riding horses correctly? That sounds more
> like what the standard analysts are doing. They tell people
> (the so-called "cranks") they're not riding horses (or doing
> math) correctly. Also, riding horses requires finding a physical
> horse and riding it. It's not something that can be done on
> Usenet, as opposed to math.
I have not seen any math "done on the internet", though it can be, and
sometimes is, transmitted via the internet.
Jesse F. Hughes
> Some posters at sci.math are labeled "cranks." Why? The
> prevailing opinion appears to be that the reason is that their
> arguments lack rigor. Then I want to have the opportunity to
> make the "cranks'" arguments more rigorous, so that they won't
> be called "cranks" anymore. There'll be no reason to call them
> "cranks" anymore if their arguments were rigorous.
But if you make arguments that are rigorous and correct, it does not
follow that *their* reasoning becomes retroactively valid. It just
means that there is a theory in which their conclusions happen to be
true.
Take Mitch, for example. He says that it's "mathematically true" (or
something like that) that there is a least number greater than zero.
Suppose you concoct a theory in which it is the case that there is a
least number greater than zero. How does this support his
non-argument? His conclusion is still wrong in the most obvious
interpretation and he has still provided no argument at all.
So, even if you succeed, you succeed only in showing that there is a
theory in which some of their claims are true. Mitch would be a crank
nonetheless.
--
Jesse F. Hughes
"The past two days I haven't had any beer or any wine."
-- Quincy P. Hughes, Age 4 and so damn European
Ioannis
[snip]
> Some posters at sci.math are labeled "cranks." Why? The
> prevailing opinion appears to be that the reason is that their
> arguments lack rigor.
I think that's roughly right. Note that mathematical/logical rigor has an
additional useful property which can be used to define something like two
'equivalence classes' in the domain of free-thinkers and ultimately separates
cranks from professionals. Let me try to demonstrate this property hopefully in
a manner which may be useful to your understanding of the issue:
A healthy, professional mathematical mind or any serious scientist, one who has
decades of training, and certainly the great majority of those who have a Ph.D.
in a relevant science, is able to cross-check his/her own reason and arguments
using this very "rigor" you hear/speak about and validate the results he
produces. This means that scientific rigor is a sort of "safety valve" for
emergent Truth.
This rigor besides being cross-checkable, has the strange property, that its
arguments are _universally acceptable_ by any serious scientist. Yes, scientists
are human, they have petty nuances, jealousy and hidden agendas, but no serious
scientist I think will deny the validity of a correct mathematical, say,
argument.
When you see someone present a theorem, a lemma, a discovery, a conjecture,
*anything* in a scientific newsgroup, such as sci.math, I'd expect close to ZERO
responses, if the argument is rigorous and correct. Silence is the best judge
that something interesting is happening there :-) Perhaps, rarely, a very short,
"good work", or something else such.
When you see someone present a discovery, conjecture or whatever and as a result
you see a barrage of responses debating the validity of the OP's claim, this
means something is seriously wrong with the OP's argument.
What *is* wrong? The most obvious culprit is that the OP has failed the
scientific rigor test. This means, that for practical purposes, the OP lacks the
capability to cross-check his/her own conjectures and work, before presenting
them.
To pick a specific example, although I am far from an expert in logic, Cantor's
diagonal argument seems 'insinctively' clear and correct to any healthy
mathematical mind. If you (the general 'you') finds a 'fault' with this
argument, this implies that there's something wrong with your reasoning
facilities.
Mathematical rigor, therefore, seems like it defines an equivalence relation
between people (for lack of better wording, perhaps), which separates people in
two classes: People who are able to cross-check their work and the work of
others against their own reason and those who cannot and as a result spout
nonsense.
Mind you, I don't think this 'self-validating' rigor is something easy to
achieve. Speaking from experience, I think it took me, roughly 15 - 20 years
after I graduated to come to terms with it.
I believe you are sufficiently equipped, as a mathematician (or amateur
mathematician, I don't really know what your credentials are, sorry), to be able
to distinguish whose work can stand self-scrutiny and whose can't. You've amply
demonstrated many times (in particular in threads about tetration which involve
me, for example) that you can and do have the basic ability to understand
scientific rigor and the ideas behind tetration, so I am at a loss to explain
WHY you want to do what you say below:
> Then I want to have the opportunity to
> make the "cranks'" arguments more rigorous, so that they won't
> be called "cranks" anymore. There'll be no reason to call them
> "cranks" anymore if their arguments were rigorous.
In my opinion that's like trying to want to "prove the Bible or the Kuran
consistent and logically rigorous". Well, you can, to a certain extent using
some variant or other of insane logic or illogic, but you have to understand
that whether you manage to do that or not, a "believer" in the Bible or Kuran
for example, will always BE "a believer in the Bible or Kuran", no matter how
rigorous the final result is from a self-consistency standpoint, hence such a
person cannot belong to the same class that a rigorous scientist belongs,
because he accepts different axioms than the majority of the scientific
community.
Why try to "de-crank" the cranks, when there are so many interesting areas in
mathematics to explore? It can only hurt your reputation, as you have seen from
some of the ratings you got on your posts, because ultimately it means that you
are attracted to falsity.
Science is not the pursuit of falsity. It is the pursuit of Truth. And if you
insist on "making various cranks' arguments rigorous" (whatever that may mean,
anyway), imo science will eventually displace you as obsolete and I don't think
you want that.
As with any potential scientific mind, that would be a tremendous waste.
Please excuse my long and boring intrusion on the subject. I simply recognize
some good potential on lwalke and I would hate it if he goes bonkers, trying to
"de-crank" the "cranks" :-)
--
Ioannis --- "There's _always_ a mistake, somewhere."