New Anti-Cantor Attack
|-|ercules
e.g.
P(N) = {
1 - {1},
2 - {1,2},
3 - {2},
4 - {1,2,3}
...
}
The set of indexes that aren't members of their subset is
{3,4}
in this finite subset example.
And voila, {3,4} is a new subset not present in P(N).
Therefore no matter how big P(N) is there is always a new element
that can be listed and therefore the size of the set P(N) is bigger than infinity.
-----------------------------------------------------------
Here's my equivalent proof of uncountable infinity.
Let's assume an enumeration of naturals exists, call it N.
N= {
1,
2,
3,
4,
...
}
Let's calculate a new natural MAX+1.
That is 4+1 = 5
in this finite subset example.
Voila 5 is a new number not in N
Therefore no matter how big N is there is always a new element
that can be listed and therefore the size of the set N is bigger than infinity.
-------------------------------------------------
To eliminate expected confusion I will make my motives explicit,
the second proof is a spoof of Cantor's proof, if you can find the
flaw in my proof, it also applies to Cantor's 2nd proof of uncountable infinity!
Herc
--
If you think you and your partner are "meant to be"
you just ticked half a dozen symptoms of psychiatric illness
http://tinyurl.com/ShrinksRape
Daryl McCullough
>
>Cantor's second proof of 'uncountable infinity' is based on trying to >enumerate
>the powerset of naturals.
[stuff deleted]
>Here's my equivalent proof of uncountable infinity.
This is an instance of a general theorem: for every correct
proof, there exists an incorrect proof that looks the same
to the mathematically incompetent.
--
Daryl McCullough
Ithaca, NY
|-|ercules
Not your usual stuff Daryl, handwaving and ad homs.
Yes I know it's such a simple yet equivalent logical deduction of 'bigger than infinity'
and really shows how dumb Cantor subscribers are to miss that, but the fundamental
mathematical truths are generally quite simple.
Herc
porky_...@my-deja.com
>
> And voila,
you mean "accordion", I presume.
PPJ.
Daryl McCullough
>
>"Daryl McCullough" <stevend...@yahoo.com> wrote ...
>> |-|ercules says...
>>>
>>>Cantor's second proof of 'uncountable infinity' is based on trying to >enumerate
>>>the powerset of naturals.
>>
>> [stuff deleted]
>>
>>>Here's my equivalent proof of uncountable infinity.
>>
>> This is an instance of a general theorem: for every correct
>> proof, there exists an incorrect proof that looks the same
>> to the mathematically incompetent.
>
>
>Not your usual stuff Daryl, handwaving and ad homs.
I'm sorry, but your post was too stupid to merit more than that.
Marshall
What is MAX? You just start using it without saying anything
about what it is, or where it comes from.
Marshall
|-|ercules
This is a serous attempt at a refutation of Cantor's powerset proof of the existence of higher infinities.
The powerset proof is exactly this:
Assume a large/infinite room full of boxes with fridge magnets in the boxes that are any natural number, and the boxes have a unique
number written on them.
"Which box contains the numbers of all the boxes that don't contain their own number?"
is proven (by Cantor) to be nonexistent.
So extrapolate how that demonstrates higher infinities for me?
Your evasion of showing a distinction between my spoof proof and Cantor's proof is noted, also your or anybody's evasion to dispute
my other 3 claims:
>> Disproof of the nonexistence of a halting algorithm,
>> disproof of the turing machine not being decomposable into simpler
>> computations, disproof that godels proof demonstrates people can
>> understand more facts than a formal system, disproof that no answer
>> to "which box contains the numbers of all the boxes that don't contain
>> their own number?" implies a set larger than infinity
You're all talk sci.math, either prove ZFC is complete and factual or address the questions.
Herc
|-|ercules
good advice!
Herc
Sylvia Else
He's just proving that the naturals are not a finite set.
Sylvia.
Charlie-Boo
That is not possible for infinite sets. You show that the set of
FINITE SUBSETS of N is in fact N1 (I say N0=finite, N1=natural
numbers, N2=Reals, etc.)
> in this finite subset example.
>
> Voila 5 is a new number not in N
>
> Therefore no matter how big N is there is always a new element
> that can be listed and therefore the size of the set N is bigger than infinity.
bigger than finite.
C-B
Charlie-Boo
wrote:
> |-|ercules says...
>
>
>
>
>
>
>
> >"Daryl McCullough" <stevendaryl3...@yahoo.com> wrote ...
> >> |-|ercules says...
>
> >>>Cantor's second proof of 'uncountable infinity' is based on trying to >enumerate
> >>>the powerset of naturals.
>
> >> [stuff deleted]
>
> >>>Here's my equivalent proof of uncountable infinity.
>
> >> This is an instance of a general theorem: for every correct
> >> proof, there exists an incorrect proof that looks the same
> >> to the mathematically incompetent.
>
> >Not your usual stuff Daryl, handwaving and ad homs.
>
> I'm sorry, but your post was too stupid to merit more than that.
The more stupider it is, the easier it is to refute so you should give
a more complete explanation of how it is stupid. If it is trivial
then you have no reason to spend the same number of keystrokes at
ridicule as is needed for the full proof.
C-B
> --
> Daryl McCullough
> Ithaca, NY- Hide quoted text -
>
> - Show quoted text -
Transfer Principle
> On May 28, 8:45 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > We'll see about that.
> Ahem.
I stand corrected.
> > Then what should I do the next time someone makes a claim
> > that is refuted by ZFC?
> You should say what you think the truth is, and why.
The posts today by Marshall Spight and others have convinced
me that Herc/Cooper isn't worth defending.
And so let me do as Spight has suggested, and say what I
think the truth about Cooper's proof attempt is, and why. I
am not trying to bully Herc by posting the following, but
merely following Spight's suggestion.
Earlier, Herc provided the following link to his proof:
Transfer Principle
> Earlier, Herc provided the following link to his proof:
Oops, Google error. I wanted to write the link to Herc's
website, where he attempted to prove that his claim
holds, at least for finite lists. But apparently, Google
fails whenever I try to provide the link or cut and
paste the lists of reals from his website. (I'm not sure
whether a full newsreader would have been able to handle
my cutting and pasting.) Suffice it to say that Cooper's
website failed to reorder the random list, leaving out
one of the reals when it didn't match the diagonal.
So Herc's claim, at least for the finite list, is false. We
were not able to reorder the list from step #1 to obtain
the diagonal chosen in step #2.
One would think that it would be easy to find a reordering
that matches the diagonal, considering that there are 60!
reorderings yet only 3^60 diagonals. (60! is 53 orders of
magnitude greater than 3^60.) And yet Cooper's algorithm
failed to find a reordering for this diagonal.
Note that this counterexample in the _finite_ case doesn't
necessary disprove the _infinite_ claim that given a ternary
real, there exists with probability 1 a reordering of the
list of computable reals whose diagonal is the given real.
So at least this conjecture is still open. Still, even if
the conjecture is true, Herc is still indefensible because
the conjecture doesn't imply that the reals are countable.
> [You have a] terrible taste in people.
There's that accusation again. I have a terrible taste in
people merely because I've decided to defend posters like
Cooper rather than posters like Spight.
I defend posters who in my opinion need defending. I don't
defend those like Spight, who often calls those who criticize
ZFC wrong, since the main posters who attack them are the
ones being called wrong. The posters who are called wrong
are the ones who need defending, because they are being
attacked left and right for being wrong.
And even though Herc/Cooper turned out to be not worth
defending in the end, that still doesn't mean that I'm going
to start defending Spight. If the only way to have a good
taste in people is to defend posters like Spight, then I'd
much rather have a bad taste in people.
Meanwhile, I noticed that a newbie poster has started a new
thread attacking Cantor a few hours ago. I just posted in that
thread right now. I hope to find out which of the four main
cases describes that poster.
Transfer Principle
> Cantor's second proof of 'uncountable infinity' is based on trying to enumerate the powerset of naturals.
> Let's assume an enumeration of naturals exists, call it N.
Herc/Cooper is hardly the first poster on sci.math to
attempt to apply the proof of the uncountability of
the set of reals to the set of naturals. (In essence,
one is attempting to prove that R is uncountable if
and only if N is uncountable.)
In previous threads, the proof attempt would entail
listing the naturals and then diagonalizing. But this
would typically produce a number with infinitely many
digits on the diagonal. But standard naturals have
only finitely many digits. Thus, the proof attempt is
considered invalid in standard theory.
But what about Herc's proof attempt? Let's see:
> N= {
> 1,
> 2,
> 3,
> 4,
> ...
> }
> Let's calculate a new natural MAX+1.
> That is 4+1 = 5
> in this finite subset example.
> Voila 5 is a new number not in N
So Herc lists the elements of N. But then one wonders,
what is the fifth element of Herc's list, given that its
first four elements are 1,2,3, and 4?
> Therefore no matter how big N is there is always a new element
> that can be listed and therefore the size of the set N is bigger than infinity.
If Cooper is assuming that N is the set {1,2,3,4}, then he
has just proved that card(N) is greater than _four_, not
countable infinity (aleph_0). But of course, standard
theory does not dispute that card(N) > 4.
We notice the similarity between Herc's proof and the
Euclid's proof of the infinitude of the primes -- in each
case, we take a finite list of elements of a set and use
those elements to produce a new element, simply by adding
one to either the product (in Euclid's case) or maximum
(in Herc's case). But as with Euclid, this only shows
that the set is _infinite_, not _uncountable_. Note that
only _finite_ sets of naturals have maxima, so Herc is
implying that the set is finite when he mentions MAX.
But let's give Cooper the benefit of the doubt, since he
does use an ellipsis, that he does intend his list to be
infinite after all. Perhaps he intended to give the list:
N =
{1,
2,
3,
4,
6,
7,
8,
9,
...
}
so this list is infinite, yet we can find a natural number,
namely 5, that's not on the list. Yet this still doesn't
establish that N is uncountable.
Before I proceed, one might wonder why I'm going through all
this effort just to prove Herc wrong, since the other thread
already established that Herc is in Case 3 (the case where
we can call him wrong). Well, Herc did ask the question:
> To eliminate expected confusion I will make my motives explicit,
> the second proof is a spoof of Cantor's proof, if you can find the
> flaw in my proof
then attacked Spight for failing to answer it:
> Your [Spight's] evasion of showing a distinction between my spoof
> proof and Cantor's proof is noted
So I'm trying to show the distinction between Herc's and
Cantor's proof, as Cooper requests of us.
To see where this lack of symmetry comes from, we notice the
standard definition of countable:
A set x is countable iff there exists a function f: omega -> x
such that f is surjective.
(Note: some mathematicians prefer "bijective" to "surjective"
in this above definition, thereby excluding finite sets from
the countable sets.)
Thus, to find the definition of uncountable, we take the
negation of the definition of countable:
A set x is uncountable iff for every function f: omega -> x,
f is not surjective.
This uses the well-known fact from standard FOL that the
negation of a formula with an existential quantifier is a
formula with a universal quantifier.
And now we see the lack of symmetry here. In order to show
that a set is countable, we only need to find _one_ list that
is complete, but to show that a set is uncountable, we must
show that _every_ list is incomplete.
So this is the difference between Cooper and Cantor. Cooper
found only _one_ list of naturals that is missing an element,
while Cantor showed that _every_ list of reals is missing at
least one element. And furthermore, we can trivially find a
list of naturals that isn't missing an element:
N =
{1,
2,
3,
4,
5,
6,
7,
8,
9,
...
}
From the definition of countable, this is all we need to
establish countability. Recall that it only takes _one_
complete list to prove that a set is countable, yet we
need _every_ list to be incomplete to prove that a set
is uncountable.
I hope that Herc won't take this post as bullying. Herc
asked a question and I gave the answer, which is more
than we can say of Spight, who evaded Herc's question.
Tim Little
> This is a serous attempt at a refutation of Cantor's powerset proof
An attempt caused by mind-affecting drugs in your serum?
Typo-jokes aside, it may have been a serious attempt, but also a
serious failure.
> The powerset proof is exactly this:
>
> Assume a large/infinite room full of boxes [...]
No it isn't. At best that is an analogy for the proof, and not a very
good one.
> So extrapolate how that demonstrates higher infinities for me?
The actual proof (not your abortive analogy) shows that the powerset
of *any* set (infinite or not) is larger in cardinality than the
original set. Since the set of natural numbers N is infinite, |2^N|
is a larger infinity.
- Tim
|-|ercules
If you take every possible computer program and input every possible natural input
then each program can be considered an element of the powerset of N.
e.g.
UTM(3,1) = 0
UTM(3,2) = 5
UTM(3,3) = 1
UTM(3,4) = 2
...
P(N)_3 = {0,5,1,2...}
This computable powerset covers a LOT of subsets of N. You can't list a subset that it
doesn't cover.
Yet you all believe there are INFINITELY MORE missing subsets than present subsets!
Back to the Cantor Spoof Proof...
The fact you can find a complete set of N doesn't refute my demonstration, merely the
proof itself. Restrict your deductions to using a subset of mathematics where you don't
know N is missing an element.
Then Cantor's logical deduction sequence used to imply higher infinities is exactly the same
as the spoof proof.
Herc
|-|ercules
> On 2010-05-30, |-|ercules <radgr...@yahoo.com> wrote:
>> This is a serous attempt at a refutation of Cantor's powerset proof
>
> An attempt caused by mind-affecting drugs in your serum?
>
> Typo-jokes aside, it may have been a serious attempt, but also a
> serious failure.
>
>
>> The powerset proof is exactly this:
>>
>> Assume a large/infinite room full of boxes [...]
>
> No it isn't. At best that is an analogy for the proof, and not a very
> good one.
Yes it is. Substitute box for set and fridge magnet for natural, it's a valid version
of Cantor's powerset proof of uncountable infinity.
The real question you should ask yourself is why you don't accept such a silly proof?
Herc
|-|ercules
> defending in the end, that still doesn't mean that I'm going
I'm going to substitute me not being worth defending with my case
not being worth defending, and leave you be with this.
Herc
Marshall
> On May 29, 7:58 pm, "|-|ercules" <radgray...@yahoo.com> wrote:
>
> Before I proceed, one might wonder why I'm going through all
> this effort just to prove Herc wrong, since the other thread
> already established that Herc is in Case 3 (the case where
> we can call him wrong). Well, Herc did ask the question:
>
> > To eliminate expected confusion I will make my motives explicit,
> > the second proof is a spoof of Cantor's proof, if you can find the
> > flaw in my proof
>
> then attacked Spight for failing to answer it:
>
> > Your [Spight's] evasion of showing a distinction between my spoof
> > proof and Cantor's proof is noted
...
> I hope that Herc won't take this post as bullying. Herc
> asked a question and I gave the answer, which is more
> than we can say of Spight, who evaded Herc's question.
For the record, the post of Herc's, quoted above, does
not appear on my news server.
Marshall
Alan Smaill
> The more stupider it is, the easier it is to refute so you should give
> a more complete explanation of how it is stupid. If it is trivial
> then you have no reason to spend the same number of keystrokes at
> ridicule as is needed for the full proof.
No comment.
> C-B
>
--
Alan Smaill
William Hughes
> Cantor's second proof of 'uncountable infinity' is based on trying to enumerate the powerset of naturals.
>
> e.g.
> P(N) = {
> 1 - {1},
> 2 - {1,2},
> 3 - {2},
> 4 - {1,2,3}
> ...
>
> }
>
> The set of indexes that aren't members of their subset is
> {3,4}
> in this finite subset example.
>
> And voila, {3,4} is a new subset not present in P(N).
>
> Therefore no matter how big P(N) is
even if the enumeration does not have a last element
> there is always a new element
> that can be listed and therefore the size of the set P(N) is bigger than infinity.
>
> -----------------------------------------------------------
>
> Here's my equivalent proof of uncountable infinity.
>
> Let's assume an enumeration of naturals exists, call it N.
Ok, I choose an enumaration that does not have a
last element.
>
> N= {
> 1,
> 2,
> 3,
> 4,
> ...
>
> }
>
> Let's calculate a new natural MAX+1.
Can't be done, there is no last element and no max.
The difference between the two "proofs" is that
in the first case you can do the proof even if the
enumeration has no last element. In the second case you can't.
- William Hughes
Charlie-Boo
DavidW
> Cantor's second proof of 'uncountable infinity' is based on trying to
> enumerate the powerset of naturals.
> e.g.
> P(N) = {
> 1 - {1},
> 2 - {1,2},
> 3 - {2},
> 4 - {1,2,3}
> ...
Stop right there. WTF is the above supposed to be that we should be able to tell
what any succeeeding item is?
Jim Burns
> "Transfer Principle" <lwa...@lausd.net> wrote
>
>> And even though Herc/Cooper turned out to be
>> not worth defending in the end,
>> that still doesn't mean that I'm going
>
> I'm going to substitute me not being worth
> defending with my case not being worth defending,
> and leave you be with this.
For once, I heartily agree with you, Herc.
(With a "Yeah! Verily, yeah!").
The point (my point, Herc's point) is that
the worth of the argument is not determined by
the worth of the arguer.
(Savor the moment, Herc.
It may not come again for a long while.)
Jim Burns
Bart Goddard
@charm.magnus.acs.ohio-state.edu:
> The point (my point, Herc's point) is that
> the worth of the argument is not determined by
> the worth of the arguer.
>
How do you propose that a worthless arguer
come up with a good argument? Maybe "determined"
is too strong a word, but at the very least,
the correlation coefficient here is .999....
B.
--
Cheerfully resisting change since 1959.
George Greene
It's NOT "equivalent", but it does have a very similar structure.
It does similarly prove that something newer and bigger exists.
The only problem is, what your proof proves the existence of IS NOT
a natural number. Your proof is a proof that EVERY natural number
FAILS to be something (i.e., the biggest).
>
> Let's assume an enumeration of naturals exists, call it N.
>
> N= {
> 1,
> 2,
> 3,
> 4,
> ...
>
> }
>
> Let's calculate a new natural MAX+1.
That is not possible, because YOU HAVE NOT SAID what "MAX" is.
Is MAX supposed to be the maximum natural number? The problem
with that is, MAX *did not show up* on your "enumeration" of the
naturals
above. Your enumeration ended with "...", and nobody really knows
what
you meant by that. If you meant the usual thing, then, obviously,
that
sequence DOES NOT HAVE a last or maximum element, so your
calculation is impossible. If you wanted to ACTUALLY BE ABLE to
start this "calculation", then your enumeration WOULD have had to look
like
> N= {
> 1,
> 2,
> 3,
> 4,
> ...
> MAX
> }
If you are going instead to say
> That is 4+1 = 5
> in this finite subset example.
Then your enumeration has to look like
> N= {
> 1,
> 2,
> 3,
> 4 = MAX
> }
WITHOUT the "..." .
But see, your problem is, you're stupid, so you couldn't
even state YOUR OWN example accurately.
> Voila 5 is a new number not in N
>
> Therefore no matter how big N is there is always a new element
> that can be listed
Right.
Up to this point, you are telling the truth as best you know how.
If you just knew WHERE TO GO from here, then everything would
be OK. You have been constructing a proof that is going to use
the inference rule called universal generalization. It is going to
say something about EACH AND EVERY natural number.
What you are (correctly) saying it says is that, for every natural
number, there is a biggER one, and therefore, for every natural
number n, n is NOT the biggest one (n =/= MAX ; in point of fact,
MAX(N) simply does not exist at all).
> and therefore the size of the set N is bigger than infinity.
Now, you're just being an idiot.
Nothing in your proof said ANYTHING WHATSOEVER
about SET SIZES! YOU WERE ONLY talking about numbers
and enumerations! You have (sort of) correctly proved that
NO enumeration naturals WITH a MAX element is an enumeration
of ALL of them, but that simply does NOT SAY anything about
"bigger than infinity"! If you WANTED to say something about set
SIZE then you wuold NEED to FIRST actually SAY something
about set SIZE, namely, "The SIZE of the set of the first N naturals
(starting with 1 and ending at n) IS n." Had you said THAT, THEN
you could say, "For no n is n the size of the set of ALL the
naturals".
This REALLY WOULD PROVE that there was a "new" size that was
"bigger" than any NATURAL size. But that would NOT be proving
that anything was "bigger than infinity" -- that would just be proving
that the size of N *is* infinity. And it really would be a whole new
kind of [cardinal] "number".
George Greene
>
> e.g.
> P(N) = {
> 1 - {1},
> 2 - {1,2},
> 3 - {2},
> 4 - {1,2,3}
> ...
>
> }
>
> {3,4}
>
>
> Therefore no matter how big P(N) is there is always a new element
> that can be listed and therefore the size of the set P(N) is bigger than infinity.
Your use of "therefore" here is incorrect. This IS NOT HOW our
version
of the proof goes. What happens after "there is always a new element
of p(N)" is, THEREFORE, THE OLD list WAS NOT COMPLETE.
We make a generalization about the LISTS.
What this proof proves is that NO list is EVER a complete list of
all the subsets. But these ARE INFINITE lists!
NO countably infinite list is EVER long enough (they are ALL too
short),
therefore, the set being listed IS BIGGER than countABLE.
This is the EXACT same thing that is going on in your proof:
No FINITE list is ever long enough (to list all the naturals),
therefore
the set of naturals IS BIGGER than any finite number (i.e. it is
infinite).
BOTH proofs create a new bigger number in a similar way.
George Greene
> Yes I know it's such a simple yet equivalent logical deduction of 'bigger than infinity'
> and really shows how dumb Cantor subscribers are to miss that,
No, it isn't.
Your second proof is a deduction of "bigger than FINITE" because
all the lists you are generalizing over ARE FINITE -- they have a last
MAX element. Even though you were too stupid to draw it that way.
If the list actually is a list of all the naturals in order then IT
HAS NO MAX,
and your proof can't even get started.
> but the fundamental mathematical truths are generally quite simple.
Of course they are, yet despite this, you are too stupid to apply
them.
George Greene
> But what about Herc's proof attempt?
Some of us have been here for over 25 years.
Maybe you should ask instead of telling.
> Let's see:
>
> > N= {
> > 1,
> > 2,
> > 3,
> > 4,
> > ...
> > }
> > Let's calculate a new natural MAX+1.
> So Herc lists the elements of N. But then one wonders,
> what is the fifth element of Herc's list, given that its
> first four elements are 1,2,3, and 4?
That is not even the point. THE POINT is that "MAX"
IS AN UNDEFINED TERM in this context: Herc's "enumeration"
DOES NOT HAVE any value "MAX". He hasn't even said what
he wants or expects MAX to mean!
> > That is 4+1 = 5
> > in this finite subset example.
> > Voila 5 is a new number not in N
OK, so NOW we know that Herc thought that MAX=4.
But in that case, he should've written
> > N= {
> > 1,
> > 2,
> > 3,
> > ...
> > 4 (=MAX)
> > }
AS OPPOSED to what he actually wrote.
It's best to cite the first error first, usually.
> > Therefore no matter how big N is there is always a new element
> > that can be listed and therefore the size of the
> > set N is bigger than infinity.
>
> If Cooper is assuming that N is the set {1,2,3,4}, then he
> has just proved that card(N) is greater than _four_, not
> countable infinity (aleph_0).
No, he's done better than that; he's proved that for ANY (natural)
value of MAX, card(N) > MAX. He really has proved the existence
of a new kind of cardinal (for the cardinality of this set), given
that
he's proved that no natural is its cardinality. Sure, I know he said
4,
but BEFORE that, he said MAX, and he didn't put any particular value
on it. So in some sense he would've had the correct proof, if he just
hadn't put the ... ' s at the end of his "enumeration" WHILE trying to
claim that it STILL had a MAX value.
In any case, seriously, it is bad (seriously, bad) for you to try to
talk
TO US about this. Please address your objections to Herc himself,
and please try to focus on one or two specific ones per message.
Otherwise you're just masturbating in public.
Jim Burns
>
>> The point (my point, Herc's point) is that
>> the worth of the argument is not determined by
>> the worth of the arguer.
>
> How do you propose that a worthless arguer
> come up with a good argument? Maybe "determined"
> is too strong a word, but at the very least,
> the correlation coefficient here is .999....
It sounds like "explain" instead of "propose"
may be what you meant above. Since I don't know
how to answer your question with "propose", I
will answer it with "explain".
I don't have to explain how a worthless arguer
could come up with a good argument. I just have
to evaluate the argument, good or poor.
Suppose that the correlation coefficient is
.999... . Is that a good counter-argument?
If it is, then I have a gift for you: the
probability that Goldbach's Conjecture is
true is > 0.999. Go and publish! You'll
be famous! (You're welcome.)
Perhaps I misunderstand you, though.
If you are using the correlation coefficient
between an arguer and their arguments to
pick which arguments to invest your time in,
then I agree with you -- it is important to
consider all the arguments of the past that
turned out to waste your time.
However, once you have decided to argue for
or against some position, you would not use
those past arguments as an "explanation"
why someone is wrong, would you?
The distinction I am drawing is between
the beginnings of arguments and the ends of
arguments.
I call choosing to enter an argument the
beginning of an argument, and I think that
there should be no broad standards for
choosing to enter or not to enter any particular
argument. Let it be a matter of taste or
intuition: it is still your own time that you
are spending. The correlation coefficient
is as good as any way to choose, and better
than most, but still up to the individual.
I call drawing the conclusion the end of
an argument. I do not know you, Bart Goddard,
but I suspect that you would be insulted
if I claimed that you claimed that Herc's
mathematical arguments were nonsense /because/
Herc is crazy, not because of some mathemmatical
fault.
It is the second, non-mathematical way of judging
that I think Transfer Principle is using,
judging from his writing upthread "even though
Herc/Cooper turned out to be not worth defending
in the end". Perhaps I am mistaken.
I decided to beat once again this argument-
turned-dead-horse because, after some reflection,
I have decided that this one point embodies
most of my personal philosophy of mathematics
and because it was a pleasant change of pace to
agree with Herc, with whom I have dealt in the past.
Jim Burns
Aatu Koskensilta
> Suppose that the correlation coefficient is .999... . Is that a good
> counter-argument? If it is, then I have a gift for you: the
> probability that Goldbach's Conjecture is true is > 0.999. Go and
> publish! You'll be famous! (You're welcome.)
What sort of probability is involved here?
> I decided to beat once again this argument- turned-dead-horse because,
> after some reflection, I have decided that this one point embodies
> most of my personal philosophy of mathematics
I'm baffled. What is it in your point that is specifically mathematical?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
> And even though Herc/Cooper turned out to be not worth defending in
> the end, that still doesn't mean that I'm going to start defending
> Spight.
Out of idle curiosity, how would you go about "defending" Marshall?
|-|ercules
This is not worth arguing because no one in sci math will recognize ANY valid new result,
but this is my last post on the topic.
You all SAY that the 'new subset' works in the infinite case and MAX only works on any finite case.
But you also show HOW you get a 'new subset' with finite examples.
It works so well with finite examples!!! That is why you are all fooled.
1 - {1,2}
2 - {3,4}
3 - {5,6}
the box containing the numbers of all the boxes that don't contain their own number is....
there is none... the numbers of all the boxes that don't contain their own number is..
{2,3}
IT'S NOT IN THE LIST
It seems to work, as demonstrated in the finite case, so you are all fooled that the logical
deduction for the infinite case is legitimate.
Your entire proof/belief/philosophy of higher sets than infinite size, relies on the fact that
there_is_no_box_containing_the_numbers_of_all_the_boxes_that_don't_contain_their_own_number!
That's your religion!
You can compute every possible digit sequence up to infinite length, you can compute every possible subset of N
up to infinite length, but that doesn't bother any of you, because it's in your text book.
Herc
Bart Goddard
@charm.magnus.acs.ohio-state.edu:
> However, once you have decided to argue for
> or against some position, you would not use
> those past arguments as an "explanation"
> why someone is wrong, would you?
Depends on what level the "why" is. Why
the argument is wrong is one thing. Why
the person keeps trotting out wrong arguments
is another.
You used the word "determine". I think a good
argument could be made that his very poor
arguing skills are exactly what determines
the fact that his argument is lousy. At
least in the sense that his bad argument
was caused by his bad agrument muscles.
You can rebut that "correlation is not
causation" but in this case, I'm pretty
sure it IS causation.
Bart Goddard
@charm.magnus.acs.ohio-state.edu:
> I call drawing the conclusion the end of
> an argument. I do not know you, Bart Goddard,
> but I suspect that you would be insulted
> if I claimed that you claimed that Herc's
> mathematical arguments were nonsense /because/
> Herc is crazy, not because of some mathemmatical
> fault.
>
No, that's exactly what I'm claiming. An insane
argument exists, and the cause of its existence is
an insane arguer.
|-|ercules
that would only hold if you were disputing ALL my claims including my other 3 proofs
1/ an effective halting algorithm
2/ computers are not less capable than people because of Godel's proof
3/ the simplest computer model http://tinyurl.com/computermodel
and 4/ a 21st Century Adam and Eve is not impossible, though you can
call the son of God crazy even though you won't examine the evidence
http://tinyurl.com/ProofOfGenesis
So yes or no Goddard, are you disputing those 3 claims too?
Herc
Jim Burns
> Jim Burns <burn...@osu.edu> writes:
>
>> Suppose that the correlation coefficient is
>> .999... . Is that a good counter-argument?
>> If it is, then I have a gift for you: the
>> probability that Goldbach's Conjecture is
>> true is > 0.999. Go and publish! You'll be
>> famous! (You're welcome.)
>
> What sort of probability is involved here?
What sort of probability do you recommend that
I involve?
Something to consider is probability-as-
degree-of-belief, leading to a Bayesian analysis
of the truth of Goldbach's conjecture, using the
vast pool of even numbers that are known to be
the sums of two primes as evidence. I expect
that there are further issues that need
to be resolved, but I did so want to leave
something for Bart Goddard to do, if he decided
to publish as I suggested.
It could also turn out that the further issues are
more difficult than a moment's reflection while typing
about something else might indicate, in which case,
my suggestion that Bart publish a probabilistic
non-proof proof might better be interpreted as pointing
out that a correlation coefficient between a poor
arguers and the arguments they produce is not really
a very good counter-argument.
But I would have thought that you would have seen
through me immediately, Aatu. I confess that
I introduced a probabilistic non-proof proof of
Goldbach's conjecture only as an example of a really,
really bad argument, and, so, what kind of probability
that I planned not to use was not an issue that
I spent much thought upon.
Perhaps you can imagine my surprise that someone
as insightful as you would apparently miss my point.
>> I decided to beat once again this argument-
>> turned-dead-horse because, after some reflection,
>> I have decided that this one point embodies most
>> of my personal philosophy of mathematics
>
> I'm baffled. What is it in your point that is
> specifically mathematical?
Uhmm? Is there something about the philosophy of
mathematics that requires it to be applied only to
mathematics? Then I am a failed philosopher of
mathematics, and I should run off to wherever
failed philosophers run off to, the Florida Keys,
perhaps.
I will have to think about why evaluating
an argument independent of its arguer seems so
obviously foundational to me. In the past,
wondering about why some obvious truth is so
obviously true has been very entertaining to me.
Perhaps I will be able to give you a better answer
at some later point in time.
Jim Burns
Jim Burns
>
>> I call drawing the conclusion the end of
>> an argument. I do not know you, Bart Goddard,
>> but I suspect that you would be insulted
>> if I claimed that you claimed that Herc's
>> mathematical arguments were nonsense /because/
>> Herc is crazy, not because of some mathemmatical
>> fault.
>
> No, that's exactly what I'm claiming. An insane
> argument exists, and the cause of its existence is
> an insane arguer.
I think our apparent disagreement is more of
a failure to communicate.
The background for my point is someone else
asserting that people who have been labeled
cranks have a harder time getting people to
agree with them -- /because/ they have been
labeled cranks. My position is that it is not
that these people are wrong because they are
cranks; it is that they are cranks because
they are wrong.
Perhaps you will surprise me, but I strongly
suspect that you would judge Herc's arguments
to be invalid even if I had asserted them,
or David Ullrich had, or Arturo Magidin had
than if Herc had asserted them.
It seems like common sense, too obvious to
need mentioning, to say that changing whose
mouth an argument comes out of does not change
the quality of the argument -- but that is,
really, what I am trying to say here.
I should also point out that this is not a
universal belief about arguments. Certainly,
it is not true in politics. That sorry excuse
for a debate about (United States) health
care reform saw senators furiously denouncing
Democrats for the same proposals that they
had applauded when they had come out of
Republican mouths.
Jim Burns
|-|ercules
maybe you should stop being such an obvious hypocrite and talk about
the argument and stop attacking the person.
Here is the latest post in the argument so far...
Jim Burns
> "Jim Burns" <burn...@osu.edu> wrote
>> Bart Goddard wrote:
>>> Jim Burns <burn...@osu.edu> wrote in
>>> news:hu3n56$jvj$1...@charm.magnus.acs.ohio-state.edu:
>>>
>>>> I call drawing the conclusion the end of
>>>> an argument. I do not know you, Bart Goddard,
>>>> but I suspect that you would be insulted
>>>> if I claimed that you claimed that Herc's
>>>> mathematical arguments were nonsense /because/
>>>> Herc is crazy, not because of some mathemmatical
>>>> fault.
>>>
>>> No, that's exactly what I'm claiming. An insane
>>> argument exists, and the cause of its existence is
>>> an insane arguer.
[...]
>> Perhaps you will surprise me, but I strongly
>> suspect that you would judge Herc's arguments
>> to be invalid even if I had asserted them,
>> or David Ullrich had, or Arturo Magidin had
>> than if Herc had asserted them.
>>
>> It seems like common sense, too obvious to
>> need mentioning, to say that changing whose
>> mouth an argument comes out of does not change
>> the quality of the argument -- but that is,
>> really, what I am trying to say here.
[...]
> maybe you should stop being such an obvious
> hypocrite and talk about the argument
> and stop attacking the person.
Am I being a hypocrite by not addressing your
argument? I don't think so: I haven't said one
word about whether your arguments are good, bad,
crazy, sane, or anything else, although I have
quoted others.
I stopped addressing your arguments, Herc,
some years ago, when you flipped out so badly
that I became afraid that you might hurt yourself
or someone else. I would have enjoyed arguing with
you some more about Cantor's diagonal theorem,
but I did not want to cause anything like that.
I haven't really paid much attention to your
mathematical arguments since then. If I did
express an opinion about any of your current
arguments, then I shouldn't have. It would have
been poorly informed.
It's possible (I don't remember) that I
expressed an opinion about those arguments of
years ago, and it sounded like I was referring to
today's arguments. If I did, I'm sorry; I
did not mean to.
> Here is the latest post in the argument so far...
I have read the rest of this post now.
I have an opinion about the argument you present,
and whether you have been taking your pills lately
or not is irrelevant to that opinion. In my own
opinion, I would rather I and my arguments were
called "crazy" than called what my opinion is
about them. Anyway, I am not going to share my
opinion with you.
So, Herc, are you still being prescribed drugs to
stop the voices? And are you taking those drugs?
(I'm deleting your argument below. I understand how that
could be seen as a way of saying it is worthless.
I don't mean that, though. Remember that I am
perfectly willing to tell you something is
worthless, if I think so. I just don't want to talk
about it.)
Jim Burns
Tim Little
> The more stupider it is, the easier it is to refute
On the contrary, the "more stupider" it is, the harder it is to
refute. Greater levels of stupidity are much more immediately
apparent to most people. If there are people who don't see what makes
it stupid, they are very likely either stupid themselves or not at all
familiar with the subject in question. It is difficult to explain
anything at all about the subject in either case.
A formal mathematical derivation of |N|=|2^N| containing an error is
very easy to refute: just point out the error. A reasonable person
familiar with the subject will recognise that it is in fact an error:
end of discussion.
A vague argument based on quoting that Cantor's proof says "Therefore
no matter how big P(N) is there is always a new element that can be
listed and therefore the size of the set P(N) is bigger than infinity"
is *precisely* an example of "an incorrect proof that looks the same
to the mathematically incompetent", as Daryl stated.
The stupidity of presenting it as a "New Anti-Cantor Attack" is
immediately evident to any reasonable and competent person who has
even a passing familiarity with what constitutes mathematical proof.
So those to whom it is not immediately evident are either
unreasonable, incompetent, or totally unfamiliar with the subject. In
none of these cases will it be easy to refute to that person's
satisfaction. And so this thread will probably run for months and
accrue hundreds or thousands of posts.
- Tim
Marshall
> Transfer Principle <lwal...@lausd.net> writes:
> > And even though Herc/Cooper turned out to be not worth defending in
> > the end, that still doesn't mean that I'm going to start defending
> > Spight.
>
> Out of idle curiosity, how would you go about "defending" Marshall?
What?! Haven you never seen "Walker, Texas Ranger?"
The obvious answer is "roundhouse kick."
Marshall
David Bernier
Following the discovery of Russell's Paradox, there was a move by
Poincare and Weyl towards predicative definitions/constructions.
I was reading an article by Feferman:
< math.stanford.edu/~feferman/papers/DasKontinuum.pdf >
about (mainly) Weyl's _Das_Kontinuum_ (1918).
Quoting Feferman on Poincare:
"As we saw, Poincare argued that such apparent definitions
are improper: an object is to be defined or determined only
in terms of prior objects, notions and totalities;
only those are predicative." [page 7]
Feferman also discusses ACA_0, a theory having to do with
arithmetic-sentence based analysis or something like that.
It's "nice" to think one can/could introduce every non-primitive object
based on combining functions and objects defined previously,
starting from N.
I've been wondering what Poincare thought of Cantor's diagonal
argument.
FWIW, once one is used to not bothering about impredicative
definitions or constructions (I include myself), it's
not so easy telling apart the predicative from the impredicative.
David Bernier
Transfer Principle
> Bart Goddard wrote:
> > No, that's exactly what I'm claiming. An insane
> > argument exists, and the cause of its existence is
> > an insane arguer.
> asserting that people who have been labeled
> cranks have a harder time getting people to
> agree with them -- /because/ they have been
> labeled cranks. My position is that it is not
> that these people are wrong because they are
> cranks; it is that they are cranks because
> they are wrong.
And of course, I am the poster to whom Burns is
referring here. I indeed asserted that if two
people posted identical arguments, one under the
name of a well-known "crank" and one under that of
a newbie, the "crank" would have a harder time
getting people to agree with them.
> I should also point out that this is not a
> universal belief about arguments. Certainly,
> it is not true in politics. That sorry excuse
> for a debate about (United States) health
> care reform saw senators furiously denouncing
> Democrats for the same proposals that they
> had applauded when they had come out of
> Republican mouths.
I'm glad that Burns mentioned politics here, since
there is another well-known political "argument" that
someone used just yesterday right here on sci.math!
Yesterday, a poster named Bergman tried to argue
that N is isomorphic to C. Then the poster Gerry
Myerson questioned the OP's defintion of "is." Then
Ostap Bender pointed out the similarity between
Myerson's questioning the definition of "is" and
former President Clinton's definition of that same
two-letter word just over a decade ago.
The point I'm trying to make is that plenty of people
use the word "is" all the time, but does Myerson ask
them to define that word? No, but he asks Bergman to
define that word, just because of his low reputation
as someone who believes in an isomorphism between the
sets N and C.
Apparently to Myerson, only those who contradict
Cantor have to define their two-letter words, not
those who agree with Cantor.
So far I can't tell whether Goddard, like Myerson,
would treat posters differently based solely on their
reputation (including Clintonian hairsplitting against
such posters), but Goddard might be leaning somewhat
in that direction.
Then again, I know that Goddard is one of the foremost
advocates of a moderated sci.math group. So if a
person like Bergman were to post, Goddard wouldn't
need to ask him to define "is" -- all he'd have to do
is block the poster.
Bart Goddard
9dec-b49...@e6g2000vbm.googlegroups.com:
> Apparently to Myerson, only those who contradict
> Cantor have to define their two-letter words, not
> those who agree with Cantor.
>
You're way out of context. Adults don't engage in
pissing contests, so the very issue about whether
this or that person "has to" define two-letter words
is irrelevant.
If Gerry Myerson asked someone to define "is", then
his motivation was to guide that person to understanding,
not to beat him over the head with hair-splittings.
>Then again, I know that Goddard is one of the foremost
>advocates of a moderated sci.math group. So if a
>person like Bergman were to post, Goddard wouldn't
>need to ask him to define "is" -- all he'd have to do
>is block the poster.
As has been stated multiple times, the proposal for
sci.math.moderated is to filter content only. So your
commit is specious.
Transfer Principle
> Transfer Principle <lwal...@lausd.net> wrote in news:e0e4acfa-fea4-4a1b-
> 9dec-b490a52b6...@e6g2000vbm.googlegroups.com:
> > Apparently to Myerson, only those who contradict
> > Cantor have to define their two-letter words, not
> > those who agree with Cantor.
> his motivation was to guide that person to understanding
I disagree. How does asking someone to define "is" lead
to understanding at all? When Clinton asked for the
definition of "is," did that lead to any understanding?
I can see how Myerson's asking for the definitions of
"isomorphic," N, and C can lead to understanding. Indeed,
if it turns out that Bergman is in Case 1 of my four-case
list and he attempts to start a new theory, _I_'d like to
understand more about it, and Bergman's definitions of
"isomorphic," N, and C can help.
But to me, asking for a definition of the non-mathematical
two-letter linking verb "is" goes too far. Now we're no
longer discussing mathematical definitions. Indeed, I
wonder what sort of response Myerson even _expects_ when
he asks Bergman for the definition of "is."
It amazes me how Clinton was ridiculed a decade ago for
asking for the definition of "is," yet posters who ask for
the definition of the same are defended, especially -- the
key point that I'm trying to make -- the one who asks for
the definition has a higher reputation on sci.math than
the person being asked to provide a definition.
Bart Goddard
bc4e-1b6...@q8g2000vbm.googlegroups.com:
> I disagree.
No, you dis-understand. Porn stars claim that
they are just actresses because actresses also "sell"
their bodies to producers. That doesn't make it
the same at all. The fact that Clinton and Myerson
both asked for definitions of "is" doesn't make
it the same either.
> the one who asks for
> the definition has a higher reputation on sci.math than
> the person being asked to provide a definition.
As I said, adults don't participate in pissing contests.
Real mathematicians prove theorems; they aren't all that
interested in proving themselves. The gaggle of camp-
followers who are obsessed with "reputation" and "status"
and "who has to define 'is' and who doesn't" might
considering growing up or going away.
George Greene
> The stupidity of presenting it as a "New Anti-Cantor Attack" is
> immediately evident to any reasonable and competent person who has
> even a passing familiarity with what constitutes mathematical proof.
Well, that is obviously not who we are dealing with, so that is almost
irrelevant.
But even more IRrelevant is the "immediately" part: Herc has been
doing this
FOR A DECADE. A lot of us here have been seeing these arguments for
TWO
decades. So anything that is immediate is not relevant: what IS
relevant is the
kinds of reactions that will be had by people with LONG experience in
reacting to these errors.
> So those to whom it is not immediately evident are either
> unreasonable, incompetent, or totally unfamiliar with the subject.
Everybody's competence varies. That is not really an admissible
insult in this context.
> In none of these cases will it be easy to refute to that person's
> satisfaction.
That's just unduly pessimistic. Lots of people understand that simple
contradictions
are contradictions. It's sort of the deep dark relevant secret that
that is ACTUALLY ALL
ANYbody NEEDS to understand to do this. The rest of it is very purely
grammar and
string-matching, two things that people have to be able to do ALREADY
just to write
coherently.
> And so this thread will probably run for months and
> accrue hundreds or thousands of posts.
Hundreds is trivial.
Thousands, I doubt it. I just don't have the stamina any more.
And as is being discussed in another thread, a lot of the good people
have left.
herbzet
Charlie-Boo wrote:
> The more stupider it is, the easier it is to refute
--
hz
Against stupidity, the gods themselves contend in vain.
-- Schiller --
Bart Goddard
This is what is the same about USENET and music. The
stupider the louder and the louder the stupider.
Marshall
>
> I indeed asserted that if two
> people posted identical arguments, one under the
> name of a well-known "crank" and one under that of
> a newbie, the "crank" would have a harder time
> getting people to agree with them.
This is exactly as it should be.
Marshall
Marshall
See also Date's Incoherence Principle:
"It is difficult to treat coherently that which is incoherent."
Marshall
Don Stockbauer
Of course you'd believe in a fallacy, the "Argument Against the Man."
Don Stockbauer
"Fantastic insight into the true nature of Reality is isomorphic to
insanity."
Marshall
Stupid people are less likely to produce useful
results than smart people, and therefor less worthy
of attention. This may appear to be a logical fallacy
to those who aren't looking carefully.
Marshall
|-|ercules
David Bernier
> Tim Little wrote:
[...]
I've been reading Poincare's "Science et méthode", (1908).
-> LIVRE II : Le raisonnement mathématique
---> Chapters III, IV, V here:
< http://www.ac-nancy-metz.fr/enseign/philo/textesph/default.htm>
(about 80% of the way down the page).
Poincare was an intuitionist: first one learns about numbers
(or Euclidean geometry) by doing it without too much concern
about axioms, and at a later stage one can develop an axiomatic
system (as Hilbert did for geometry). Poincare says that it's
not the axioms and logic that convince, but the whole of the proof
taken together.
In Chapter V he mentions three paradoxes of the day:
1- Burali-Forti's paradox
< http://en.wikipedia.org/wiki/Burali-Forti_paradox >
2- Zermelo-Konig paradox
Konig's paradox is a cousin of Berry's paradox:
< http://mathgate.info/cebrown/notes/vonHeijenoort.php > --->
Zermelo-Konig "paradox" related to Richard's paradox).
3 - Richard's paradox .
< http://mathgate.info/cebrown/notes/vonHeijenoort.php#Richard >
Poincare died in 1912. So (after 1908) he didn't live many years
in which ZFC was further developed.
I think what's commonly considered valid in mathematics has
changed throughout history:
[ discoverer of irrationals perished by drowning:
--> http://en.wikipedia.org/wiki/Hippasus#Irrational_numbers ]
Every natural number can be given a finite description, a numeral.
This is not so for a typical subset of N, P(N), P(P(N)), etc.
So in a sense generic subsets of N or R are not so tangible.
David Bernier
Aatu Koskensilta
> Perhaps you can imagine my surprise that someone
> as insightful as you would apparently miss my point.
I didn't miss your point. I was hoping to prod you into presenting your
thoughts on an issue I find interesting. As you note, when people say
that probably P is not NP, probably the Riemann hypothesis is true, and
so on, usually they're thinking of degrees of belief, or, at least, it
is difficult to come up with any other coherent way of making sense of
such assertions; that is, they mean just that most experts would be
willing to bet a not insignificant sum that no refutation of these
conjectures is forthcoming, and indeed we find people are happy to use
algorithms, in the real world, in production systems, the correctness of
which depend on the truth of such conjectures. This interpretation makes
less sense in case of baffling claims such as "we have good evidence
that PA is probably consistent" and so on sometimes put forth by
logically innocent mathematicians.
> Uhmm? Is there something about the philosophy of mathematics that
> requires it to be applied only to mathematics?
Is there something about the philosophy of horology that requires that
it be applied only to horology? I'm unsure what sort of philosophy we're
talking about here. I was baffled because there seemed to be nothing
specifically about mathematics in the in-itself perfectly reasonable
attitude or philosophy you described, and I didn't, and still don't,
quite understand in what sense it is a philosophy of
mathematics. Perhaps you meant it only in the sense people speak of
their "philosophy of chess" or "philosophy of cooking", i.e. a general
attitude or approach they apply when going about some activity or
pastime?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechan kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
> Feferman also discusses ACA_0, a theory having to do with
> arithmetic-sentence based analysis or something like that.
ACA_0 is a conservative extension of PA in the language of second-order
arithmetic -- that is, all of its theorems which can be expressed in the
language of PA are provable in PA, and conversely. It has both number
and set variables, and as axioms induction stated as a single sentence:
For all sets X, if 0 is in X, and n+1 is in X whenever n is, then all
numbers are in X.
the usual axioms for successor, addition and multiplication, and a
predicative comprehension schema:
For any formula P containing no bound set variables, the universal
closure of
There exists a set X such that a number x is in X if and only if P(x).
is an axiom.
This theory is predicatively justified, in the sense that we can make
predicativist sense of its axioms, and can provide arguments for them
that are compelling on the predicativist conception of mathematics. (It
is also finitely axiomatizable, unlike PA itself.)
Aatu Koskensilta
> Poincare was an intuitionist
Sure, but not in the sense the term is used today in foundations and
mathematical logic.
> Every natural number can be given a finite description, a numeral.
> This is not so for a typical subset of N, P(N), P(P(N)), etc.
Right, hence the qualms of various predicatively and vaguely
constructively inclined mathematicians at the turn of the 20th century.
MoeBlee
> It amazes me how Clinton was ridiculed a decade ago for
> asking for the definition of "is,"
He didn't ask for a definitionof 'is'. He correctly and legitimately
pointed out that the answer to the question posed to him depends on
which sense of the word 'is' was meant.
MoeBlee
David R Tribble
> Here's my equivalent proof of uncountable infinity.
> Let's assume an enumeration of naturals exists, call it N.
> N= { 1, 2, 3, 4, ... }
>
> Let's calculate a new natural MAX+1.
> That is 4+1 = 5
> in this finite subset example.
> Voila 5 is a new number not in [this finite subset of] N
>
> Therefore no matter how big N is there is always a new element
> that can be listed and therefore the size of the set N is bigger than infinity.
Ah, but how much bigger than infinity? One more bigger?
Twice as big? Infinity times bigger? Infinity squared bigger?
Oh, and is MAX the same as the size of N, or is it something
different?
Details, man, details. It's the precise details that matter.
David Bernier
> David Bernier<davi...@videotron.ca> writes:
>
>> Feferman also discusses ACA_0, a theory having to do with
>> arithmetic-sentence based analysis or something like that.
>
> ACA_0 is a conservative extension of PA in the language of second-order
> arithmetic -- that is, all of its theorems which can be expressed in the
> language of PA are provable in PA, and conversely. It has both number
> and set variables, and as axioms induction stated as a single sentence:
>
> For all sets X, if 0 is in X, and n+1 is in X whenever n is, then all
> numbers are in X.
>
> the usual axioms for successor, addition and multiplication, and a
> predicative comprehension schema:
>
> For any formula P containing no bound set variables, the universal
> closure of
>
> There exists a set X such that a number x is in X if and only if P(x).
>
> is an axiom.
>
> This theory is predicatively justified, in the sense that we can make
> predicativist sense of its axioms, and can provide arguments for them
> that are compelling on the predicativist conception of mathematics. (It
> is also finitely axiomatizable, unlike PA itself.)
>
Thanks for your explanation. Poincare wasn't a slouch, and neither
were Weyl or Brouwer. So I try to grasp their point of view,
to the extent possible.
Admittedly, there are many existence proofs today which use the
Axiom of Choice, and reals can't all be listed.
The other side of the coin for me goes like this:
I like to think (picturing the Cantor set), that any ternary
expression 0.b_1 b_2 b_3 ... b_j ... 1<=j< oo ,
b_j in {0, 2} makes sense. Then there is measure theory according
to which every countable subset X of the reals has Lebesgue measure zero.
(And so on).
All of Lebesgue measure theory in R^n, compactness and connectedness
(general topology), Baire category, make sense to me.
So I'm sort of disappointed with what I read of Poincare in that
for me, it doesn't seem to give a fully satisfactory
account or description of the continuum (the reals, the real number line):
what is there to find in the real number line? What reals exist?
And, BTW, if you happen to know what Weyl and/or Poincare thought
of Dedekind cuts and/or Cauchy's development of Cauchy sequences,
what with "every Cauchy sequence converges", I'd be happy to
hear about it.
David Bernier