The modern mathematical concept of infinity is indefensible
Albrecht
Modern mathematics implies that the number of the natural numbers is
greater than every natural number. This opinion is indefensible since
the natural numbers count themselves.
On the other hand: No natural number is able to denote the number of
the natural numbers. In consequence there is no number of the natural
numbers and hence there is no cardinality of the set of the natural
numbers.
The natural numbers can be listed in unary system with the symbol "O"
as follows:
O
OO
OOO
OOOO
OOOOO
...
Any natural number is the number of its predecessors, inclusive the
considered number self.
The sequence
O
O
OO
O
O
OOO
O
O
O
OOOO
O
O
O
O
OOOOO
...
clearly shows this fact.
The number of natural numbers can't be greater than every natural
number since there are not more natural numbers than there are natural
numbers.
The concept of an actual infinity as in set theories like ZFC is
indefensible.
Best regards
Albrecht S. Storz
Mannheim, Germany
mike3
> Modern mathematics implies that the number of the natural numbers is
> greater than every natural number. This opinion is indefensible since
> the natural numbers count themselves.
>
> On the other hand: No natural number is able to denote the number of
> the natural numbers. In consequence there is no number of the natural
> numbers and hence there is no cardinality of the set of the natural
> numbers.
>
It just means that it is not a natural number. It doesn't mean it is
not a "number"
unless you define "number" to be only natural numbers.
<snip>
Joshua Cranmer
> The number of natural numbers can't be greater than every natural
> number since there are not more natural numbers than there are natural
> numbers.
But for each natural number, there is a larger natural number. This is
indisputable fact; if you combine it with your claims, you get Zeno's
Paradox in a different form. If you accept the idea of infinity, the
paradox disappears.
--
Beware of bugs in the above code; I have only proved it correct, not
tried it. -- Donald E. Knuth
downisr...@gmail.com
best of the best:
sexypicturess.blogspot.com
William Hughes
> Any natural number is the number of its predecessors, inclusive the
> considered number self.
Correct, however the number of *all* natural numbers, say w, is not a
natural
number. w is not the number of its predecessors inclusive w.
<snip>
> The number of natural numbers can't be greater than every natural
> number since there are not more natural numbers than there are natural
> numbers.
This is only true if "the number of natural numbers" has to be
a natural number. Note that "the number of natural numbers"
is not a natural number.
- William Hughes
Herbert Newman
This one caught my attention, since it is especially silly:
> Albrecht wrote:
>>
>> The number of natural numbers can't be greater than every natural number,
>> since there are not more natural numbers than there are natural numbers.
>> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>>
Since that latter is (more or less) a "tautology", it "proves" nothing.
(The argument has the same logical form as the following one:
The sun CAN'T be a star, SINCE 1 = 1.)
Actually, the claim
Since ~(#N > #N), P
can be "reduced" to the claim
P.
But P is false in this case, since -as is well known-
aleph_0 is the (cardinal) number of (the set of) natural numbers,
and for every natural number n: n < aleph_0
(at least in the context of set theory, say ZFC).
With other words:
An e N: n < #N,
i.e. the number of natural numbers _is_ (and hence -contrary to A's claim-
_can_ be) greater than every natural number (at least in the context of set
theory, say ZFC).
Hence A's "argument" from above is neither sound, nor valid.
To make a long story short: no need to appeal to Zeno's Paradox here. A's
stuff is idiotic enough even without that (i.e. self-sufficiently).
Herb
P.S. Of course you are right when stating:
"But for each natural number, there is a larger natural number. This is
indisputable fact,..."
Which leads to the idea that there (actually) are infinitely many natural
numbers. (Thought some finitists might object, and urge for replacing
"actually" with "potentially" in my statement. :-)
Still this does not (automatically) mean that there IS a number of natural
numbers. Though it was (first) shown by Cantor and Frege that the number of
natural numbers is a sensible concept.
Herbert Newman
> Albrecht:
>>
>> No natural number is able to denote the number of the natural numbers.
>> In consequence there is no number of the natural numbers [...].
>>
What an incredibly silly "argument".
Right. No NATURAL number is able "to denote the number of the natural
numbers". But since the number of natural numbers isn't a NATURAL number,
we can hardly "conclude" the nonexistence of this number from that. :-)
Though again A's "argument" has been shown to be neither sound, nor valid.
Hint:
>
> [...] it is not a natural number.
>
Again:
Let N = {0, 1, 2, ...} and #N = card(N) ("the number of natural numbers").
Then
An e N: n =/= #N ( ~En e N: n = #N ).
"No natural number is able to denote the number of natural numbers." (A.)
Or with other words:
#N !e N.
"The number of natural numbers isn't a natural number."
Still (in ZFC at least) we have:
Ex(x = #N).
"There is a set which is the number of natural numbers."
Or with other words: "The number of natural numbers exists."
Herb
lwa...@lausd.net
> Albrecht wrote:
> > The number of natural numbers can't be greater than every natural
> > number since there are not more natural numbers than there are natural
> > numbers.
> But for each natural number, there is a larger natural number. This is
But Albrecht never disputed this fact. I admit that some
so-called "cranks," such as WM, do believe in a largest
natural, but there's no evidence in this post that
Albrecht is one of them. As far as I can tell, Albrecht
accepts every theorem of PA.
> If you accept the idea of infinity [....]
Clearly Albrecht _doesn't_ accept the idea of Infinity.
Albrecht, as a finitist, is working in a theory such as
ZF-Infinity (and possibly with ~Infinity, the negation
of Infinity, as well). In this theory, one can prove
that for every natural n, a larger natural n+1 exists
(this requires only the Pairing and Union axioms), but
one can't prove that the set N of all naturals exists
(and if we include ~Infinity, we can prove that the set
N _can't_ exist).
As for Zeno's paradoxes, I'd think that with since the
paradoxes deal with the infinite, no infinity means no
paradox, so there would be no Zeno's paradoxes in a
theory like ZF-Infinity(+~Infinity).
lwa...@lausd.net
> On Tue, 06 Jan 2009 21:32:03 -0500 Joshua Cranmer wrote:
> > This one caught my attention, since it is especially silly:
> stuff is idiotic enough even without that (i.e. self-sufficiently).
I don't find it idiotic. Just because Albrecht is a
finitist working in a theory like ZF-Infinity
(possibly +~Infinity), it doesn't mean that he is
an idiot. Of course, I would not be surprised to
find that many adherents of ZFC consider most
finitists to be idiots.
> Which leads to the idea that there (actually) are infinitely many natural
> numbers. (Thought some finitists might object, and urge for replacing
> "actually" with "potentially" in my statement. :-)
So ironically, right after he calls Albrecht's
finitist arguments "idiotic," Newman acknowledges
the finitists and their beliefs.
Right, Newman, some finitists might object to the
existence of an infinite set of naturals -- and
indeed, Albrecht is obviously such a finitist.
lwa...@lausd.net
> Though again A's "argument" has been shown to be neither sound, nor valid.
ZF-Infinity+~Infinity is neither sound nor valid?
Since Infinity is independent of the other axioms
of ZF, if ZF-Infinity is unsound and invalid, then
ZF itself is also unsound and invalid.
David C. Ullrich
>On Jan 7, 12:19 am, Herbert Newman <nomail@invalid> wrote:
>> Though again A's "argument" has been shown to be neither sound, nor valid.
>
>ZF-Infinity+~Infinity is neither sound nor valid?
There you go yet again, defending something utterly
other than what the guy said.
ZF-Infinity+-Infinity is logical theory that has
certain well-defined axioms. In the OP there was
nothing about what follows from this axiom or
that axiom, the OP _stated_ that [blah blah]
was "indefensible". He "proved" this, and his
proof was nonsense, since it assumed that
"number" means "natural number", which is
clearly not what the word means in the context
he was discussing.
And in particular his argument said nothing
whatever about ZF-Infinity+-Infinity, or
any other axiomatic system. Indeed, the fact
that he seems to be missing the notion the
math is based on axioms seems to be the problem.
>Since Infinity is independent of the other axioms
>of ZF, if ZF-Infinity is unsound and invalid, then
>ZF itself is also unsound and invalid.
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.)
David C. Ullrich
>On Jan 7, 12:02 am, Herbert Newman <nomail@invalid> wrote:
>> On Tue, 06 Jan 2009 21:32:03 -0500 Joshua Cranmer wrote:
>> > This one caught my attention, since it is especially silly:
>> To make a long story short: no need to appeal to Zeno's Paradox here. A's
>> stuff is idiotic enough even without that (i.e. self-sufficiently).
>
>I don't find it idiotic. Just because Albrecht is a
>finitist working in a theory like ZF-Infinity
>(possibly +~Infinity), it doesn't mean that he is
>an idiot.
He's not a finitist working in that theory, or if
he is there's no evidence of it in his post. He's
_asserting_ that Infinity is _false_. More than
that, he's claiming to _prove_ that Infinity is
false.
>Of course, I would not be surprised to
>find that many adherents of ZFC consider most
>finitists to be idiots.
>
>> Which leads to the idea that there (actually) are infinitely many natural
>> numbers. (Thought some finitists might object, and urge for replacing
>> "actually" with "potentially" in my statement. :-)
>
>So ironically, right after he calls Albrecht's
>finitist arguments "idiotic," Newman acknowledges
>the finitists and their beliefs.
Nothing ironic about it.
>Right, Newman, some finitists might object to the
>existence of an infinite set of naturals -- and
>indeed, Albrecht is obviously such a finitist.
David C. Ullrich
David C. Ullrich
wrote:
>
>Modern mathematics implies that the number of the natural numbers is
>greater than every natural number. This opinion is indefensible since
>the natural numbers count themselves.
>
>On the other hand: No natural number is able to denote the number of
>the natural numbers. In consequence there is no number of the natural
>numbers
That follows _if_ "number" means "natural number". In fact
that's not what it means in this context.
The actual content of your post, stripping the assertions
based on misunderstanding that words mean what we say
they mean, boils down to the fact that there are infinitely
many natural numbers. It's good of you to point this out.
David C. Ullrich
Han de Bruijn
> that he seems to be missing the notion the
> math is based on axioms seems to be the problem.
You've never summarized the "axioms of math" in your book "Complex made
Simple" and _yet_ it _is_ mathematics all over the place. Right ?
Mathematics is mathematics, based on axiom systems or not. And what the
OP does _is_ mathematics, whether you like it or not. Axiom systems aid
in understanding, perhaps, but cannot replace it.
Han de Bruijn
John Jones
> On Jan 6, 9:08 pm, Albrecht <albst...@gmx.de> wrote:
>
>> Any natural number is the number of its predecessors, inclusive the
>> considered number self.
>
>
> Correct, however the number of *all* natural numbers, say w, is not a
> natural
> number. w is not the number of its predecessors inclusive w.
That puts an unbearable strain on 'all'. 'All' is either referring to a
a new emergent property, or it is not. That is, you can say
1) 'all' natural numbers: where 'all' refers to the standard property
or way of talking about 'all', in, for example, 'all the numbers from
one to ten'.
Or, and which you seem to be doing, you can say it like this
2) 'all natural numbers': which is an emergent, new property.
Unfortunately, you have not said what this property is. All you have
said about it is that it allows us to say its number is not a natural
number.
Toni...@yahoo.com
*********************************************************
New year, new trollings. In particular that new-year's pearl
"mathematics is mathematics, based on axiom systems or not"...
You sound like my old auntie telling us "moral is moral, now, future,
past, here and everywhere!", with that biblical look in her eyes.
But I'll tell you what: I'll grant you the benefit of doubt about the
above, and I won't compare you to my auntie, if you are gracious
enough as to tell us WHAT is mathematics. Fair enough? Then we'll know
what is mathematics, axioms or not.
Regards
Tonio
Toni...@yahoo.com
> On Jan 7, 12:02 am, Herbert Newman <nomail@invalid> wrote:
>
> > On Tue, 06 Jan 2009 21:32:03 -0500 Joshua Cranmer wrote:
> > > This one caught my attention, since it is especially silly:
> > To make a long story short: no need to appeal to Zeno's Paradox here. A's
> > stuff is idiotic enough even without that (i.e. self-sufficiently).
>
> I don't find it idiotic. Just because Albrecht is a
> finitist working in a theory like ZF-Infinity
> (possibly +~Infinity), it doesn't mean that he is
> an idiot. Of course, I would not be surprised to
> find that many adherents of ZFC consider most
> finitists to be idiots.
>
At least most "finitists" here in sci.math. And I write "finitists"
because finitists would be way too much of a honour to give them,
taking into consideration that the huge majority of them (well, in
fact all of them, as far as I can recall right now) are NOT
mathematicians.
Albrecht began his new year's nonsense with the following:
"Modern mathematics implies that the number of the natural numbers is
greater than every natural number. This opinion is indefensible since
the natural numbers count themselves."
From this very unique parraph one can know we have here a HUGE crank:
Modern mathematics don't imply anything of the like as he wrote, and
of course not that "the number of natural numbers" (what is that?) is
anything.
What does it mean that "the natural numbers count themselves"? I
haven't yet seen a number 7 counting itself, or other natural number:
perhaps he meant something else, but what?
You see, lwalk? You love to defend the indefensible, at least in
mathematics. This guy, as ALL other "finitists" I can recall right now
here in sci.math, don't really know what they're talking about, but
they make lots of noise...just as you do.
Regards
Tonio
Han de Bruijn
> On Jan 7, 1:26 pm, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
>
>>David C. Ullrich wrote:
>>
>>>that he seems to be missing the notion the
>>>math is based on axioms seems to be the problem.
>>
>>You've never summarized the "axioms of math" in your book "Complex made
>>Simple" and _yet_ it _is_ mathematics all over the place. Right ?
>>
>>Mathematics is mathematics, based on axiom systems or not. And what the
>>OP does _is_ mathematics, whether you like it or not. Axiom systems aid
>>in understanding, perhaps, but cannot replace it.
>
> *********************************************************
>
> New year, new trollings. In particular that new-year's pearl
> "mathematics is mathematics, based on axiom systems or not"...
> You sound like my old auntie telling us "moral is moral, now, future,
> past, here and everywhere!", with that biblical look in her eyes.
>
> But I'll tell you what: I'll grant you the benefit of doubt about the
> above, and I won't compare you to my auntie, if you are gracious
> enough as to tell us WHAT is mathematics. Fair enough? Then we'll know
> what is mathematics, axioms or not.
Isn't mathematics: what mathematicians do ?
Han de Bruijn
Han de Bruijn
> On Jan 7, 10:42 am, lwal...@lausd.net wrote:
>
>>On Jan 7, 12:02 am, Herbert Newman <nomail@invalid> wrote:
>>
>>>On Tue, 06 Jan 2009 21:32:03 -0500 Joshua Cranmer wrote:
>>>
>>>>This one caught my attention, since it is especially silly:
>>>
>>>To make a long story short: no need to appeal to Zeno's Paradox here. A's
>>>stuff is idiotic enough even without that (i.e. self-sufficiently).
>>
>>I don't find it idiotic. Just because Albrecht is a
>>finitist working in a theory like ZF-Infinity
>>(possibly +~Infinity), it doesn't mean that he is
>>an idiot. Of course, I would not be surprised to
>>find that many adherents of ZFC consider most
>>finitists to be idiots.
>
> ************************************************************
>
> At least most "finitists" here in sci.math. And I write "finitists"
> because finitists would be way too much of a honour to give them,
> taking into consideration that the huge majority of them (well, in
> fact all of them, as far as I can recall right now) are NOT
> mathematicians.
No. Some of them are just humble theoretical physicists. Some of them
have solved systems of non-linear, inhomogeneous, partial differential
equations, numerically. But they are still _no_ "real" mathematicians.
Keep dreaming ..
> Albrecht began his new year's nonsense with the following:
>
> "Modern mathematics implies that the number of the natural numbers is
> greater than every natural number. This opinion is indefensible since
> the natural numbers count themselves."
>
> From this very unique parraph one can know we have here a HUGE crank:
> Modern mathematics don't imply anything of the like as he wrote, and
> of course not that "the number of natural numbers" (what is that?) is
> anything.
> What does it mean that "the natural numbers count themselves"? I
> haven't yet seen a number 7 counting itself, or other natural number:
> perhaps he meant something else, but what?
>
> You see, lwalk? You love to defend the indefensible, at least in
> mathematics. This guy, as ALL other "finitists" I can recall right now
> here in sci.math, don't really know what they're talking about, but
> they make lots of noise...just as you do.
Sure you're a hanger-on. And never slap the hand that feeds you, right ?
Han de Bruijn
victor_me...@yahoo.co.uk
>
> No. Some of them are just humble theoretical physicists.
There is no such thing as a humble theoretical physicist.
David Bernier
Either an entity is a natural number, or it is not.
Would you agree?
David Bernier
David Bernier
An anecdote is that Freeman Dyson behaved in a gentlemanly manner
when Hugh Montgomery visited IAS in Princeton, where
Montgomery showed Dyson his work on correlations in spacings
of the zeros of the Riemann zeta function on the
critical line.
David Bernier
Han de Bruijn
How about a humble programmer ?
http://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD340.html
Han de Bruijn
Alan Smaill
> On Jan 6, 6:32 pm, Joshua Cranmer <Pidgeo...@verizon.invalid> wrote:
> > Albrecht wrote:
> > > The number of natural numbers can't be greater than every natural
> > > number since there are not more natural numbers than there are natural
> > > numbers.
> > But for each natural number, there is a larger natural number. This is
> > indisputable fact [....]
>
> But Albrecht never disputed this fact. I admit that some
> so-called "cranks," such as WM, do believe in a largest
> natural,
Stop right there!
WM has specifically *denied* that he belives that there is a largest
natural, and called that idea "nonsense".
That makes sense for followers of the idea of the potentially infinite.
--
Alan Smaill email: A.Smaill at ed.ac.uk
School of Informatics tel: 44-131-650-2710
University of Edinburgh
george
> greater than every natural number. This opinion is indefensible
It is NOT, however, inCONSISTENT, which, this being sci.logic,
is the ONLY thing anybody in HERE is going to care about.
> since the natural numbers count themselves.
That is not even grammatical.
If a set has a size or a cardinality, or if its members can be
counted,
then they are counted BY *ONE* number, NOT by "the natural numbers"
PLURAL. You can allege that a set-with-more-than-one-member "counts"
another when it is in mutual ONE-TO-ONE-CORRESPONDENCE with another,
and, indeed the natural numbers (like EVERY set) ARE bijectible with
themselves, but that does NOT preclude (though you idiotically claim
it does) the number-of-natural-numbers being bigger than every natural
number.
> On the other hand: No natural number is able to denote the number of
> the natural numbers.
That is NOT the OTHER hand! That is the SAME hand!
In light of this truth, what you said ABOVE is just WRONG!
Yes, the natural numbers "count themselves", BUT NO INDIVIDUAL
natural number counts ALL of them; that count of all of them REALLY IS
BIGGER than all of them!
> In consequence there is no number of the natural numbers
> and hence there is no cardinality of the set of the natural
> numbers.
Your capacity to determine logical consequence is severely
flawed. You have not employed any recognizable rule of inference.
What logically follows from what you have said is that there is
no NATURAL number that is "the number of the natural numbers".
There COULD STILL BE OTHER kinds of numbers!
Toni...@yahoo.com
****************************************************************************
Ah, great definition! Just like "moral is what moralists preach", uh?
Ok....got it.
Regards
Tonio
george
> > Any natural number is the number of its predecessors, inclusive the
> > considered number self.
This is not even grammatical.
People in general are going to have to get used to the fact that
0 IS a natural number. That means that every natural number is
the number of smaller natural numbers. Since 0 is first, there are
no smaller natural numbers, so that winds up defining 0 as the
cardinality
of the empty set.
>
> > The number of natural numbers can't be greater than every natural
> > number since there are not more natural numbers than there are natural
> > numbers.
This is just bullshit. EVERY natural number is greater than
than every one of the numbers in the set that IT "numbers",
IF you do it right: 7={0,1,2,3,4,5,6} and is greater than all of them.
Daryl McCullough
>
>
>Modern mathematics implies that the number of the natural numbers is
>the natural numbers count themselves.
>
>the natural numbers.
That's what it *means* to say that "the cardinality of the set of naturals
is greater than any natural number". So you are *agreeing* with the
"modern mathematics", you are just getting hung up on the terminology.
Let's go through the details, and you can say exactly which one
you find objectionable:
1. Define a partial ordering <= on sets as follows: A <= B if there
is an injection from A to B. (An injection from A to B is a function
f from A to B such that if x and y are unequal, then f(x) is unequal
to f(y).
2. Define an equivalence relation ~ on sets as follows: A ~ B if
A <= B and B <= A.
Now, what is cardinality? It's just a convention for naming
the equivalence classes of the equivalence relation ~. Mathematically,
cardinality should be a function "card" on sets with the following
property:
If A ~ B, then card(A) = card(B). It doesn't really matter how
you define this function as long as it satisfies this requirement.
A standard way is this:
3. If A is a set, and there is a natural number n such that
A ~ {0,1,2,...,n-1}, (the set of all naturals less than n),
then we say card(A) = n.
4. If A is a set, and A ~ { 0, 1, 2, ... } (the set of all
naturals), then we say card(A) = omega.
5. If A is a set, and A ~ R (the set of all reals), then
we say card(A) = C (the continuum).
Most sets that come up naturally in analysis have one of
those cardinalities.
So the set of all naturals has cardinality omega *BY* *DEFINITION*.
There is nothing to prove.
--
Daryl McCullough
Ithaca, NY
Daryl McCullough
george says...
I think that Albrecht is making the following, truly ridiculous
point:
Instead of considering the sets {}, {0}, {0,1}, {0,1,2}, he wants
to consider the sets {1}, {1,2}, {1,2,3}. Each of these sets has the
property that its cardinality is equal to its largest element. The
set { 1, 2, 3, ... } of *all* positive naturals has no largest element.
So if the pattern "cardinality = largest element" holds, then this
set has no cardinality. Why Albrecht thinks that the pattern should
hold for infinite sets, I have no idea. Like most crackpots, he's
not very good at mathematics or logic.
Daryl McCullough
>I don't find it idiotic. Just because Albrecht is a
>finitist working in a theory like ZF-Infinity
>(possibly +~Infinity), it doesn't mean that he is
>an idiot.
Well, he is an idiot. He isn't a finitist, and he
isn't working in ZF-Infinity. He's a crackpot.
He's making claims that are false, and he is
making fallacious arguments for them.
If someone wants to work in the theory ZF-Infinity +
"all sets are finite", that's perfectly fine. That
theory is equivalent to working in first-order
Peano Arithmetic. That isn't what Albrecht is doing.
He's not proving theorems in any such theory. He's
making diatribes against "modern mathematics".
Ralf Bader
> lwa...@lausd.net writes:
>
>> On Jan 6, 6:32 pm, Joshua Cranmer <Pidgeo...@verizon.invalid> wrote:
>> > Albrecht wrote:
>> > > The number of natural numbers can't be greater than every natural
>> > > number since there are not more natural numbers than there are
>> > > natural numbers.
>> > But for each natural number, there is a larger natural number. This is
>> > indisputable fact [....]
>>
>> But Albrecht never disputed this fact. I admit that some
>> so-called "cranks," such as WM, do believe in a largest
>> natural,
>
> Stop right there!
> WM has specifically *denied* that he belives that there is a largest
> natural, and called that idea "nonsense".
>
> That makes sense for followers of the idea of the potentially infinite.
Mückenheim asserts that
1) the universe is finite, consisting of some 10^80 elementary particles
2) a number can only exist if the ressources of the universe are sufficient
to denote it
3) therefore, for example a number requiring 10^10^10 digits in random
succession does not exist
4) using abbreviative notations like exponentials, there are nevertheless
arbitrarily large numbers which do exist
5) the sequence of natural numbers therefore has gaps
See his piece of junk "The physical constraints of numbers" in the arxiv.
Mückenheim does not answer the question why the finiteness of the universe
does not limit the availability of "abbreviative notations".
Mückenheim does not answer the question why it is not an "abbreviative
notation" to say e.g. "take the number whose decimal expansion consists of
the first 10^10^10 digits of pi"; on the contrary, he gives something like
this as an example of a number which does not exist. That is, he has no
well-defined notion of "abbreviative notation" and does not even see the
necessity to define that notion.
Mückenheim does not answer the question how that natural number sequence
with gaps should work, concerning e.g. induction.
I have no idea how such nonsense should make sense to any intelligent
person. I do see sense in, e.g., the ultrafinitistic endeavors of Van
Bendegem, although I don't share that position. But Mückenheim's "theory"
is the most idiotic nonsense I have ever seen.
Ralf
Virgil
<9d09d751-4058-4400...@q30g2000prq.googlegroups.com>,
Albrecht <albs...@gmx.de> wrote:
> Modern mathematics implies that the number of the natural numbers is
> greater than every natural number. This opinion is indefensible since
> the natural numbers count themselves.
Then you must be claiming that the number of natural numbers is less
than or equal to some natural number?
Can you exhibit some such natural as large as the number of naturals?
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
>David C. Ullrich wrote:
>
>> that he seems to be missing the notion the
>> math is based on axioms seems to be the problem.
>
>You've never summarized the "axioms of math" in your book "Complex made
>Simple"
No.
>and _yet_ it _is_ mathematics all over the place. Right ?
Yes. Based on a set of axioms. The _standard_ set, which is why
there's no need to summarize it explicitly.
>Mathematics is mathematics, based on axiom systems or not. And what the
>OP does _is_ mathematics, whether you like it or not. Axiom systems aid
>in understanding, perhaps, but cannot replace it.
>
>Han de Bruijn
David C. Ullrich
Virgil
> The number of natural numbers can't be greater than every natural
> number since there are not more natural numbers than there are natural
> numbers.
That implies that there can be no numbers other than natural numbers.
Which makes mathematics largely useless.
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
Most mathematician eat and breath and do other things that are generally
not regarded as being mathematics.
>
> Han de Bruijn
Virgil
Alan Smaill <sma...@collins.inf.ed.ac.uk> wrote:
> lwa...@lausd.net writes:
> > But Albrecht never disputed this fact. I admit that some
> > so-called "cranks," such as WM, do believe in a largest
> > natural,
>
> Stop right there!
> WM has specifically *denied* that he belives that there is a largest
> natural, and called that idea "nonsense".
But WM has also claimed things which necessitate a largest natural.
Mariano Suárez-Alvarez
With such interesting anecdotes, you must be
quit a success at parties and other social
events! :/
-- m
Toni...@yahoo.com
> In article <01fbd644-cc50-4ac6-8324-228fa8a02...@a29g2000pra.googlegroups.com>,
> george says...
>
>
>
>
>
>
>
> >> On Jan 6, 9:08 pm, Albrecht <albst...@gmx.de> wrote:
> >> > Any natural number is the number of its predecessors, inclusive the
> >> > considered number self.
>
> >This is not even grammatical.
> >People in general are going to have to get used to the fact that
> >0 IS a natural number. That means that every natural number is
> >the number of smaller natural numbers. Since 0 is first, there are
> >no smaller natural numbers, so that winds up defining 0 as the
> >cardinality
> >of the empty set.
>
> >> > The number of natural numbers can't be greater than every natural
> >> > number since there are not more natural numbers than there are natural
> >> > numbers.
>
> >This is just bullshit. EVERY natural number is greater than
> >than every one of the numbers in the set that IT "numbers",
> >IF you do it right: {0,1,2,3,4,5,6} and is greater than all of them.
>
> I think that Albrecht is making the following, truly ridiculous
> point:
>
*****************************************************************************
What ridiculous, usually unbelievably stupid and nonsensical, point is
Albrecht trying to do at any fixed point in time is, most of the
times, anyone's guess.
This time he's excelled himself though, perhaps due to the influence
of all that christmas and new year's alcohol, and it seems to be clear
that he means to crash on infinite in mathematics.
Regards
Tonio
> Instead of considering the sets {}, {0}, {0,1}, {0,1,2}, he wants
> to consider the sets {1}, {1,2}, {1,2,3}. Each of these sets has the
> property that its cardinality is equal to its largest element. The
> set { 1, 2, 3, ... } of *all* positive naturals has no largest element.
> So if the pattern "cardinality = largest element" holds, then this
> set has no cardinality. Why Albrecht thinks that the pattern should
> hold for infinite sets, I have no idea. Like most crackpots, he's
> not very good at mathematics or logic.
>
> --
> Daryl McCullough
> Ithaca, NY- Hide quoted text -
>
> - Show quoted text -
John Jones
How can I agree with that? If a proposed entity E isn't 'this thing',
then I am quite entitled to question whether there is any entity E that
is 'that thing'.
David Bernier
Fine, thank you.
How should emergence be represented?
David Bernier
Herbert Newman
>
> Albrecht began his new year's nonsense with the following:
>
> "Modern mathematics implies that the number of the natural numbers is
> greater than every natural number. This opinion is indefensible since
> the natural numbers count themselves."
>
> From this very unique paragraph one can know we have here a HUGE crank:
> Modern mathematics don't imply anything of the like as he wrote, and
> of course not that "the number of natural numbers" (what is that?) is
> anything.
>
Well, the number of natural number refers to the cardinality of the set of
natural numbers. Actually, this claim can be formulated the following way:
An e N: #N > n (with #N = card(N)).
>
> What does it mean that "the natural numbers count themselves"? I
> haven't yet seen a number 7 counting itself, or other natural number:
> perhaps he meant something else, but what?
>
He meant the following (rather trivial) fact:
An e N\{0}: #{1, ..., n} = n.
But he "concludes" some strange things concerning #N from this fact.
The complete lack of any coherent argumentation is rather typical for a
crackpot/crank though.
"Modern astronomy implies that the number of the planets is
greater than one. This opinion is indefensible since the
earth is a planet."
(I guess A. would add: Hence the number of planets is one!)
Herb
John Jones
Emergence is a new property. Like a bouquet emerges from a collection of
flowers. The property of the bouquet is not the property of the
collection of flowers.
John Jones
yes, but {} is a seventh element. {0,1,2,3,4,5,6} has seven elements. It
is better written as 'set 0 1 2 3 4 5 6'. That's seven.
John Jones
You are equivocating over your use of 'all'. This is not the 'all' of
'all the numbers from one to ten'.
> So if the pattern "cardinality = largest element" holds, then this
David Bernier
I've read a bit on the Web about emergence in complex systems. So I have
more questions. In the world around us, or reality, are there
properties that you would say are non-emergent? If yes, how does
one know that there is no lower layer from which the phenomena or
property could emerge?
David Bernier
Virgil
John Jones <jonesc...@aol.com> wrote:
There is an important distinction between saying that a set has seven
elements and that the set has seven as an element.
Virgil
John Jones <jonesc...@aol.com> wrote:
Not at all. To say that "all these" may be different from "all those" is
hardly equivocal, even when "these" and "those" are allowed to overlap.
Toni...@yahoo.com
wrote:
> Albrecht wrote:
> > Modern mathematics implies that the number of the natural numbers is
> > greater than every natural number. This opinion is indefensible since
> > the natural numbers count themselves.
>
> the OP ever answering. This one is an able troll. :-)
****************************************************************
Not more able than any other regular crank. I supose we're used to
this routine: a cranks writes some nonsense, and many of sci.math
members get vacuumed into the thread.
Nothing wrong with that, imo: I think many people here, just like me,
take this mostly as an entertainment, relaxing time, though sometimes
one can learn stuff here.
Regards
Tonio
lwa...@lausd.net
> On Wed, 7 Jan 2009 00:45:13 -0800 (PST), lwal...@lausd.net wrote:
> >ZF-Infinity+~Infinity is neither sound nor valid?
> There you go yet again, defending something utterly
> other than what the guy said.
I've noticed that my Google ranking is now down to
one star. I'm now officially a "crank," and have
passed the point of no return. Of course, I knew
going in, when I first started posting to sci.math,
that by defending theories other than ZFC, it would
happen sooner or later that I'd be labelled "crank"
and have a one-star rating, just like others who
propose theories other than ZFC.
I officially became a "crank" by trying to defend
the finitist Albrecht. Ullrich explains:
> ZF-Infinity+-Infinity is logical theory that has
> certain well-defined axioms. In the OP there was
> nothing about what follows from this axiom or
> that axiom, the OP _stated_ that [blah blah]
> was "indefensible".
Note that for the proposed ideas of many so-called
cranks, such as AP, tommy1729, and others, we don't
yet know whether there exists a rigorous theory
that incorporates all of their ideas and desired
properties of a theory. But for Albrecht, we know
that there _is_ a theory, ZF-Infinity+~Infinity,
that _does_ have all of the properties that he
wants it to have. In particular, it's a theory in
which we don't have infinite sets. I'd have thought
that the established mathematicians would at least
be more sympathetic to ZF-Infinity+~Infinity than to
the theories of other "cranks," since the former,
unlike the latter, is already known to be possible
in a rigorous theory. But I guess I was wrong.
> He "proved" this, and his
> proof was nonsense, since it assumed that
> "number" means "natural number", which is
> clearly not what the word means in the context
> he was discussing.
Here Ullrich criticizes Albrecht's proof because
of the use of the word "number," in the phrase
"the number of natural numbers." Obviously, the
first word "number" refers to how many naturals
there are -- that is, the _cardinality_ of the
set of naturals. Thus Ullrich points out that
Albrecht is confusing the two distinct concepts
of _cardinal_ number and _natural_ number.
But in ZF-Infinity+~Infinity, the only possible
cardinals are the finite cardinals -- which are
exactly the natural numbers! Thus, in the theory
ZF-Infinity+~Infinity, every cardinal number _is_
a natural number, and so Albrecht can validly use
them interchangeably in that theory!
> And in particular his argument said nothing
> whatever about ZF-Infinity+-Infinity, or
> any other axiomatic system.
Au contraire. Albrecht's argument did say something
about ZF-Infinity+~Infinity -- since he used the
concepts of cardinal number and natural number
interchangeably (as Ullrich pointed out), which one
_can_ validly do in ZF-Infinity+~Infinity!
Therefore, Albrecht's argument is a sound and valid
proof in ZF-Infinity+~Infinity.
Han de Bruijn
You act as if *HdB* made it up, but I didn't. Believe it or not: quite a
few mathematicians find that mathematics is what mathematicians do.
Han de Bruijn
lwa...@lausd.net
> On Jan 7, 10:42 am, lwal...@lausd.net wrote:
> > I don't find it idiotic. Just because Albrecht is a
> > finitist working in a theory like ZF-Infinity
> > (possibly +~Infinity), it doesn't mean that he is
> > find that many adherents of ZFC consider most
> > finitists to be idiots.
> At least most "finitists" here in sci.math. And I write "finitists"
> because finitists would be way too much of a honour to give them,
> taking into consideration that the huge majority of them (well, in
> fact all of them, as far as I can recall right now) are NOT
> mathematicians.
No true Scotsman would put sugar on his porridge?
No true mathematician could ever be a finitist?
Or maybe it would be more accurate to say, no true
finitist would argue the way Albrecht argues?
If so, then where are these alleged "true" finitists
whose arguments are more mathematical than Albrecht's?
> Albrecht began his new year's nonsense with the following:
> "Modern mathematics implies that the number of the natural numbers is
> greater than every natural number. This opinion is indefensible since
> the natural numbers count themselves."
> From this very unique parraph one can know we have here a HUGE crank:
> Modern mathematics don't imply anything of the like as he wrote, and
> of course not that "the number of natural numbers" (what is that?) is
> anything.
McCullough and Newman have already explained what Albrecht
most likely meant. Most likely, he meant:
ZFC ("Modern mathematics") implies that the cardinality of
the set of natural numbers is greater than every natural number.
ZFC |- An (neN -> (card(N) > n))
> What does it mean that "the natural numbers count themselves"? I
> haven't yet seen a number 7 counting itself, or other natural number:
> perhaps he meant something else, but what?
According to McCullough, Albrecht is counting starting
with 1 rather than 0, as is standardly done with the
von Neumann naturals, so that he meant something like:
card({1,2,3,4,5,6,7}) = 7
So Albrecht's argument is that since the cardinality of
a set which contains 1, and contains the predecessor of
all its elements except 1, is the largest element of
the set, the same must be true for N. And since N has
no largest element, it therefore can't exist.
Obviously this argument is invalid in ZF, but it's valid
in ZF-Infinity+~Infinity.
> You see, lwalk? You love to defend the indefensible, at least in
> mathematics.
ZF-Infinity+~Infinity is indefensible? Finitism is indefensible?
> This guy, as ALL other "finitists" I can recall right now
> here in sci.math, don't really know what they're talking about, but
> they make lots of noise...just as you do.
The fact that _all_ the other finitists Tonio can recall here
on sci.math act just Albrecht, to me, is evidence that this
is how "true" finitists act. I have no more reason to believe
the finitists who post on sci.math aren't representative of
the "true" finitists, any more than I have reason to believe
that the adherents of ZFC who post here aren't typical among
adherents of ZFC.
Han de Bruijn
Most engineers eat and breath and do other things that are generally not
not regarded as being engineering. Engineering is not what engineers do.
And art is not what artists do. Sure.
Han de Bruijn
lwa...@lausd.net
> lwal...@lausd.net writes:
> > But Albrecht never disputed this fact. I admit that some
> > so-called "cranks," such as WM, do believe in a largest
> > natural,
> Stop right there!
> WM has specifically *denied* that he belives that there is a largest
> natural, and called that idea "nonsense".
So neither 10^100, nor especially 10^10^100, is not an
upper bound on the set of naturals that WM accepts?
If they aren't upper bounds, then why does WM repeatedly
mention googol(plex), or the number of particles in the
physical universe, in his arguments against Infinity?
lwa...@lausd.net
> In article <01fbd644-cc50-4ac6-8324-228fa8a02...@a29g2000pra.googlegroups.com>,
> george says...
> >than every one of the numbers in the set that IT "numbers",
> >IF you do it right: {0,1,2,3,4,5,6} and is greater than all of them.
> to consider the sets {1}, {1,2}, {1,2,3}. Each of these sets has the
> property that its cardinality is equal to its largest element. The
> set { 1, 2, 3, ... } of *all* positive naturals has no largest element.
I've noticed that many so-called "cranks" prefer to start
counting from 1 rather than 0.
Traditionally, in schools, one learns that {1,2,3,...} is
the set of _natural_ numbers, while {0,1,2,3,...} is the
set of _whole_ numbers. Thus zero is the only _whole_
number that isn't a _natural_ number.
Of course, set theorists (nor most mathematicians past
grade school), don't use the term "whole numbers." And
they usually include zero as a natural number, since it
makes the definition of a natural number easier via the
von Neumann ordinals. {1,2,3,...} can't be a von Neumann
ordinal -- only {0,1,2,3,...} can.
Also, it makes it easier to identify each natural as the
set of naturals less than itself, a set whose cardinality
is the natural itself. To do the same for the naturals
starting from 1, as Albrecht would prefer, would result
in, say, 7 = {1,2,3,4,5,6,7}, which clearly violates
ZF's Axiom of Foundation/Regularity.
But none of this explains why the so-called "cranks" are
the ones who prefer to start from 1 rather than 0. Many
of the arguments between "cranks" and the established set
theorist could be resolved simply by agreeing on whether
to start naturals from 0 or 1.
lwa...@lausd.net
> Tonic...@yahoo.com wrote:
> > Ah, great definition! Just like "moral is what moralists preach", uh?
> > Ok....got it.
> few mathematicians find that mathematics is what mathematicians do.
To many people, there are some things that mathematicians
_don't_ do. And one of those things is espouse finitism,
at least the type of finitism that Albrecht, or even HdB
himself, proposes.
Toni...@yahoo.com
> Tonic...@yahoo.com wrote:
> > On Jan 7, 3:14 pm, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
>
> >>Tonic...@yahoo.com wrote:
>
> >>>On Jan 7, 1:26 pm, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
>
> >>>>David C. Ullrich wrote:
>
> >>>>>that he seems to be missing the notion the
> >>>>>math is based on axioms seems to be the problem.
>
> >>>>You've never summarized the "axioms of math" in your book "Complex made
> >>>>Simple" and _yet_ it _is_ mathematics all over the place. Right ?
>
> >>>>Mathematics is mathematics, based on axiom systems or not. And what the
> >>>>OP does _is_ mathematics, whether you like it or not. Axiom systems aid
> >>>>in understanding, perhaps, but cannot replace it.
>
> >>>*********************************************************
>
> >>>New year, new trollings. In particular that new-year's pearl
> >>>"mathematics is mathematics, based on axiom systems or not"...
> >>>You sound like my old auntie telling us "moral is moral, now, future,
> >>>past, here and everywhere!", with that biblical look in her eyes.
>
> >>>But I'll tell you what: I'll grant you the benefit of doubt about the
> >>>above, and I won't compare you to my auntie, if you are gracious
> >>>enough as to tell us WHAT is mathematics. Fair enough? Then we'll know
> >>>what is mathematics, axioms or not.
>
> >>Isn't mathematics: what mathematicians do ?
>
> >>Han de Bruijn-
>
>
> > Ah, great definition! Just like "moral is what moralists preach", uh?
> > Ok....got it.
>
> > Regards
> > Tonio
>
> You act as if *HdB* made it up, but I didn't. Believe it or not: quite a
> few mathematicians find that mathematics is what mathematicians do.
>
****************************************************************
Hey, I will accept that as a description of what mathematics is,
though it is rather unsatisfying.
I though asked from you a DEFINITION (which, btw, I would never, ever
evewn dream to dare to give at all), and all this why? Because you
wrote that "mathematics is mathematics, with axioms or without them",
so it looked like you're pretty sure of what mathematics is...
Sp let us assume you meant ONLY to give an informal, loose description
of what mathematics is by saying "it is what mathematicians do"...aha,
so now allow me to ask you: who is, according to you, a mathematician?
I hope you won't answer me, cyclically, that a mathematician is
whoever does maths...
Regards
Tonio
Toni...@yahoo.com
> On Jan 7, 5:07 am, Tonic...@yahoo.com wrote:
>
> > On Jan 7, 10:42 am, lwal...@lausd.net wrote:
> > > I don't find it idiotic. Just because Albrecht is a
> > > finitist working in a theory like ZF-Infinity
> > > (possibly +~Infinity), it doesn't mean that he is
> > > an idiot. Of course, I would not be surprised to
> > > find that many adherents of ZFC consider most
> > > finitists to be idiots.
> > At least most "finitists" here in sci.math. And I write "finitists"
> > because finitists would be way too much of a honour to give them,
> > taking into consideration that the huge majority of them (well, in
> > fact all of them, as far as I can recall right now) are NOT
> > mathematicians.
>
> No true Scotsman would put sugar on his porridge?
> No true mathematician could ever be a finitist?
>
> Or maybe it would be more accurate to say, no true
> finitist would argue the way Albrecht argues?
>
> If so, then where are these alleged "true" finitists
> whose arguments are more mathematical than Albrecht's?
>
You have this rare quality of mixing it all, confusing it all and
misunderstanding it all...or much of it.
Who EVEN hinted that "no true mathematician could ever be a finitist"?
No, I don't think that any true finitist could argue the way Albrecht
does: I'd say that no true mathematician, or even reasoning person,
could.
About your last question: I don't have the faintest idea....what
should I?
**********************************************************
> > Albrecht began his new year's nonsense with the following:
> > "Modern mathematics implies that the number of the natural numbers is
> > greater than every natural number. This opinion is indefensible since
> > the natural numbers count themselves."
> > From this very unique parraph one can know we have here a HUGE crank:
> > Modern mathematics don't imply anything of the like as he wrote, and
> > of course not that "the number of natural numbers" (what is that?) is
> > anything.
>
> McCullough and Newman have already explained what Albrecht
> most likely meant. Most likely, he meant:
>
> ZFC ("Modern mathematics") implies that the cardinality of
> the set of natural numbers is greater than every natural number.
>
> ZFC |- An (neN -> (card(N) > n))
>
*************************************************************
Again "guessing" what others meant, lwalk? This is a rather disgusting
and annoying habit of yours.
*************************************************************
*******************************************************************
I don't have the palest doubt that you see evidence of things where
they don't exist at all, and you've here already decided that Albrecht
with his nonsenses, as many other NON-MATHEMATICIANS with theirs, are
"true" representatives of the finitist movement in mathematics.
I hope the "NON-MATHEMATICIANS" (and this does involve here IGNORANCE
of mathematics, and in most cases lack of logical reasoning habits as
well) did NOT go unnoticed by you...
Regards
Tonio
Herbert Newman
> John Jones wrote:
>>
>> george wrote:
>>>
>>> This is just bullshit. EVERY natural number is greater than
>>> than every one of the numbers in the set that IT "numbers",
>>> IF you do it right: 7 = {0,1,2,3,4,5,6} and is greater than
>>> all of them.
>>>
>> yes, but {} is a seventh element.
>>
Right, since 0 = {}.
>>
>> {0,1,2,3,4,5,6} has seven elements.
>>
Right! (What an insight.)
>>
>> It is better written as 'set 0 1 2 3 4 5 6'.
>>
Why, since
{0,1,2,3,4,5,6}
just means
[the] set [consisting of the elements] 0, 1, 2, 3, 4, 5, 6.
?
With other words,
{0,1,2,3,4,5,6} = the set consisting of 0, 1, 2, 3, 4, 5, 6.
>>
>> That's seven.
>>
Right. If we stick to von Neumann's definition of the natural numbers.
>
> There is an important distinction between saying that a set has seven
> elements and that the set has seven as an element.
>
Right. For example the set
{0,1,2,3,4,5,6}
has seven elements, but does not have 7 as an element. On the other hand,
the set
{0,1,2,3,4,5,6,7}
has eight (!) elements, and _does_ have 7 as an element. Finally, the set
{1,2,3,4,5,6,7}
has seven elements, and does have the 7 as an element. Not though (JJ) that
these three sets are (pairwise) distinct:
{0,1,2,3,4,5,6} =/= {0,1,2,3,4,5,6,7},
{0,1,2,3,4,5,6} =/= {1,2,3,4,5,6,7},
{0,1,2,3,4,5,6,7} =/= {1,2,3,4,5,6,7}.
Herb
Herbert Newman
>
> On Jan 8, 11:13 am, lwal...@lausd.net wrote:
>>
>> McCullough and Newman have already explained what Albrecht
>> most likely meant. Most likely, he meant:
>>
>> ZFC ("Modern mathematics") implies that the cardinality of
>> the set of natural numbers is greater than every natural
>> number.
>>
With other words, in ZFC we have
An e N: card(N) > n.
If this is what A. meant, A's right. But concerning his "conclusions" he's
wrong. Little wonder, since his "arguments" are complete nonsense/bullshit.
>>>
>>> What does it mean that "the natural numbers count themselves"? I
>>> haven't yet seen a number 7 counting itself, or other natural number:
>>> perhaps he meant something else, but what?
>>>
>> According to McCullough, Albrecht is counting starting
>> with 1 rather than 0, as is standardly done with the
>> von Neumann naturals, so that he meant something like:
>>
>> card({1,2,3,4,5,6,7}) = 7
>>
Again, this might very well be true. So what? :-o
>>
>> So Albrecht's argument is that since the cardinality of
>> a set which contains 1, and contains the predecessor of
>> all its elements except 1, is the largest element of
>> the set, the same must be true for N. And since N has
>> no largest element, it therefore can't exist.
>>
>> Obviously this argument is invalid in ZF, [...]
>>
Indeed. That's just what I said. Albrecht's "arguments" are neither sound
nor valid. Period.
Note, lwalk, that the context of Albrecht's claims IS "modern mathematics",
i.e. ZFC (and not any other system of set theory you might dream up).
>>>
>>> This guy, as ALL other "finitists" I can recall right now
>>> here in sci.math, don't really know what they're talking about,
>>> but they make lots of noise...just as you do.
>>>
>> The fact that _all_ the other finitists Tonio can recall here
>> on sci.math act just Albrecht, to me, is evidence that this
>> is how "true" finitists act.
>>
Well, since "finitists" in this context can safely be read as "cranks" your
statement amounts to:
The fact that _all_ the other cranks Tonio can recall here
on sci.math act just [like] Albrecht, to me, is evidence
that this is how "true" cranks act.
Right. That's indeed the case.
The problem of course is...
>
> ... that you see evidence of things where they don't exist at all,
> and you've here already decided that Albrecht with his nonsenses,
> as many other NON-MATHEMATICIANS with theirs, are "true" representatives
> of the finitist movement in mathematics.
>
Especially when they in fact are representatives of the mathematical crank
movement in Usenet.
Clearly lwalk is himself one of them. Though he prefers to hide behind the
other guys (he might consider this a clever strategy).
>
> I hope the "NON-MATHEMATICIANS" (and this does involve here IGNORANCE
> of mathematics, and in most cases lack of logical reasoning habits as
> well) did NOT go unnoticed by you...
>
Well, you just mentioned it..., the "lack of logical reasoning habits".
Herb
Herbert Newman
>>>
>>> To make a long story short: no need to appeal to Zeno's Paradox here.
>>> A's stuff is idiotic enough even without that (i.e. self-sufficiently).
>>>
>> I don't find it idiotic. [lwalk]
>>
So what? It's still a fact.
ANY _mathematician_
>>
>> ... working in a theory like ZF-Infinity [lwalk]
>>
WOULD AGREE THAT even in his theory
Infinity -> An e N: card(N) > n
would hold.
But...
>
> Albrecht began his new year's nonsense with the following:
>
> "Modern mathematics implies that the number of the natural numbers is
> greater than every natural number. This opinion is indefensible since
> the natural numbers count themselves."
>
Huh?!
Herb
David C. Ullrich
>On Jan 7, 3:15 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> On Wed, 7 Jan 2009 00:45:13 -0800 (PST), lwal...@lausd.net wrote:
>> >ZF-Infinity+~Infinity is neither sound nor valid?
>> There you go yet again, defending something utterly
>> other than what the guy said.
>
>I've noticed that my Google ranking is now down to
>one star. I'm now officially a "crank," and have
>passed the point of no return. Of course, I knew
>going in, when I first started posting to sci.math,
>that by defending theories other than ZFC, it would
>happen sooner or later that I'd be labelled "crank"
>and have a one-star rating, just like others who
>propose theories other than ZFC.
The problem is not that he's proposing a theory
other than ZFC and you're defending it. The
problem is that he's simply _asserting_ that
ZFC is _wrong_. (That's _his_ problem;
your problem is that you can't seem to see the
difference even after it's pointed out.)
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.)
Daryl McCullough
>According to McCullough, Albrecht is counting starting
>with 1 rather than 0, as is standardly done with the
>von Neumann naturals, so that he meant something like:
>
>card({1,2,3,4,5,6,7}) = 7
>
>So Albrecht's argument is that since the cardinality of
>a set which contains 1, and contains the predecessor of
>all its elements except 1, is the largest element of
>the set, the same must be true for N. And since N has
>no largest element, it therefore can't exist.
>
>Obviously this argument is invalid in ZF, but it's valid
>in ZF-Infinity+~Infinity.
Albrecht's argument is mathematically incompetent,
whether or not you are working in ZF-Infinity+~Infinity.
It's nonsense regardless of what axioms you choose.
The axiom ~Infinity *says* "There is no set containing
all natural numbers". So if he is working in such a
theory, then the correct way to prove that N does not
exist is "by axiom". Albrecht's way is idiotic. He
hasn't defined his terms, he hasn't said what his
axioms are. He hasn't said what rules of inference
he is using. His argument is logically and mathematically
incompetent.
Daryl McCullough
>
>On Jan 7, 3:15=A0am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> On Wed, 7 Jan 2009 00:45:13 -0800 (PST), lwal...@lausd.net wrote:
>> >ZF-Infinity+~Infinity is neither sound nor valid?
>> There you go yet again, defending something utterly
>> other than what the guy said.
>
>I've noticed that my Google ranking is now down to
>one star. I'm now officially a "crank," and have
>passed the point of no return. Of course, I knew
>going in, when I first started posting to sci.math,
>that by defending theories other than ZFC, it would
>happen sooner or later that I'd be labelled "crank"
>and have a one-star rating, just like others who
>propose theories other than ZFC.
You are completely off base here. The complaints
about WM and Albrecht is *not* that they are working
within some theory other than ZFC. The complaints
are that their arguments are logically incoherent
nonsense.
>I officially became a "crank" by trying to defend
>the finitist Albrecht.
Albrecht is first and foremost an idiot. He may
also be a finitist, but that's not the chief complaint
about him.
>Note that for the proposed ideas of many so-called
>cranks, such as AP, tommy1729, and others, we don't
>yet know whether there exists a rigorous theory
>that incorporates all of their ideas and desired
>properties of a theory. But for Albrecht, we know
>that there _is_ a theory, ZF-Infinity+~Infinity,
>that _does_ have all of the properties that he
>wants it to have. In particular, it's a theory in
>which we don't have infinite sets. I'd have thought
>that the established mathematicians would at least
>be more sympathetic to ZF-Infinity+~Infinity than to
>the theories of other "cranks," since the former,
>unlike the latter, is already known to be possible
>in a rigorous theory. But I guess I was wrong.
Albrecht is *not* proving theorems in ZF-Infinity+~Infinity.
Herbert Newman
It seems to me that lwalk is either an idiot or himelf a crank (or both).
>>
>> So Albrecht's argument is that since the cardinality of
>> a set which contains 1, and contains the predecessor of
>> all its elements except 1, is the largest element of
>> the set, the same must be true for N. And since N has
>> no largest element, it therefore can't exist.
>>
>> Obviously this argument is invalid in ZF, but it's valid
>> in ZF-Infinity+~Infinity.
>>
Obviously lwalk's claim is bullshit. Since in his pet theory
ZF-Infinity+~Infinity, there IS no set N of all natural numbers.
Hence any "arguments" about what must be true for N (!) or not
is simply meaningless.
If there definitely IS NO infinite set in a certain theory, well, there's
certainly no problem related to ANY infinite set. :-)
Well, so let's consider to ZF-Infinity, which look slightly more favorable
(in this context). There the "argument" just isn't valid. Period. (Hint in
this case we may _reasonably_ assume that there is a infinite set of all
natural numbers, N, and consider the consequences of this assumption.)
Herb
Herbert Newman
It seems to me that lwalk is either an idiot or himelf a crank (or both).
>>
>> So Albrecht's argument is that since the cardinality of
>> a set which contains 1, and contains the predecessor of
>> all its elements except 1, is the largest element of
>> the set, the same must be true for N. And since N has
>> no largest element, it therefore can't exist.
>>
>> Obviously this argument is invalid in ZF, but it's valid
>> in ZF-Infinity+~Infinity.
>>
Obviously lwalk's claim is bullshit. Since in his pet theory
ZF-Infinity+~Infinity, there IS no set N of all natural numbers.
Hence any "arguments" about what must be true for N (!) or not
is simply meaningless.
If there definitely IS NO infinite set in a certain theory, well, there's
certainly no problem related to ANY infinite set. :-)
Well, so let's consider ZF-Infinity instead, which look slightly more
Daryl McCullough
>But none of this explains why the so-called "cranks" are
>the ones who prefer to start from 1 rather than 0. Many
>of the arguments between "cranks" and the established set
>theorist could be resolved simply by agreeing on whether
>to start naturals from 0 or 1.
Cranks insist on idiosyncratic conventions and definitions
because it's the only way to prove their conclusions. That's
evidence that their conclusions are invalid; a valid conclusion
does not depend on particular conventions.
The convention that an ordinal X is equal to the set of all
ordinals less than X is very convenient because it generalizes
naturally to transfinite ordinals. In contrast, other conventions
don't generalize to infinite ordinals. But that doesn't mean
that one can't do set theory with other conventions, just that
it is more awkward. You have to have definitions that say:
If the ordinal is finite, then ... otherwise, if it is infinite,
then ... The conventions make the arguments smoother, but the
arguments go through whether or not you adopt those conventions.
In contrast, crackpot arguments only appear to work because of
their conventions. They can't be formulated using alternative
conventions.
Herbert Newman
> lwa...@lausd.net says...
>>
>> But none of this explains why the so-called "cranks" are
>> the ones who prefer to start from 1 rather than 0. Many
>> of the arguments between "cranks" and the established set
>> theorist could be resolved simply by agreeing on whether
>> to start naturals from 0 or 1.
>>
> Cranks insist on idiosyncratic conventions and definitions
> because it's the only way to prove their conclusions. That's
> evidence that their conclusions are invalid; a valid conclusion
> does not depend on particular conventions.
>
Right. In this case the crucial statement seems to be the claim that "the
natural numbers count themselves".
In symbols: An e N: {1, ..., n} = n.
Well, so what? :-o
Since #N (i.e. in axiomatic set theory, N) is no natural number, this
trivial fact (stated above) is of no relevance concerning the question if
An e N: #N > n
or not.
On the other hand:
>
> The convention that an ordinal X is equal to the set of all
> ordinals less than X is very convenient because it generalizes
> naturally to transfinite ordinals. In contrast, other conventions
> don't generalize to infinite ordinals. But that doesn't mean
> that one can't do set theory with other conventions, just that
> it is more awkward. You have to have definitions that say:
> If the ordinal is finite, then ... otherwise, if it is infinite,
> then ... The conventions make the arguments smoother, but the
> arguments go through whether or not you adopt those conventions.
>
> In contrast, crackpot arguments only appear to work because of
> their conventions. They [very often] can't be formulated using
> alternative conventions.
>
Right.
Herb
amy666
> On 8 Jan 2009 03:42:45 -0800 Daryl McCullough wrote:
>
> > lwa...@lausd.net says...
> >>
> >> But none of this explains why the so-called
> "cranks" are
> >> the ones who prefer to start from 1 rather than 0.
> Many
> >> of the arguments between "cranks" and the
> established set
> >> theorist could be resolved simply by agreeing on
> whether
> >> to start naturals from 0 or 1.
> >>
> > Cranks insist on idiosyncratic conventions and
> definitions
> > because it's the only way to prove their
> conclusions. That's
> > evidence that their conclusions are invalid; a
> valid conclusion
> > does not depend on particular conventions.
> >
> Right. In this case the crucial statement seems to be
> the claim that "the
> natural numbers count themselves".
you forgot some important details :
x counts the natural numbers AND itself.
the AND is the seperation between idiots and revolutionairs ...
regards
tommy1729
Dik T. Winter
You should better read what WM writes. There is (in WM's view) a limit on
the *number* of natural numbers that can co-exist. If a new natural number
is created another natural number vanishes from existence (or more than one).
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Dik T. Winter
...
> Traditionally, in schools, one learns that {1,2,3,...} is
> the set of _natural_ numbers, while {0,1,2,3,...} is the
> set of _whole_ numbers. Thus zero is the only _whole_
> number that isn't a _natural_ number.
Anglo-saxon schools. Try French schools for a change.
Jesse F. Hughes
> I've noticed that my Google ranking is now down to
> one star. I'm now officially a "crank," and have
> passed the point of no return.
You are now officially a very silly person for saying such things.
As others have pointed out, it makes no particular sense to think that
Albrecht is making a claim about the theory of finite sets. What
would the claim be? That in the theory of finite sets, there is no
infinite set?
Albrecht's thesis is not hard to see. It's right there in the subject
line.
And no one has been called a crank for working in the theory of finite
sets or for finding the theory interesting. They have been called
cranks for presenting silly arguments that the theory ZF is
inconsistent or otherwise bad.
--
Jesse F. Hughes
"I don't know if you noticed but I had a tremendous drop in confidence
concomittant [sic] with a dramatic grip of existential crisis."
--- James S. Harris even has better diseases than you
Denis Feldmann
> On Jan 7, 5:07 am, Tonic...@yahoo.com wrote:
>> On Jan 7, 10:42 am, lwal...@lausd.net wrote:
>>> I don't find it idiotic. Just because Albrecht is a
>>> finitist working in a theory like ZF-Infinity
>>> (possibly +~Infinity), it doesn't mean that he is
>>> an idiot. Of course, I would not be surprised to
>>> find that many adherents of ZFC consider most
>>> finitists to be idiots.
>> At least most "finitists" here in sci.math. And I write "finitists"
>> because finitists would be way too much of a honour to give them,
>> taking into consideration that the huge majority of them (well, in
>> fact all of them, as far as I can recall right now) are NOT
>> mathematicians.
>
> No true Scotsman would put sugar on his porridge?
> No true mathematician could ever be a finitist?
>
> Or maybe it would be more accurate to say, no true
> finitist would argue the way Albrecht argues?
How long will you keep misreading?
>
> If so, then where are these alleged "true" finitists
> whose arguments are more mathematical than Albrecht's?
Everywhere : look at "finitism"
>
>> Albrecht began his new year's nonsense with the following:
>> "Modern mathematics implies that the number of the natural numbers is
>> greater than every natural number. This opinion is indefensible since
>> the natural numbers count themselves."
>> From this very unique parraph one can know we have here a HUGE crank:
>> Modern mathematics don't imply anything of the like as he wrote, and
>> of course not that "the number of natural numbers" (what is that?) is
>> anything.
>
> McCullough and Newman have already explained what Albrecht
> most likely meant. Most likely, he meant:
Most likely, he meant nothing coherent
>
> ZFC ("Modern mathematics") implies that the cardinality of
> the set of natural numbers is greater than every natural number.
No, no, no. I) This is implied by any reasonable definition of
cardinality. The only way out of it is to assume N is not a set (well,
not the only way. As usual, you are so weak at real math that you never
realize that IST is another perfectly good possibility : the "class" of
standarde integers is not a set either !)
>
> ZFC |- An (neN -> (card(N) > n))
ZF (or in fact any szet theory where N is a set) entails this
>
>> What does it mean that "the natural numbers count themselves"? I
>> haven't yet seen a number 7 counting itself, or other natural number:
>> perhaps he meant something else, but what?
>
> According to McCullough, Albrecht is counting starting
> with 1 rather than 0, as is standardly done with the
> von Neumann naturals, so that he meant something like:
>
> card({1,2,3,4,5,6,7}) = 7
>
> So Albrecht's argument is that since the cardinality of
> a set which contains 1, and contains the predecessor of
> all its elements except 1, is the largest element of
> the set, the same must be true for N.
Yes. There is the place where he goes into pure crank realm, * and so
do you*!!!
And since N has
> no largest element, it therefore can't exist.
>
> Obviously this argument is invalid in ZF, but it's valid
> in ZF-Infinity+~Infinity.
No, it is nowher'e valid: it supposes what is true or all n is true for
N . *This* is what mkes all of us protest...
>
>> You see, lwalk? You love to defend the indefensible, at least in
>> mathematics.
>
> ZF-Infinity+~Infinity is indefensible? Finitism is indefensible?
With friends like you, they dont need ennemies.
>
>> This guy, as ALL other "finitists" I can recall right now
>> here in sci.math, don't really know what they're talking about, but
>> they make lots of noise...just as you do.
>
> The fact that _all_ the other finitists Tonio can recall here
> on sci.math act just Albrecht, to me, is evidence that this
> is how "true" finitists act. I have no more reason to believe
> the finitists who post on sci.math aren't representative of
> the "true" finitists, any more than I have reason to believe
> that the adherents of ZFC who post here aren't typical among
> adherents of ZFC.
Try reading books for once...
Denis Feldmann
MoeBlee
> I've noticed that my Google ranking is now down to
> one star. I'm now officially a "crank," and have
> passed the point of no return. Of course, I knew
> going in, when I first started posting to sci.math,
> that by defending theories other than ZFC, it would
> happen sooner or later that I'd be labelled "crank"
> and have a one-star rating, just like others who
> propose theories other than ZFC.
"Past the point of no return". It's good that you're not the least bit
melodramatic about your postings. Good too that you make use of the
Google poster "ranking" system. There should be at least one person
who does.
> I officially became a "crank" by trying to defend
> the finitist Albrecht. Ullrich explains:
>
> > ZF-Infinity+-Infinity is logical theory that has
> > certain well-defined axioms. In the OP there was
> > nothing about what follows from this axiom or
> > that axiom, the OP _stated_ that [blah blah]
> > was "indefensible".
Ullrich might or might not think you're a crank, but at least in what
you quoted, he didn't say so.
> Note that for the proposed ideas of many so-called
> cranks, such as AP, tommy1729, and others, we don't
> yet know whether there exists a rigorous theory
> that incorporates all of their ideas and desired
> properties of a theory. But for Albrecht, we know
> that there _is_ a theory, ZF-Infinity+~Infinity,
> that _does_ have all of the properties that he
> wants it to have.
How do you know that?
And if it is, then I guess that one of the things he doesn't care
about having is some form of analysis or calculus for the sciences.
> In particular, it's a theory in
> which we don't have infinite sets. I'd have thought
> that the established mathematicians would at least
> be more sympathetic to ZF-Infinity+~Infinity than to
> the theories of other "cranks," since the former,
> unlike the latter, is already known to be possible
> in a rigorous theory. But I guess I was wrong.
It's not a matter of "sympathy", or lack of it, for ZF-I+~I.
> Au contraire. Albrecht's argument did say something
> about ZF-Infinity+~Infinity -- since he used the
> concepts of cardinal number and natural number
> interchangeably (as Ullrich pointed out), which one
> _can_ validly do in ZF-Infinity+~Infinity!
Albrecht made NO MENTION WHATSOEVER of having particular axioms of ZF-I
+~I. In particular, Albrecht made no mention whatsoever about:
extensionality
union
pairing
power set
replacement
And too bad you didn't READ Albrecht's post. He said:
"The concept of an actual infinity as in set theories like ZFC is
indefensible."
You see, he claims that the existence of infinite sets is indefensible
in ***ZFC***, not in ZF-I+~1.
No one disputes that in ZF-I+~I we have that there are no infinite
sets
(Wait though, I seem to keep forgetting, does ZF-I+~I entail that
there are no infinite sets? If "I" stands for the ordinary axiom of
inifinity (the existence of a successor inductive set), then how do we
show in ZF-I that "there exist no successor inductive sets implies
there exist no infinite sets"? Did Aatu show that a while ago using
replacement? I forgot the argument.)
> Therefore, Albrecht's argument is a sound and valid
> proof in ZF-Infinity+~Infinity.
So what? He's not claiming JUST for ZF-I+~I, but for ***ZFC***.
Just as Z prove there are infinite sets, by way of an axiom, so also
does ZF-I+~I prove that there are no infinite sets, by way of an
axiom. THAT is not at issue with any reasonable person. But what
ALBRECHT argues is something more, and he is incorrect: His argument
does NOT show that ZFC is indefensible for proving the existence of
infinite sets.
MoeBlee
Michael Stemper
>I've noticed that many so-called "cranks" prefer to start
>counting from 1 rather than 0.
>
>Traditionally, in schools, one learns that {1,2,3,...} is
>the set of _natural_ numbers, while {0,1,2,3,...} is the
>set of _whole_ numbers. Thus zero is the only _whole_
>number that isn't a _natural_ number.
For a slightly different twist on this, my junior high algebra text
defines {1, 2, 3, ...} as the "Counting Numbers (C)", which neatly
dodges the entire issue of "what are the naturals?" (The Whole Numbers
were defined as you suggest.)
>But none of this explains why the so-called "cranks" are
>the ones who prefer to start from 1 rather than 0.
As an engineer, I almost hate to propose this, but Fortan v. C?
--
Michael F. Stemper
#include <Standard_Disclaimer>
The name of the story is "A Sound of Thunder".
It was written by Ray Bradbury. You're welcome.
Albrecht
mike3 schrieb:
> On Jan 6, 7:08�pm, Albrecht <albst...@gmx.de> wrote:
> > Modern mathematics implies that the number of the natural numbers is
> > greater than every natural number. This opinion is indefensible since
> > the natural numbers count themselves.
> >
> > On the other hand: No natural number is able to denote the number of
> > the natural numbers. In consequence there is no number of the natural
> > numbers and hence there is no cardinality of the set of the natural
> > numbers.
> >
>
> It just means that it is not a natural number. It doesn't mean it is
> not a "number"
> unless you define "number" to be only natural numbers.
> <snip>
No natural number is able to denote the number (or quantity) of the
natural numbers. And anything, what is like a number in any sense, and
is greater than any natural number is _too great_ to denote the
number (or quantity) of the natural numbers.
Best regards
Albrecht S. Storz
MoeBlee
> anything, what is like a number in any sense, and
> is greater than any natural number is _too great_ to denote the
> number (or quantity) of the natural numbers.
Of course, that follows from the Axiom of TooGreatedness. We all know
that.
MoeBlee
Albrecht
Joshua Cranmer schrieb:
> Albrecht wrote:
> > The number of natural numbers can't be greater than every natural
> > number since there are not more natural numbers than there are natural
> > numbers.
>
> But for each natural number, there is a larger natural number. This is
> indisputable fact; if you combine it with your claims, you get Zeno's
> Paradox in a different form. If you accept the idea of infinity, the
> paradox disappears.
>
Yes, for each natural number, there is a larger natural number. For
that reason there is no number or quantity which is larger than every
natural or any other number.
Infinity is paradox. And there is no method or "trick" to despose of
the paradox.
MoeBlee
> Yes, for each natural number, there is a larger natural number. For
> that reason there is no number or quantity which is larger than every
> natural or any other number.
There is no NATURAL number that is greater than all natural numbers.
But it's only your dogma that dictates that there can't be a notion of
number, such as that of cardinal number, where there is a cardinal
number that is greater than all natural numbers. Purely dogma on your
part to preclude such a sense of number.
MoeBlee
William Hughes
> No natural number is able to denote the number (or quantity) of the
> natural numbers.
You spend a lot of time proving this very simple fact.
Try just stating it and only providing a proof if someone asks
for one (unlikely).
> And anything, what is like a number in any sense, and
> is greater than any natural number is _too great_ to denote the
> number (or quantity) of the natural numbers.
On the other hand you spend no time trying to prove this.
Perhaps because it is false (e.g. equivalence class under
bijection, cardinality, is certainly like a number, and
the cardinality of the natural numbers is not too great
to denote the quantity of the natural numbers).
- William Hughes
Albrecht
William Hughes schrieb:
> On Jan 6, 9:08 pm, Albrecht <albst...@gmx.de> wrote:
>
> > Any natural number is the number of its predecessors, inclusive the
> > considered number self.
>
>
> Correct, however the number of *all* natural numbers, say w, is not a
> natural
> number. w is not the number of its predecessors inclusive w.
>
>
> <snip>
>
> > The number of natural numbers can't be greater than every natural
> > number since there are not more natural numbers than there are natural
> > numbers.
>
> This is only true if "the number of natural numbers" has to be
> a natural number. Note that "the number of natural numbers"
> is not a natural number.
>
>
In German, we have the word "Nummer" and the word "Anzahl" which both
could be translated into the english word "number". Perhaps the word
"Anzahl" should better be translated with the word "quantity". I don't
know. Maybe the phrase "discrete quantity" is still better.
I think, discrete quantity should be equal to cardinality.
So I try to rephrase:
The discrete quantity of the natural numbers can't be greater than
every natural
number since there are not more natural numbers than there are natural
numbers.
The unary notation of the natural numbers shows it directly:
O
OO
OOO
OOOO
...
Any line contains a natural number, a cardinal number, a discrete
quantity (in unary notation). Now, you are claiming that the O's in
the first column (OOO...) are no natural number. Okay? What is that in
the first column? Do you accept that it should be a discrete quantity?
Or what?
I'm interested in your opinion.
Albrecht
george schrieb:
> > On Jan 6, 9:08 pm, Albrecht <albst...@gmx.de> wrote:
> > > Any natural number is the number of its predecessors, inclusive the
> > > considered number self.
>
> This is not even grammatical.
> People in general are going to have to get used to the fact that
> 0 IS a natural number. That means that every natural number is
> the number of smaller natural numbers. Since 0 is first, there are
> no smaller natural numbers, so that winds up defining 0 as the
> cardinality
> of the empty set.
>
> >
>
> > > The number of natural numbers can't be greater than every natural
> > > number since there are not more natural numbers than there are natural
> > > numbers.
>
> This is just bullshit. EVERY natural number is greater than
> than every one of the numbers in the set that IT "numbers",
> IF you do it right: 7={0,1,2,3,4,5,6} and is greater than all of them.
I'm sure that the expression "natural number" isn't well defined in
modern math. The advent of set theory had led to a disregard of some
other aspects in math, it seems to me. Modern mathematicians are free
to include or exclude the zero in/from the natural numbers.
Interesting, since in almost all other aspects math has to be exact
and definite.
I claim that the natural numbers are well definable and that the zero
is no natural number. It's so because the natural numbers should be
representable in any sufficiently strong number system. The unary
system is a sufficiently strong number system. In unary system you
can't have zero. So, zero is no natural number:
O
OO
OOO
...
Gus Gassmann
> In German, we have the word "Nummer" and the word "Anzahl" which both
> could be translated into the english word "number". Perhaps the word
> "Anzahl" should better be translated with the word "quantity". I don't
> know. Maybe the phrase "discrete quantity" is still better.
> I think, discrete quantity should be equal to cardinality.
So why don't you use "cardinality" right away?
> So I try to rephrase:
> The discrete quantity of the natural numbers can't be greater than
> every natural
> number since there are not more natural numbers than there are natural
> numbers.
Or better: "The cardinality of the natural numbers can't be greater
than every natural number since there are not more natural numbers
than there are natural numbers." (A. Storz)
What does the first half of your sentence have to do with the second?
The "since" is supposed to start an argument supporting your assertion
in the first part, but for the life of me, I can't see it.
Jesse F. Hughes
> Yes, for each natural number, there is a larger natural number. For
> that reason there is no number or quantity which is larger than every
> natural or any other number.
You'll have to spell out how
For all natural numbers n, there is a natural number m such that
m > n. (1)
entails
For all quantities x, it is not the case that for all natural
numbers n, x > n.
Let me try my hand at this line of reasoning:
Yes, for every rational number less than 1, there is a larger
rational number less than 1. For that reason, there is no rational
number which is larger than every rational number less than 1.
Which property of natural numbers are you appealing to? It can't just
be property (1), evidently.
--
Jesse F. Hughes
"That's the base tautological space where by tautological space I mean
a region of truth." -- James S. Harris does philosophy of mathematics.
JSH is a renaissance man.
Virgil
<9b7c0a84-1803-4f4e...@i20g2000prf.googlegroups.com>,
Albrecht <albs...@gmx.de> wrote:
> No natural number is able to denote the number (or quantity) of the
> natural numbers. And anything, what is like a number in any sense, and
> is greater than any natural number is _too great_ to denote the
> number (or quantity) of the natural numbers.
Albrecht's proselytizing for his faith seems singularly ineffective
among those who are less limited than he.
David R Tribble
> Modern mathematics implies that the number of the natural numbers is
> greater than every natural number. This opinion is indefensible since
> the natural numbers count themselves.
>
> On the other hand: No natural number is able to denote the number of
> the natural numbers. In consequence there is no number of the natural
> numbers and hence there is no cardinality of the set of the natural
> numbers.
Oh, brother, here we go again. I see I've missed any
attempt of nipping this in the bud by at least 80+ posts.
What a dismal way to start out the new year...
Virgil
<0202cfc0-b284-419b...@w1g2000prm.googlegroups.com>,
Albrecht <albs...@gmx.de> wrote:
> Joshua Cranmer schrieb:
> > Albrecht wrote:
> > > The number of natural numbers can't be greater than every natural
> > > number since there are not more natural numbers than there are natural
> > > numbers.
> >
> > But for each natural number, there is a larger natural number. This is
> > indisputable fact; if you combine it with your claims, you get Zeno's
> > Paradox in a different form. If you accept the idea of infinity, the
> > paradox disappears.
> >
>
> Yes, for each natural number, there is a larger natural number. For
> that reason there is no number or quantity which is larger than every
> natural or any other number.
Non sequitur! While one can properly conclude that there is no natural
larger than every natural, that does not justify that there is no
non-natural larger than every natural any more than that there is no
non-negative rational less than every unit fraction.
>
> Infinity is paradox.
But infiniteness (of a set) is not.
David R Tribble
> Albrecht, as a finitist, is working in a theory such as
> ZF-Infinity (and possibly with ~Infinity, the negation
> of Infinity, as well).
[Sorry, late to the party. Disregard if this has already been
mentioned.]
You're giving Albrecht way too much credit, trying to read
his intent into what he wrote. What he actually wrote is
that there is no number that represents the number of
naturals. Which means that he is "working in a theory"
that rejects non-finite cardinal numbers.
It's not the fact that he rejects infinity (which he probably
does) so much as that he accepts only a ridiculously limited
concept of "number".
Virgil
<82277eb6-756a-4d70...@r37g2000prr.googlegroups.com>,
Albrecht <albs...@gmx.de> wrote:
You have not the power to enforce your claim. Natural numbers, as well
as any other mathematics objects, are what their definers say they are,
no more and no less.
That allows you to exclude actual infinities from your own mathematical
world, but prohibits you from doing it in anyone else's. At least
without their permission.> proselytizing
> It's so because the natural numbers should be
> representable in any sufficiently strong number system. The unary
> system is a sufficiently strong number system. In unary system you
> can't have zero.
In a unary system, there is an empty string, since one can always delete
the last character from any string.
Otherwise, "OOOO" could be one or two or three or four strings, since
without a beginning or ending marker required, who knows where one
begins or ends?
David R Tribble
logical sense now (my changes in brackets):
Albrecht wrote [with corrections]:
> Modern mathematics implies that the [cardinality] of the natural numbers
> is greater than every natural number.
> No natural number is able to denote the number of the natural numbers.
> In consequence there is no [natural] number of the natural numbers
> [that exceeds all naturals] and hence there is no cardinality of the set of
> the natural numbers [that is a natural number].
FTFY.
William Hughes
Yes, but all these discrete quantities are the discrete quantities
of natural numbers, say "natural discrete quantities".
There are other discrete quantities that
are not "natural discrete quantities".
> Now, you are claiming that the O's in
> the first column (OOO...) are no natural number. Okay? What is that in
> the first column? Do you accept that it should be a discrete quantity?
Yes. However, note that there are
discrete quantities that are not "natural discrete quantities".
> Or what?
>
> I'm interested in your opinion.
>
All you have done is replace "number" with "discrete quantity". True
there is no "natural discrete quantity" that is the discrete quantity
of all natural numbers. So what? The discrete quantity of all
natural
numbers is not a "natural discrete quantity".
- William Hughes
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
> Virgil wrote:
>
> > In article <df010$4964aaa9$82a1e228$15...@news1.tudelft.nl>,
> > Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
> >
> >>Isn't mathematics: what mathematicians do ?
> >
> > Most mathematician eat and breath and do other things that are generally
> > not regarded as being mathematics.
>
> Most engineers eat and breath and do other things that are generally not
> not regarded as being engineering. Engineering is not what engineers do.
> And art is not what artists do. Sure.
>
> Han de Bruijn
There is a significant and critical difference between
"mathematicians do mathematics",
and " mathematics is what mathematicians do".
David R Tribble
>> You've never summarized the "axioms of math" in your book
>> "Complex made Simple"
>
David C. Ullrich wrote:
> No.
Han de Bruijn wrote:
>> and _yet_ it _is_ mathematics all over the place. Right ?
>
David C. Ullrich wrote:
> Yes. Based on a set of axioms. The _standard_ set, which is why
> there's no need to summarize it explicitly.
Wait, you mean you didn't bother to start with a whole
chapter devoted to explaining the obvious standard set
of axioms and definitions you used in the rest of your
book? You just assumed that the reader of your book
would know these things?
You're getting lazy. I'll bet you didn't even tell the reader
that your book was a mathematics book.
-drt
Albrecht
MoeBlee schrieb:
It is no dogma but pure logic that in the sequence
O
O
OO
O
O
OOO
O
O
O
OOOO
O
O
O
O
OOOOO
...
you never get an infinite row of O's (OOO...) neither horizontally nor
vertically. At the other hand, any arbitrary part of
O
OO
OOO
OOOO
OOOOO
...
which starts from the first O (from the 1) contains an unary number
which denotes the number (or discrete quantity, or cardinal number) of
this part. Not more and not less. The whole (which starts from 1), if
there would be a whole, cannot override this logic truth. If there
would be a whole, it must contain a number which denotes the discrete
quantity of this whole.
And: A cardinal number, which is larger than any natural number (and
cardinal number, too), is too large to denote the number (discrete
quantity) of the natural numbers since there are not more natural
numbers as there are natural numbers. Else, in the sequence
O
O
OO
O
O
OOO
O
O
O
OOOO
O
O
O
O
OOOOO
...
there would be an element which has more O's in it's vertical row than
in it's horizontal row.
By definition, the natural numbers consist only of natural numbers,
and so, the natural numbers are more than any single of them denotes
but can't be more than all of them denote. That's the paradox of
infinity.
george
wrote:
> I think that Albrecht is making the following, truly ridiculous
> point:
I think we are in full agreement about that, but MY point is,
it really does MATTER where you start. You need to come around
AND START there (where *I* start).
> Instead of considering the sets {}, {0}, {0,1}, {0,1,2}, he wants
> to consider the sets {1}, {1,2}, {1,2,3}. Each of these sets has the
> property that its cardinality is equal to its largest element.
Clearly. Glad we agree.
So he wants to generalize THAT to the infinite case.
*I*, on the other hand, want to generalize the result you get
BY STARTING WITH ZERO to the infinite case.
HE has the greater problem because HE (and WM) are the ones
asserting that generalizing from infinitely many finite cases to the
one infinite case IS A VALID INFERENCE! Therefore, THEY HAVE
TO ACCEPT *MY* generalization as well as their own! But the two
are in conflict! So THEIR proposed mode of inference LEADS TO
CONTRADICTION. Yet for some odd reason, they would rather
say that OURS does!
> The set { 1, 2, 3, ... } of *all* positive naturals has no largest element.
But all its elements are naturals, so again, the restriction "largest
natural element" STILL APPLIES.
> So if the pattern "cardinality = largest element" holds, then this
> set has no cardinality.
Again, no NATURAL cardinality.
THAT actually WOULD be valid.
> Why Albrecht thinks that the pattern should
> hold for infinite sets, I have no idea.
Oh, of course you do. Lots of things DO hold for infinite sets.
And WM's point was that the infinite set IS MADE OUT OF finite
parts, so basically, he wants to know, if 1 liter of water is wet,
and adding another leaves 2 liters wet, and adding a third leaves
3 liters wet, how is adding infinitely many supposed to leave you
DRY?!?
This would be a reasonable question if it weren't for the fact that
continually
adding more finite liters NEVER *EVER* GETS YOU *ANY CLOSER* to
infinity;
there are STILL INFINITELY many natural numbers SEPARATING you from
infinity, NO MATTER HOW HIGH you go. In other words, WM et al simply
fail to understand that what happens with ITERATED BINARY union or
addition
SIMPLY HAS NO RELEVANCE AT ALL to the infinite case.
Albrecht
Gus Gassmann schrieb:
The natural numbers count themselves. How could they be more than they
count?
David R Tribble
> I've noticed that my Google ranking is now down to
> one star. I'm now officially a "crank," and have
> passed the point of no return.
You're not a crank, you're a crank defender.
Of course, this is a lot like saying you're not a thief,
you're a lawyer who defends thieves.
> Of course, I knew
> going in, when I first started posting to sci.math,
> that by defending theories other than ZFC, it would
> happen sooner or later that I'd be labelled "crank"
> and have a one-star rating, just like others who
> propose theories other than ZFC.
Plenty of non-cranks have posted stuff about theories other
than ZFC. You don't appear to have accepted that fact,
though.
You keep missing the telling distinction:
Cranks reject modern math. Cantor's theorems, infinity,
set theory, axiomatic systems, whatever, there is always
something in standard math that they think is broken.
It would be fine if they were proposing new alternative
theories to ZFC. But they're not, they're rejecting ZFC as
wrong and evil.
-drt
Jesse F. Hughes
> LWalker wrote:
>> I've noticed that my Google ranking is now down to
>> one star. I'm now officially a "crank," and have
>> passed the point of no return.
>
> You're not a crank, you're a crank defender.
>
> Of course, this is a lot like saying you're not a thief,
> you're a lawyer who defends thieves.
Surely, it is good to have lawyers who defend thieves.
But it seems to me that you are implying otherwise.
--
Jesse F. Hughes
"All information is subject to change without notice."
-- California Alternative High School
Toni...@yahoo.com
> george schrieb:
>
>
>
>
>
> > > On Jan 6, 9:08 pm, Albrecht <albst...@gmx.de> wrote:
> > > > Any natural number is the number of its predecessors, inclusive the
> > > > considered number self.
>
> > This is not even grammatical.
> > People in general are going to have to get used to the fact that
> > 0 IS a natural number. That means that every natural number is
> > the number of smaller natural numbers. Since 0 is first, there are
> > no smaller natural numbers, so that winds up defining 0 as the
> > cardinality
> > of the empty set.
>
> > > > The number of natural numbers can't be greater than every natural
> > > > number since there are not more natural numbers than there are natural
> > > > numbers.
>
> > This is just bullshit. EVERY natural number is greater than
> > than every one of the numbers in the set that IT "numbers",
> > IF you do it right: 7={0,1,2,3,4,5,6} and is greater than all of them.
>
> I'm sure that the expression "natural number" isn't well defined in
> modern math. The advent of set theory had led to a disregard of some
> other aspects in math, it seems to me. Modern mathematicians are free
> to include or exclude the zero in/from the natural numbers.
> Interesting, since in almost all other aspects math has to be exact
> and definite.
>
And it is: some authors consider 0 a natural, and some others don't,
and most mathematicians don't really give much of a damn about this,
since it doesn't really matter for most applications, as long as some
given author specifies clearly what is his definition.
Nothing loose or "disregarding": it just is not that important.
***************************************************************
> I claim that the natural numbers are well definable and that the zero
> is no natural number. It's so because the natural numbers should be
> representable in any sufficiently strong number system.
*************************************************************
Why? Because you, or you guru WM, say so? And what do you call
"sufficiently strong number system" to? What does it involve? On what
grounds is it based? And what does it mean for you that the natural
numbers are representable, and why do you think this is not so in ZF,
say?
Tonio
************************************************************
The unary
> system is a sufficiently strong number system. In unary system you
> can't have zero. So, zero is no natural number:
>
> O
> OO
> OOO
> ...
>
> Best regards
> Albrecht S. Storz- Hide quoted text -
>
> - Show quoted text -