this has no proof, therefore formal compilers don't work
|-|erc
the sky is falling, long live Godel!
TOTAL BULLSHIT mathematics today
have any of you actually seen where ZFC derives uncountable infinity.
I haven't, I bet most people here haven't they simply regurgitate it,
but I can guarantee one thing, it doesn't derive higher infinity,
what axiom would that come from? what it does do is blindly calculates
an extra number to the list of reals, with no idea what a number line
looks like, then human operaters say "oh gee just like Cantors proof,
the only meaning is.. TADA uncountable infinity". TOTAL BULLSHIT
throw away your chicken shit century old math texts
anything that quotes Godel or Cantor as mathematicians is BS
================GODEL DISPROOF===============
IT: in certain systems there are true statements that are unprovable.
Assume ~IT
In all systems, all true statements have some proof. #
let G = G has no proof
Assume G is false
G has no proof is false
G has a proof
G is true
CONTRADICTION
Assume G is true
G has a proof (from #)
G is true
G
G has no proof
CONTRADICTION
G is undecidable
IT is not provable from Godels proof
So it seems the incompleteness theorem does not stand up to proof by resolution.
==================CANTOR DISPROOF======================
The anti-diagonal is not computable, therefore it doesn't exist.
let UTM(real, digit) mod 10 calculate the list of reals. (where real is deceptively an integer)
the antidiagonal is UTM(digit, digit) + 1 mod 10
if this is computable, then some TM computes it and some emulated TM also computes it,
UTM(ad, digit) mod 10 = UTM(digit, digit) + 1 mod 10
when digit = ad,
UTM(ad, ad) mod 10 = UTM(ad, ad) + 1 mod 10
Contradiction!
Therefore antidiag is an invalid formula.
This is a more sensible conclusion than higher infinities exist as I've basically just
rearranged Cantor's proof. One can argue its a valid specification of a sequence
but it doesn't actually compute a new sequence of digits.
Herc
--
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Jan Burse
> have any of you actually seen where ZFC derives uncountable infinity.
> I haven't, I bet most people here haven't they simply regurgitate it,
> but I can guarantee one thing, it doesn't derive higher infinity,
> what axiom would that come from? what it does do is blindly calculates
From the Power Set axiom. And by the insight
that P(X) > X for X > 1. On the other hand ZFC
has a countable model, which is kind of a
paradox. (And for some cranks like elmesiak
an inconquerable obstacle)
But you might also read the original paper
by cantor, which came before the development
of ZFC, where cantor proofs uncountability
of the line.
Bye
Jan Burse
> that P(X) > X for X > 1.
Err, can strengthen this to:
P(X) > X for X > 0.
Don Stockbauer
As far as doing anything the least little bit useful and
constructivistic and contributing to our comfortable survival, there
are only two infinities:
1. Potential
2. Actual
Alan Smaill
ju...@diegidio.name
> |-|erc schrieb:
>
> > have any of you actually seen where ZFC derives uncountable infinity.
> > I haven't, I bet most people here haven't they simply regurgitate it,
> > but I can guarantee one thing, it doesn't derive higher infinity,
> > what axiom would that come from? what it does do is blindly calculates
>
> From the Power Set axiom.
So you just make an axiom of it. Apart from the arbitrariness, it's
more and more clear that this leads to unavoidable contradictions and
then, ex falso quod libet: what you like holds, and what you don't
like does not hold.
> And by the insight
> that P(X) > X for X > 1.
That insight is of no use as soon as we start talking about infinite
sets.
-LV
ju...@diegidio.name
> On 2 Aug, 16:20, Jan Burse <janbu...@fastmail.fm> wrote:
>
> > |-|erc schrieb:
>
> > > have any of you actually seen where ZFC derives uncountable infinity.
> > > I haven't, I bet most people here haven't they simply regurgitate it,
> > > but I can guarantee one thing, it doesn't derive higher infinity,
> > > what axiom would that come from? what it does do is blindly calculates
>
> > From the Power Set axiom.
>
> So you just make an axiom of it. Apart from the arbitrariness, it's
> more and more clear that this leads to unavoidable contradictions and
> then, ex falso quod libet: what you like holds, and what you don't
> like does not hold.
>
> > And by the insight
> > that P(X) > X for X > 1.
>
> That insight is of no use as soon as we start talking about infinite
> sets.
>
> -LV
>
> > On the other hand ZFC
> > has a countable model, which is kind of a
> > paradox.
It's full of paradoxes: what else should we expect?
-LV
Virgil
"|-|erc" <h@r.c> wrote:
> oh no, the statement said it had no proof, mathematics is incomplete
> the sky is falling, long live Godel!
>
> TOTAL BULLSHIT mathematics today
>
> have any of you actually seen where ZFC derives uncountable infinity.
Yes!
> I haven't, I bet most people here haven't they simply regurgitate it,
Did you expect everyone to re-invent the proof all by themelves?
That is not normally how mathematics works. Once one person proves
something, others are allowed to use the same proof without paying
royalties.
> but I can guarantee one thing, it doesn't derive higher infinity,
Herkimer's guarantees are not money in the bank.
> what axiom would that come from?
Several of them.
> what it does do is blindly calculates
> an extra number to the list of reals
If Herkimer is talking about Cantor's diagonal construction, the number
it constructs is definitely a member of the standard set of reals.
And therefore there is no "list of all reals" possible.
> with no idea what a number line looks like,
What has any sort of number line to do with anything?
> then human operaters say
That would seem to argue that those like Herkimer and WM. et al, are not
human. Which might explain a lot.
>
> ================GODEL DISPROOF===============
>
> IT: in certain systems there are true statements that are unprovable.
>
> Assume ~IT
> In all systems, all true statements have some proof. #
>
>
> let G = G has no proof
>
>
> Assume G is false
> G has no proof is false
> G has a proof
> G is true
> CONTRADICTION
>
>
> Assume G is true
> G has a proof (from #)
> G is true
> G
> G has no proof
> CONTRADICTION
Non sequitur
>
>
> G is undecidable
> IT is not provable from Godels proof
>
>
> So it seems the incompleteness theorem does not stand up to proof by
> resolution.
That it may not satisfy the forms of illogic that herkimer argues by
does not invalidate it in the eyes of those with better logic.
>
>
> ==================CANTOR DISPROOF======================
>
> The anti-diagonal is not computable, therefore it doesn't exist.
>
> let UTM(real, digit) mod 10 calculate the list of reals. (where real is
> deceptively an integer)
>
> the antidiagonal is UTM(digit, digit) + 1 mod 10
No it is not. Herkimer again proves he does not understand what he is
criticising. Thus the res tof his pseudo-analysis is all non sequitur.
Jan Burse
Yes, the distinction is very simple.
Potential Infinite
==================
For example in the pythagorean triangle equation:
x^2 + y^2 = z^2
The x, y and z range over reals. This is a potential
infinity. Although the variables will only hold
a value from the line, they may assume an infinitude
of different values.
Actual Infinite
===============
Now in
P(X) > X for X > 0,
X is ranging over a set. And sets are thought of as
not simply being points from an infinite whole, but
as being infinite in them selfs. Thus actual infinite.
Cantor was struggeling with the actual infinite.
But to my taste the poor guy was trapped into a
religios tightness of the epoch, maybe fearing to
call demons into being when dealing with infinity.
Anyway
I recommend:
Michael Hallett: Cantorian Set Theory and
Limitation of Size, Oxford Logic Guides 10,
1984
Jan Burse
Oops. Hm.
P(0) = {{}}
0 = {}
Yep
Jan Burse
>> And by the insight
>> that P(X) > X for X > 1.
>
> That insight is of no use as soon as we start talking about infinite
> sets.
We ARE talking about finite sets my dear.
The above holds for all sets finite and not finite.
bye
Jan Burse
> It's full of paradoxes: what else should we expect?
A little green man will bite your balls??
ju...@diegidio.name
> ju...@diegidio.name schrieb:
>
> >> And by the insight
> >> that P(X) > X for X > 1.
>
> > That insight is of no use as soon as we start talking about infinite
> > sets.
>
> We ARE talking about finite sets my dear.
Are we?
> The above holds for all sets finite and not finite.
So, are you even?
Of course the above holds for finite sets. What happens for not finite
sets is at stake here... if I am not mistaken.
-LV
> bye
Jan Burse
But existence of an infinite set needs the infinity axiom.
Power set axiom alone is not enough...
Bye
ju...@diegidio.name
A little green man is enough... ;)
-LV
Dave Seaman
Power set alone is enough to prove that |P(X)| > X for all cardinals X.
Even without the axiom of infinity, it's still possible that infinite
sets might exist. We just can't prove that they exist. However, the
power set axiom alone is still enough to prove that |P(X)| > X for all
cardinals X, finite or infinite.
--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
Virgil
<2fe206fe-4a39-4088...@d45g2000hsc.googlegroups.com>,
Don Stockbauer <donsto...@hotmail.com> wrote:
For set theories in which sets identical must have the same members in
order to be called eaual, there is no such thing as a potentially
infinite set.
Don Stockbauer
> In article
> <2fe206fe-4a39-4088-8c62-357a7f58d...@d45g2000hsc.googlegroups.com>,
> Don Stockbauer <donstockba...@hotmail.com> wrote:
>
>
>
> > On Aug 2, 10:23 am, Jan Burse <janbu...@fastmail.fm> wrote:
> > > Jan Burse schrieb:
>
> > > > that P(X) > X for X > 1.
>
> > > Err, can strengthen this to:
>
> > > P(X) > X for X > 0.
>
> > As far as doing anything the least little bit useful and
> > constructivistic and contributing to our comfortable survival, there
> > are only two infinities:
>
> > 1. Potential
>
> > 2. Actual
>
> For set theories in which sets identical must have the same members in
> order to be called eaual, there is no such thing as a potentially
> infinite set.
I'm just saying there is a cybernetic way of looking at infinity. The
potential infinity, where you set up an endless loop knowing someday
it will fail, say in spacecraft from a failed battery. Then there is
the actualized infinity, aleph null, which is a mathematical concept
and cannot be used for real world problems. From what I can tell
reading these discussions once you try to derive different types,
"flavors" of the actual infinity it seems you go down a slippery slope
of endless discussion, perhaps because infinity by it's very nature
lends itself to endless discussions. That's all I mean, just a bit of
philosophy at this point.
Virgil
<c8e54691-334b-41e1...@l64g2000hse.googlegroups.com>,
ju...@diegidio.name wrote:
> On 2 Aug, 16:20, Jan Burse <janbu...@fastmail.fm> wrote:
> > |-|erc schrieb:
> >
> > > have any of you actually seen where ZFC derives uncountable infinity.
> > > I haven't, I bet most people here haven't they simply regurgitate it,
> > > but I can guarantee one thing, it doesn't derive higher infinity,
> > > what axiom would that come from? what it does do is blindly calculates
> >
> > From the Power Set axiom.
>
> So you just make an axiom of it. Apart from the arbitrariness, it's
> more and more clear that this leads to unavoidable contradictions and
> then, ex falso quod libet: what you like holds, and what you don't
> like does not hold.
That seems to be your favorite form of argument, julio.
>
> > And by the insight
> > that P(X) > X for X > 1.
>
> That insight is of no use as soon as we start talking about infinite
> sets.
>
Since it is provable in, among other systems, ZFC, why not?
For any set X, P(X) is the set of all subsets of X, it is easily seen
that there is an injection from X to P(X), namely, for each x in X,
x --> {x} is such an injection.
It is also clear that any f:X -> P(X) must fail to have
{x in X: not x in f(x)} in its range, so no surjection can exist.
Virgil
<fba7a962-e16a-4fb4...@w7g2000hsa.googlegroups.com>,
Don Stockbauer <donsto...@hotmail.com> wrote:
> On Aug 2, 12:20 pm, Virgil <Vir...@gmale.com> wrote:
> > In article
> > <2fe206fe-4a39-4088-8c62-357a7f58d...@d45g2000hsc.googlegroups.com>,
> > Don Stockbauer <donstockba...@hotmail.com> wrote:
> >
> >
> >
> > > On Aug 2, 10:23 am, Jan Burse <janbu...@fastmail.fm> wrote:
> > > > Jan Burse schrieb:
> >
> > > > > that P(X) > X for X > 1.
> >
> > > > Err, can strengthen this to:
> >
> > > > P(X) > X for X > 0.
> >
> > > As far as doing anything the least little bit useful and
> > > constructivistic and contributing to our comfortable survival, there
> > > are only two infinities:
> >
> > > 1. Potential
> >
> > > 2. Actual
> >
> > For set theories in which sets identical must have the same members in
> > order to be called eaual, there is no such thing as a potentially
> > infinite set.
>
> I'm just saying there is a cybernetic way of looking at infinity. The
> potential infinity, where you set up an endless loop knowing someday
> it will fail, say in spacecraft from a failed battery.
There are 'programs' which , when executed, will not end of themselves,
but that is not at all the same thing as a set which is infinite within
the axioms and definitions of some set theory.
herbzet
Don Stockbauer wrote:
> Jan Burse wrote:
> > Jan Burse schrieb:
> >
> > > that P(X) > X for X > 1.
> >
> > Err, can strengthen this to:
> >
> > P(X) > X for X > 0.
>
> As far as doing anything the least little bit useful and
> constructivistic and contributing to our comfortable survival, there
> are only two infinities:
>
> 1. Potential
>
> 2. Actual
How many infinities are there if we don't so constrain ourselves?
--
hz
herbzet
Virgil wrote:
> In article <OD_kk.25023$IK1....@news-server.bigpond.net.au>,
> "|-|erc" <h@r.c> wrote:
[...]
> > ================GODEL DISPROOF===============
> >
> > IT: in certain systems there are true statements that are unprovable.
> >
> > Assume ~IT
> > In all systems, all true statements have some proof. #
>
> >
> >
> > let G = G has no proof
> >
> >
> > Assume G is false
> > G has no proof is false
> > G has a proof
> > G is true
We're assuming all statements with a proof are true? OK.
> > CONTRADICTION
> >
> >
> > Assume G is true
> > G has a proof (from #)
> > G is true
> > G
> > G has no proof
> > CONTRADICTION
>
> Non sequitur
??? Seems alright so far. "G has no proof" contradicts "G has a proof"
three lines previous.
> > G is undecidable
??? _This_ seems like a non-sequitur.
> > IT is not provable from Godels proof
All you've shown is that the assumption of ~IT leads to a contradiction.
Therefore, it's false.
> > So it seems the incompleteness theorem does not stand up to proof by
> > resolution.
Um, whatever.
--
hz
|-|erc
There were 2 assumptions. 1 of them is wrong.
~IT and G = G has no proof
They are incompatible.
Actually my negation of the imcompleteness theorem is too general.
~IT = there exists a theory% where all true statements have a proof
% = a sound recursively enumerable extension of Robinson Arithmetic
What I have shown is a consistent theory where Godel statements are not theorems.
i.e. a system where ~IT.
Therefore the Incompleteness Theorem is disproven.
Herc
herbzet
So you're saying that the statement G cannot exist on the assumption of ~IT.
But G exists. Therefore ...
> Actually my negation of the imcompleteness theorem is too general.
> ~IT = there exists a theory% where all true statements have a proof
That is not the negation of "in certain systems there are true
statements that are unprovable."
> % = a sound recursively enumerable extension of Robinson Arithmetic
>
> What I have shown is a consistent theory where Godel statements are not theorems.
> i.e. a system where ~IT.
> Therefore the Incompleteness Theorem is disproven.
Whatever, dude.
--
hz
MoeBlee
> On 2 Aug, 16:20, Jan Burse <janbu...@fastmail.fm> wrote:
>
> > |-|erc schrieb:
>
> > > have any of you actually seen where ZFC derives uncountable infinity.
> > > I haven't, I bet most people here haven't they simply regurgitate it,
> > > but I can guarantee one thing, it doesn't derive higher infinity,
> > > what axiom would that come from? what it does do is blindly calculates
>
> > From the Power Set axiom.
>
> So you just make an axiom of it. Apart from the arbitrariness,
(1) The question is whether ZFC proves that there are uncountable
sets. And ZFC does prove that there are unountable sets. (2) That
certain axioms are arbitrary or not is a separate question from what
the theorems are from those axioms. (3) Read Boolos's 'Logic Logic
Logic' for a discussion of whether the axioms of ZF are arbitrary.
> it's
> more and more clear that this leads to unavoidable contradictions
You've never shown a contradition in ZFC.
> > And by the insight
> > that P(X) > X for X > 1.
Not just for X>1. For ANY X.
> That insight is of no use as soon as we start talking about infinite
> sets.
Whether it's of "use" or not, the fact is that just a few simple
principles prove that there is no surjection from a set onto its power
set.
MoeBlee
MoeBlee
> On Sat, 02 Aug 2008 19:11:15 +0200, Jan Burse wrote:
> > Jan Burse schrieb:
> >> ju...@diegidio.name schrieb:
> >>>> And by the insight
> >>>> that P(X) > X for X > 1.
>
> >>> That insight is of no use as soon as we start talking about infinite
> >>> sets.
>
> >> We ARE talking about finite sets my dear.
>
> >> The above holds for all sets finite and not finite.
>
> >> bye
> > But existence of an infinite set needs the infinity axiom.
> > Power set axiom alone is not enough...
>
> Power set alone is enough to prove that |P(X)| > X for all cardinals X.
And if we don't have the power set axiom, then statements about PX are
pretty much irrelevent anyway. So, given that PX is meaningful (in the
sense of properly referring), then all that is needed to prove that
there is not a surjection from X onto PX is intuitionistic logic and
the axiom schema of separation. I.e., GIVEN the power set axiom, all
else we need is intuitionistic logic and the axiom schema of
separation. And if we are NOT given the power set axiom, then the
matter is pretty much irrelevent anyway.
MoeBlee
MoeBlee
> On Aug 2, 12:20 pm, Virgil <Vir...@gmale.com> wrote:
> > In article
> > <2fe206fe-4a39-4088-8c62-357a7f58d...@d45g2000hsc.googlegroups.com>,
> > Don Stockbauer <donstockba...@hotmail.com> wrote:
> > > 1. Potential
>
> > > 2. Actual
>
> > For set theories in which sets identical must have the same members in
> > order to be called eaual, there is no such thing as a potentially
> > infinite set.
>
> I'm just saying there is a cybernetic way of looking at infinity. The
> potential infinity, where you set up an endless loop knowing someday
> it will fail, say in spacecraft from a failed battery. Then there is
> the actualized infinity, aleph null, which is a mathematical concept
> and cannot be used for real world problems.
It is used to axiomatize mathematics used for scientific problems.
> From what I can tell
> reading these discussions once you try to derive different types,
> "flavors" of the actual infinity it seems you go down a slippery slope
> of endless discussion, perhaps because infinity by it's very nature
> lends itself to endless discussions. That's all I mean, just a bit of
> philosophy at this point.
Potentially endless discussions, right? Anyway, it's not much of a
philosophical position that a notion is to be rejected because it is
claimed that the notion has "in its nature" that it can be discussed
endlessly.
MoeBlee
Balthasar
wrote:
>>>
>>> But existence of an infinite set needs the infinity axiom.
>>> Power set axiom alone is not enough...
>>>
>> Power set alone is enough to prove that |P(X)| > |X| for all X.
>>
> And if we don't have the power set axiom, then statements about PX are
> pretty much irrelevant anyway. So, given that PX is meaningful (in the
> sense of properly referring), then all that is needed to prove that
> there is not a surjection from X onto PX is intuitionistic logic and
> the axiom schema of separation. I.e., GIVEN the power set axiom, all
> else we need is intuitionistic logic and the axiom schema of
> separation. And if we are NOT given the power set axiom, then the
> matter is pretty much irrelevant anyway.
>
Well, I would like to disagree, if you don't mind.
We could still prove the following theorem:
Az(z e x <-> z c y) -> |x| > |y|.
With other words, if there were a set x, such that x contained all and
only the subsets of some set y then the cardinality of that set x would
be larger than the cardinality of the set y.
And this result could still be applied for concrete cases. Say for x_0 =
{{}, {a}, {b}, {a,b}} and y_0 = {a, b}. After all, it's easy to verify
that we have
Az(z e x_0 <-> z c y_0)
in this case.
B.
--
"For every line of Cantor's list it is true that this line does not
contain the diagonal number. Nevertheless the diagonal number may
be in the infinite list." (WM, sci.logic)
Balthasar
>
> We could still prove the following theorem:
>
> Az(z e x <-> z c y) -> |x| > |y|.
>
> With other words, if there were a set x, such that x contained all and
> only the subsets of some set y then the cardinality of that set x would
> be larger than the cardinality of the set y.
>
This approach -of course- is inspired by Russell's remark:
"Mathematics may be defined as the subject where we never know what we
are talking about, nor whether what we are saying is true."
Virgil
Balthasar <nomail@invalid> wrote:
> On Mon, 4 Aug 2008 10:48:29 -0700 (PDT), MoeBlee <jazz...@hotmail.com>
> wrote:
>
> >>>
> >>> But existence of an infinite set needs the infinity axiom.
> >>> Power set axiom alone is not enough...
> >>>
> >> Power set alone is enough to prove that |P(X)| > |X| for all X.
> >>
> > And if we don't have the power set axiom, then statements about PX are
> > pretty much irrelevant anyway. So, given that PX is meaningful (in the
> > sense of properly referring), then all that is needed to prove that
> > there is not a surjection from X onto PX is intuitionistic logic and
> > the axiom schema of separation. I.e., GIVEN the power set axiom, all
> > else we need is intuitionistic logic and the axiom schema of
> > separation. And if we are NOT given the power set axiom, then the
> > matter is pretty much irrelevant anyway.
> >
> Well, I would like to disagree, if you don't mind.
>
> We could still prove the following theorem:
>
> Az(z e x <-> z c y) -> |x| > |y|.
That depends on whether the "c" in your "z c y" means any subset or
proper subset, at lest for y = {}.
MoeBlee
> > And if we don't have the power set axiom, then statements about PX are
> > pretty much irrelevant anyway. So, given that PX is meaningful (in the
> > sense of properly referring), then all that is needed to prove that
> > there is not a surjection from X onto PX is intuitionistic logic and
> > the axiom schema of separation. I.e., GIVEN the power set axiom, all
> > else we need is intuitionistic logic and the axiom schema of
> > separation. And if we are NOT given the power set axiom, then the
> > matter is pretty much irrelevant anyway.
>
> Well, I would like to disagree, if you don't mind.
>
> We could still prove the following theorem:
>
> Az(z e x <-> z c y) -> |x| > |y|.
>
> With other words, if there were a set x, such that x contained all and
> only the subsets of some set y then the cardinality of that set x would
> be larger than the cardinality of the set y.
Of course I agree with that. Indeed it's the kind of thing I would
point to in any situation in which the supporting axiom is taken away.
There is always still the HYPOTHETICAL - IF there is a set of such a
nature, THEN such and such. I didn't mean to preclude that kind of
thing by my remark (which is misleading and not well stated) about
'irrelevance'.
In fact, you can see that I also argued that the power set axiom is
not even needed for the proof in this form: Given any set Y that is a
set that has as members all subsets of X, it is not the case that
there is surjection from Y onto X. That is provable with just
intuitionistic logic and the axiom schema of separation.
MoeBlee
Balthasar
wrote:
>>
>> We could still prove the following theorem:
>>
>> Az(z e x <-> z c y) -> |x| > |y|.
>>
>> With other words, if there were a set x, such that x contained all and
>> only the subsets of some set y then the cardinality of that set x would
>> be larger than the cardinality of the set y.
>>
> Of course I agree with that. Indeed it's the kind of thing I would
> point to in any situation in which the supporting axiom is taken away.
> There is always still the HYPOTHETICAL - IF there is a set of such a
> nature, THEN such and such.
>
Right. Actually, we could reformulate _any_ theorem Phi in ZFC the
following way
A_1 & ... & A_n -> Phi
where A_i are the axioms used in the proof of Phi.
Of course, this is not the usual way to do math. On the other hand, this
shows that those endless discussions with cranks are rather pointless:
what has been shown beyond any doubt just is: IF this and that, THEN ...
The question IF A_1, ..., A_n actually "do hold", is rather moot. While
cranks, on the other hand, have a tendency to assert that this and that
IS THE CASE, and/or CANNOT BE THE CASE, etc.
(Ok, I assumed that our framework is classical logic here. But that's
fine, I'm talking about _classical mathematics_ here.)
>
> In fact, you can see that I also argued that the power set axiom is
> not even needed for the proof in this form: Given any set Y that is a
> set that has as members all subsets of X, it is not the case that
> there is surjection from Y onto X. That is provable with just
> intuitionistic logic and the axiom schema of separation.
>
Right.
MoeBlee
> (Ok, I assumed that our framework is classical logic here. But that's
> fine, I'm talking about _classical mathematics_ here.)
My only point about that was that Cantor's theorem doesn't even NEED
full classical logic, but rather only the intuitionistic subpart.
(Even just the minimal logic subpart?).
MoeBlee