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Set Theory: Should You Believe

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Rupert

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Jun 27, 2006, 6:31:04 PM6/27/06
to
Norman Wildberger, an Associate Professor at my university (the
University of New South Wales), has written a discussion of the
foundations of mathematics called "Set Theory: Should You Believe"
which is current available on his website at
http://web.maths.unsw.edu.au/~norman/

I am sure he would appreciate any feedback. He can be reached at
n.wild...@unsw.edu.au

Peter Webb

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Jun 27, 2006, 9:56:30 PM6/27/06
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"Rupert" <rupertm...@yahoo.com> wrote in message
news:1151447464.7...@u72g2000cwu.googlegroups.com...

The more I read about this guy, the more I think he is a crank that somehow
scored an Associate Professorship.

His "rational geometry" appears to be based only defing a rational angle as
the sine of an "normal" angle. This makes some calculations easier (because
it eliminates a "sine" term) but makes common operations - like adding two
angles - far more complex. It adds absolutely zero mathematically - his
underlying structure is still Euclidean geometry - and as a practical
technique for surveying, physics, building etc it is way more cumbersome
than normal trig.

I couldn't find the paper you mention, but the closest was on multi-sets.
His summary states "For more than a century, mathematicians have been
hypnotized by the allure of set theory. Unfortunately, the theory has at
least two crucial failings. First of all, infinite set theory doesn't make
proper logical sense. Secondly, the fundamental data structures in
mathematics ought to be the same ones that are the most important in
computer science, science and ordinary life". Pure crank territory.

The four new structures he proposes to replace set theory are presented
without axioms, and indeed just use set theory machinery to prove his
"theorems". The whole thing could be formalised quite easily, in much the
same way as n-tuples can be defined in terms of sets (indeed, n-tuples are
one of the new "basic structures" he proposes). Maybe he didn't do this
because it would destroy his basic premise that these four "new" structures
(known for a 100 years) are more fundamental than set theory.

Strip away his title, and the guy looks like a "b-grade" internet crank to
me. My advice is to avoid his classes entirely.

Peter Webb


david petry

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Jun 27, 2006, 11:34:49 PM6/27/06
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Peter Webb wrote:
> "Rupert" <rupertm...@yahoo.com> wrote in message
> news:1151447464.7...@u72g2000cwu.googlegroups.com...
> > Norman Wildberger, an Associate Professor at my university (the
> > University of New South Wales), has written a discussion of the
> > foundations of mathematics called "Set Theory: Should You Believe"
> > which is current available on his website at
> > http://web.maths.unsw.edu.au/~norman/
> >
> > I am sure he would appreciate any feedback. He can be reached at
> > n.wild...@unsw.edu.au

> I couldn't find the paper you mention,

http://web.maths.unsw.edu.au/~norman/views2.htm

Also, this guy lists R. Srinivasan, who is frequent contributer to this
newsgroup, as one of his co-authors.

Kevin Karn

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Jun 28, 2006, 12:54:26 AM6/28/06
to

Excellent paper, Rupert. Thanks for posting it. Loved this bit:

***********
In perpetuating these notions, modern mathematics takes on many of the
aspects of a religion. It has its essential creed---namely Set Theory,
and its unquestioned assumptions, namely that mathematics is based on
`Axioms', in particular the Zermelo-Fraenkel `Axioms of Set Theory'. It
has its anointed priesthood, the logicians, who specialize in studying
the foundations of mathematics, a supposedly deep and difficult subject
that requires years of devotion to master. Other mathematicians learn
to invoke the official mantras when questioned by outsiders, but have
only a hazy view about how the elementary aspects of the subject hang
together logically.

Training of the young is like that in secret societies---immersion in
the cult involves intensive undergraduate memorization of the standard
thoughts before they are properly understood, so that comprehension
often follows belief instead of the other (more healthy) way around. A
long and often painful graduate school apprenticeship keeps the cadet
busy jumping through the many required hoops, discourages critical
thought about the foundations of the subject, but then gradually yields
to the gentle acceptance and support of the brotherhood. The
ever-present demons of inadequacy, failure and banishment are however
never far from view, ensuring that most stay on the well-trodden path.

The large international conferences let the fellowship gather together
and congratulate themselves on the uniformity and sanity of their world
view, though to the rare outsider that sneaks into such events the
proceedings no doubt seem characterized by jargon, mutual
incomprehensibility and irrelevance to the outside world. The official
doctrine is that all views and opinions are valued if they contain
truth, and that ultimately only elegance and utility decide what gets
studied. The reality is less ennobling---the usual hierarchical
structures reward allegiance, conformity and technical mastery of the
doctrines, elevate the interests of the powerful, and discourage
dissent.
***********

Couldn't have said it better myself. "Higher" set theory is a cult.
Infinity is their God.
"God exists" is Axiom #6.

Probably the best way to short-circuit this nonsense is to go after the
funding. Go straight for the trough the pigs are feeding in.

R. Srinivasan

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Jun 28, 2006, 1:47:45 AM6/28/06
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I didn't co-author any paper with Prof. Wildberger (there are many
people with the name "Srinivasan", so could be somebody else). But
where is this reference to "Srinivasan" as a co-author? I didn't find
it anywhere in his website.

Regards, RS

R. Srinivasan

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Jun 28, 2006, 2:04:07 AM6/28/06
to

I found it, on his "Research" page, at
<http://web.maths.unsw.edu.au/~norman/research.htm>

The "R. Srinivasan" he lists there as a co-author is not me. I do
recollect a Prof. by that name in Bangalore (maybe from Indian
Institute of Science, Bangalore, maybe not). I plan to go through
Wildberger's objections to set theory and see if it has any connection
with my own views.

Regards, RS

Rupert

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Jun 28, 2006, 2:37:33 AM6/28/06
to

Peter Webb wrote:
> "Rupert" <rupertm...@yahoo.com> wrote in message
> news:1151447464.7...@u72g2000cwu.googlegroups.com...
> > Norman Wildberger, an Associate Professor at my university (the
> > University of New South Wales), has written a discussion of the
> > foundations of mathematics called "Set Theory: Should You Believe"
> > which is current available on his website at
> > http://web.maths.unsw.edu.au/~norman/
> >
> > I am sure he would appreciate any feedback. He can be reached at
> > n.wild...@unsw.edu.au
> >
>
> The more I read about this guy, the more I think he is a crank that somehow
> scored an Associate Professorship.
>
> His "rational geometry" appears to be based only defing a rational angle as
> the sine of an "normal" angle.

Sine squared, actually.

> This makes some calculations easier (because
> it eliminates a "sine" term) but makes common operations - like adding two
> angles - far more complex. It adds absolutely zero mathematically - his
> underlying structure is still Euclidean geometry - and as a practical
> technique for surveying, physics, building etc it is way more cumbersome
> than normal trig.
>

It gives a new perspective on the subject, giving it a more algebraic
flavour. All his laws work over arbitrary fields of characteristic not
equal to two. In surveying etc. it enables you to use exact values
rather than approximations.

Rupert

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Jun 28, 2006, 2:38:30 AM6/28/06
to

My apologies, I should have been more specific. The paper is at

http://web.maths.unsw.edu.au/~norman/views.htm

Peter Webb

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Jun 28, 2006, 3:12:19 AM6/28/06
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"Rupert" <rupertm...@yahoo.com> wrote in message
news:1151476710.7...@d56g2000cwd.googlegroups.com...

This guy is a professor? Of mathematics? At a University?

God help you.

abo

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Jun 28, 2006, 5:12:43 AM6/28/06
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Rupert wrote:

>
> My apologies, I should have been more specific. The paper is at
>
> http://web.maths.unsw.edu.au/~norman/views.htm

I think you can formalize his vision of the natural numbers in the
following way. Consider second-order PA without the Successor Axiom.
Define a "second-order natural number" (SONN) to be a function (that
is, a second-order relationship satisfying the normal conditions) from
some initial segment of the naturals to {0,1}, that is the binary
strings. These are the natural numbers which can be "written down".
One can define addition, multiplication, etc for the SONN. But one
cannot prove that unique prime factorization holds.

abo

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Jun 28, 2006, 6:48:18 AM6/28/06
to

Sorry, that was too simple. You will be able to prove unique prime
factorization. So I can't really add anything useful.

Daryl McCullough

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Jun 28, 2006, 6:58:09 AM6/28/06
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Rupert says...

>Norman Wildberger, an Associate Professor at my university (the
>University of New South Wales), has written a discussion of the
>foundations of mathematics called "Set Theory: Should You Believe"
>which is current available on his website at
>http://web.maths.unsw.edu.au/~norman/

Frankly, I think his paper is crap. He sounds like a less loony
version of David Petry. Most of his objections to set theory are
without any content whatsoever, and seem either uninformed or
willfully dishonest. For example, he writes, about the axioms
of ZFC:

However even to a mathematician it should be obvious that these
statements are awash with difficulties. What is a property?
What is a parameter? What is a function? What is a family of sets?

These comments make it seem that the axioms of set theory are
dependent on a number of concepts, "property", "parameter",
"function", etc. that are left undefined or vaguely defined in set
theory. That's completely false. Those terms don't *appear* in the axioms
of set theory, they only appear in his particular *paraphrase* of
those axioms. They were used under the assumption that people
already *have* a notion of what those statements mean.

Another silly and content-free comment: He writes about the axiom of
infinity:

6. Axiom of infinity: There exists an infinite set.

...

One might as well declare that: There is an all-seeing Leprechaun!
or There is an unstoppable mouse!

What a ridiculous comment. First of all, the axiom of infinity
doesn't actually say that there exists an infinite set. It says
that there exists a set containing all natural numbers. It's called
the axiom of infinity because earlier definitions of "infinite set"
(for example, by Dedekind) could be seen to imply that the set of
naturals was an infinite set.

Wildberger's talk about modern mathematics as religion, with priests
and secret knowledge, etc. is just stupid. It's a remark that a
smart-aleck teenager would make, but has no serious point.

I certainly agree with Professor Wildberger that the axioms of ZFC
are not intuitively obvious to beginners, but that really wasn't
their point---they weren't invented as teaching tools, but as a way
of investigating foundations.

Maybe Wildberger is right, that a different foundation other
than set theory might be more intuitive and useful for students.
If he has such an alternative in mind, great. But this particular
paper trashing set theory makes no contribution along those lines.
The paper is complete crap. (In my opinion)

--
Daryl McCullough
Ithaca, NY

Kevin Karn

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Jun 28, 2006, 9:29:09 AM6/28/06
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Daryl McCullough wrote:

> Another silly and content-free comment: He writes about the axiom of
> infinity:
>
> 6. Axiom of infinity: There exists an infinite set.
>
> ...
>
> One might as well declare that: There is an all-seeing Leprechaun!
> or There is an unstoppable mouse!
>
> What a ridiculous comment. First of all, the axiom of infinity
> doesn't actually say that there exists an infinite set. It says
> that there exists a set containing all natural numbers. It's called
> the axiom of infinity because earlier definitions of "infinite set"
> (for example, by Dedekind) could be seen to imply that the set of
> naturals was an infinite set.

You're a mush-mouth, and you're not fooling anybody. Why don't you
start over again and say that in predicate calculus so we can check the
logic.

> Wildberger's talk about modern mathematics as religion, with priests
> and secret knowledge, etc. is just stupid. It's a remark that a
> smart-aleck teenager would make, but has no serious point.

Infinity is a synonym for God, and always has been. At least Spinoza
and Aquinas had the integrity to try to prove God's existence. You
fakers just assume it. Personally, when somebody comes to my door
selling the idea that there is an "infinite being" which "exists" and
is "greater than anything I can imagine" and that the existence of this
being is "self-evident", I take a pass on the brain-washing.

Frederick Williams

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Jun 28, 2006, 10:03:32 AM6/28/06
to
Daryl McCullough wrote:
>
> ...

>
> Wildberger's talk about modern mathematics as religion, with priests
> and secret knowledge, etc.

Is JSH one of his students?

--
Remove "antispam" and ".invalid" for e-mail address.

Daryl McCullough

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Jun 28, 2006, 10:09:09 AM6/28/06
to
Kevin Karn says...

>Daryl McCullough wrote:

>> What a ridiculous comment. First of all, the axiom of infinity
>> doesn't actually say that there exists an infinite set. It says
>> that there exists a set containing all natural numbers. It's called
>> the axiom of infinity because earlier definitions of "infinite set"
>> (for example, by Dedekind) could be seen to imply that the set of
>> naturals was an infinite set.
>
>You're a mush-mouth, and you're not fooling anybody. Why don't you
>start over again and say that in predicate calculus so we can check the
>logic.

Okay:

Definition 1: A set S is Dedekind-infinite if there exists a
function f from S to S such that (1) f is one-to-one, but
(2) f is not a surjection.

Do you want definitions of "function", "one-to-one", "surjection",
or can I assume that you know some mathematics already?

Claim: If we let S = the natural numbers, and let f be defined
by f(x) = x+1, then (1) f is one-to-one, and (2) f is not a
surjection (because 0 is not in the image of f).

Do you want a proof that f(x) = x+1 is one-to-one, or that 0
is not in the image of f?

Conclusion: The set of natural numbers is Dedekind infinite.

>Infinity is a synonym for God, and always has been.

Well, it isn't in set theory.

I hope you enjoyed this little exchange, because I will not
respond again to your posts.

Peter Niessen

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Jun 28, 2006, 4:35:18 PM6/28/06
to

Bullshit like Mückenheim.
--
Mit freundlichen Grüssen
Peter Nießen

Rupert

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Jun 28, 2006, 7:15:12 PM6/28/06
to

In the presence of the other axioms, it is equivalent to the
proposition "There exists an infinite set." In some presentations of
the axioms (the one at the start of Jech's textbook "Set Theory", for
example), the axiom is given as "There exists an infinite set." My
guess is that Wildberger read one of these presentations of the axioms.

I think that what he needs to do is acquaint himself with the work that
has already been done on alternative foundations and try and situate
his views within the context of that work, explaining how they are
similar and how they differ.

Rupert

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Jun 28, 2006, 7:16:47 PM6/28/06
to

You may not like his philosophical work very much. To judge whether he
deserves his post as an Associate Professor, I think you should look to
his mathematical publications.

MoeBlee

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Jun 28, 2006, 8:09:42 PM6/28/06
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Daryl McCullough wrote:
> I certainly agree with Professor Wildberger that the axioms of ZFC
> are not intuitively obvious to beginners,

What level of beginner? I had no interest in mathematics before I
learned basic first order predicate logic, and then I discovered that
set theory is a related subject. After a reasonable amount of thinking
about and working with the set theory axioms, I found that they are
easy to understand. For me, it's the opposite of what Wildberger
claims. It's mathematics without set theory (or some formal
axiomatization of a particular theory) that is hard for me to grasp.
Set theory and formal axiomatiztion, on the other hand, provide me with
an entrance into mathematics, and an understanding of mathematics both
in its details and in the bigger picture, that I can't imagine having,
let alone even being interested in, without mathematical logic and set
theory.

That paper by Wildberger just reads like a rant about foundations by
someone who just seems not to be very interested in foundations.
Whether he os genuinely interested in foundations, I can't say; but
that paper sure doesn't reflect such an interest, even as a critique of
standard foundations.

MoeBlee

david petry

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Jun 28, 2006, 8:45:21 PM6/28/06
to

R. Srinivasan wrote:
> R. Srinivasan wrote:
> > david petry wrote:
> > > Peter Webb wrote:
> > > > "Rupert" <rupertm...@yahoo.com> wrote in message
> > > > news:1151447464.7...@u72g2000cwu.googlegroups.com...
> > > > > Norman Wildberger, an Associate Professor at my university (the
> > > > > University of New South Wales), has written a discussion of the
> > > > > foundations of mathematics called "Set Theory: Should You Believe"
> > > > > which is current available on his website at
> > > > > http://web.maths.unsw.edu.au/~norman/
> > > > >
> > > > > I am sure he would appreciate any feedback. He can be reached at
> > > > > n.wild...@unsw.edu.au
> > >
> > > > I couldn't find the paper you mention,
> > >
> > > http://web.maths.unsw.edu.au/~norman/views2.htm
> > >
> > > Also, this guy lists R. Srinivasan, who is frequent contributer to this
> > > newsgroup, as one of his co-authors.
> >
> > I didn't co-author any paper with Prof. Wildberger (there are many
> > people with the name "Srinivasan", so could be somebody else). But
> > where is this reference to "Srinivasan" as a co-author? I didn't find
> > it anywhere in his website.
>
> I found it, on his "Research" page, at
> <http://web.maths.unsw.edu.au/~norman/research.htm>
>
> The "R. Srinivasan" he lists there as a co-author is not me.

Then I owe you an apology. I was guessing it was probably you, since
there seem to be similarities in interests, besides having the same
name. I really should be more careful.

Peter Webb

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Jun 28, 2006, 9:49:23 PM6/28/06
to

"Rupert" <rupertm...@yahoo.com> wrote in message
news:1151536512.7...@b68g2000cwa.googlegroups.com...


Yes, but he says mathemetics is flawed because ZFC requires an axiom that
says "there is an infinite set" without defining infinity. ZFC has no such
axiom. It has an axiom that says There exists a set S such that the null set
is an element of S, and for all x an element of S, x union {x} is and
element of x. It is called the axiom of infinity because under the von
Neuman construction of numbers, this states that there is a set of Natural
Numbers. (pretty reasonable I would have thought). Saying that we haven't
defined "infinity" is like saying we haven't defined "Extensionality" or
"Unordered Pair" or "Foundation" or "Regularity" or any of the other words
that are used to NAME the axioms. These are just names that are invented for
the axioms, not the axioms themselves.


> In some presentations of
> the axioms (the one at the start of Jech's textbook "Set Theory", for
> example), the axiom is given as "There exists an infinite set." My
> guess is that Wildberger read one of these presentations of the axioms.
>

If the guy is an Associate Professor of Mathematics, publishing papers on
the foundations of mathematics and the failure of the axiomatic system, I
would have hoped that he knew the axioms of Set Theory, not just the names
of the axioms.


> I think that what he needs to do is acquaint himself with the work that
> has already been done on alternative foundations and try and situate
> his views within the context of that work, explaining how they are
> similar and how they differ.
>

Yeah, learning some basic mathematics would certainly be a good idea.
Perhaps you should explain to your professor that he should do this before
he publishes any more papers on the subject. (Common advice to cranks in
this forum). As I said before, until he does this, I would avoid his
classes.

Ralf Bader

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Jun 28, 2006, 10:27:14 PM6/28/06
to
Peter Niessen wrote:

It definitely has a very Mückenheimian flavour, but at least its author
recognized that there /is/ a largest natural number in a Mückenheimian
setting - something that Mückenheim (of course) didn't grasp when I
tried to explain it to him, using essentially the same argument as in
Wildberger's paper.

I think that some of Wildberger's observations are to the point. Set
theory really isn't the essence of mathematics, although most, maybe
all, of mathematics can be reconstructed within set theory.
Understanding this requires familiarity with "real" mathematics, not
just foundations of mathematics.

However, other parts of Wildberger's paper are, plainly spoken, just
dull. What he is saying about category theory reveals just one thing:
Besides a handful of basic vocabulary, Wildberger knows nothing about
the subject. His explanation "Why real numbers are a joke" is a joke,
as are his remarks about mathematics and computer science. There has
been substantial work -in fact, this is one of the great mathematical
achievements of the 20th century- to clarify the notion of
computability. Restricting mathematics to the computable realm, for
example to computable real numbers as Wildberger suggests, because
anything non-computable is kind of intractable, makes it impossible to
characterize and clarify the notion of computability, because there is
no longer any "background" one could view computability in contrast to.
Computability would become a notion written in white letters on a white
sheet of paper (or screen).

There are lots of other points. For example, Wildberger mixes up logical
and didactical contexts. Constructing the rational numbers as quotient
field of the ring of integers is not intended as a way to explain to
10-year-olds what a fraction is, and the explanations of fractions
accessible to 10-year-olds are not suitable if one wants to do hard
number theory.


Ralf

Kevin Karn

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Jun 28, 2006, 10:55:07 PM6/28/06
to
Daryl McCullough wrote:
> Kevin Karn says...
>
> >Daryl McCullough wrote:
>
> >> What a ridiculous comment. First of all, the axiom of infinity
> >> doesn't actually say that there exists an infinite set. It says
> >> that there exists a set containing all natural numbers. It's called
> >> the axiom of infinity because earlier definitions of "infinite set"
> >> (for example, by Dedekind) could be seen to imply that the set of
> >> naturals was an infinite set.
> >
> >You're a mush-mouth, and you're not fooling anybody. Why don't you
> >start over again and say that in predicate calculus so we can check the
> >logic.
>
> Okay:
>
> Definition 1: A set S is Dedekind-infinite if there exists a
> function f from S to S such that (1) f is one-to-one, but
> (2) f is not a surjection.
>
> Do you want definitions of "function", "one-to-one", "surjection",
> or can I assume that you know some mathematics already?
>
> Claim: If we let S = the natural numbers, and let f be defined
> by f(x) = x+1, then (1) f is one-to-one, and (2) f is not a
> surjection (because 0 is not in the image of f).
>
> Do you want a proof that f(x) = x+1 is one-to-one, or that 0
> is not in the image of f?
>
> Conclusion: The set of natural numbers is Dedekind infinite.

Sorry, that's not predicate calculus. That's English. And you left out
the good bits, i.e. the predicate calculus versions of:

"That's ridiculous." and


"First of all, the axiom of infinity doesn't actually say that there
exists an infinite set."

In fact, while we're at it, I'd like to see the predicate calculus
formalizations of the following as well:

"Bullshit like Mückenheim."
"The guy looks like a "b-grade" internet crank to me."

Yup, the foundation of math is logic. The foundation of logic is the
axioms. And the foundation of the axioms is insults.

> >Infinity is a synonym for God, and always has been.
>
> Well, it isn't in set theory.
>
> I hope you enjoyed this little exchange, because I will not
> respond again to your posts.

Wow, I'm truly devastated. Sniff sniff
Just so you know, I'll be tattooing your malarkey with snide remarks
whenever the feeling moves me. :-)

Rupert

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Jun 29, 2006, 12:48:14 AM6/29/06
to

I did actually take one of his courses once, back in my undergrad days.
It was in combinatorics. It was a good course.

Patrick

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Jun 29, 2006, 7:28:03 AM6/29/06
to

Do you doubt that f is one-to-one or not onto?

Given that you are in such strong agreement with
Wildberger's paper why should you or anyone care to
translate mathematical statements into the language
of the anointed priesthood?


William of Ockham

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Jun 29, 2006, 3:56:15 PM6/29/06
to

Naturally I agree 100% with the sentiments of the paper. But the
further you go on, the flimsier the arguments get. In particular, he
seems to give the axiom of infinity as "there exists an infinite set"
and nothing more. I say "seems" because both my browsers had a problem
with his page. All the standard versions of the axiom, even the bad
ones, have more than this. Did anyone else have the same difficulty?
Zermelo's original 1908 axiom says ""There exists in the domain at
least one set Z that contains the null set as an element and is so
constituted that to each of its elements a there corresponds a further
element of the form {a}, in other words, that with each of its elements
a it also contains the corresponding set {a} as element" i.e. it does
not claim there is an "infinite" set, only that there is a set with the
aforesaid properties. The Von Neumann is obviously different, but,
again, is explicity. Where is the author getting this version of the
axiom from? If it is not my browser - a lot of the formulae were not
coming out.

On the objection "what is a property" historically Zermelo's original
formulation had a flaw that was spotted by Russell. Grattan Guiness
has a good history of this.

As I say I am 100% with the man, but it is one thing to agree with a
conclusion, another thing to agree whether the conclusion follows from
the stated assumptions.

I didn't understand the objections above to his remarks about
computabilty. What exactly is the problem with what he says? Not that
he says much.

William of Ockham

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Jun 29, 2006, 3:58:10 PM6/29/06
to

William of Ockham wrote:
> On the objection "what is a property" historically Zermelo's original
> formulation had a flaw that was spotted by Russell. Grattan Guiness
> has a good history of this.

I should have added: the flaw was corrected.

Kevin Karn

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Jul 1, 2006, 10:19:36 AM7/1/06
to

The theory of computability begins with a false premise, i.e. that
there is such a thing as an infinite tape, or (equivalently) a tape to
which you can always add one more cell. There is no such machine, and
no such tape. In short, Turing Machines (as defined) do not exist.

So computability theory is basically an achievement in fiction writing,
different in no essential way from a Harry Potter novel. Assuming the
existence of an infinite tape is no different from assuming that people
can levitate, or that pigs can talk, or that you can feed a million
people with one fish. It may be entertaining to explore the
consequences of such false assumptions, but those consequences are not
true (except vacuously), and they are not useful. In short, it's a form
of idle speculation which deserves to be ridiculed, not admired. No
different really from great achievements of the 13th century, like
Aquinas' elegantly reasoned demonstration that we won't have to crap in
the afterlife.

The reality is that even computable functions are not computable.
Computability is not determined by a definition on a piece of paper.
It's determined by the ability to mobilize physical resources in the
real world.

Patricia Shanahan

unread,
Jul 1, 2006, 11:10:05 AM7/1/06
to
Kevin Karn wrote:
...

> The theory of computability begins with a false premise, i.e. that
> there is such a thing as an infinite tape, or (equivalently) a tape to
> which you can always add one more cell. There is no such machine, and
> no such tape. In short, Turing Machines (as defined) do not exist.
>
> So computability theory is basically an achievement in fiction writing,
> different in no essential way from a Harry Potter novel. Assuming the
> existence of an infinite tape is no different from assuming that people
> can levitate, or that pigs can talk, or that you can feed a million
> people with one fish. It may be entertaining to explore the
> consequences of such false assumptions, but those consequences are not
> true (except vacuously), and they are not useful. In short, it's a form
> of idle speculation which deserves to be ridiculed, not admired. No
> different really from great achievements of the 13th century, like
> Aquinas' elegantly reasoned demonstration that we won't have to crap in
> the afterlife.
>
> The reality is that even computable functions are not computable.
> Computability is not determined by a definition on a piece of paper.
> It's determined by the ability to mobilize physical resources in the
> real world.

Yes, infinite memory computers cannot really be built. Every real
computer is a finite state automaton.

However, for many purposes infinite memory models are a more useful
abstraction for reasoning about real computers than finite state
automata would be. Most real computers have far too many states for it
to be practical to make any use of the finiteness of the set of states.

Bounded memory, with the associated risk of running out of memory, is a
nasty complication. It is often better to split a problem into pieces,
and deal with the pieces separately. Turing machines, and other infinite
memory models, let us put the finite memory complications on one side,
and deal with them separately from basic algorithm analysis.

Knowing, for example, that the halting problem is not Turing-decidable
is very useful data. It means that any practical solution would have to
be based in some way on the boundedness of the number of states of a
real computer.

Practical applications of mathematics involve abstractions that ignore
some aspects of the real world. The key to applying mathematics well is
knowing which aspects of the real world are important for a particular
piece of analysis, and so must be reflected in the mathematical models,
and which should be ignored or handled separately.

Patricia

Peter Webb

unread,
Jul 1, 2006, 12:49:16 PM7/1/06
to
The theory of computability begins with a false premise, i.e. that
there is such a thing as an infinite tape, or (equivalently) a tape to
which you can always add one more cell. There is no such machine, and
no such tape. In short, Turing Machines (as defined) do not exist.

*** Not true in one essential regard. All problems that can be solved -
produce a valid output - do so after a finite number of steps, and hence
only use a finite amount of tape. The problems you can't solve are those
that require an infinite number of steps to compute - ie they never halt.
Infinite tape is only needed for problems which can't be solved, and that is
basically why they can't be. You can't write 1/3 in base 10, because a
Turing Machine which pumped out 0.333... would require an infintely long
tape for the answer. Therefore it is impossible. No Turing machine that
solves a problem could actually ever use an infintely long tape, because it
would never halt. The infintely long tape is only needed for things you
can't do.

So computability theory is basically an achievement in fiction writing,
different in no essential way from a Harry Potter novel. Assuming the
existence of an infinite tape is no different from assuming that people
can levitate, or that pigs can talk, or that you can feed a million
people with one fish. It may be entertaining to explore the
consequences of such false assumptions, but those consequences are not
true (except vacuously), and they are not useful. In short, it's a form
of idle speculation which deserves to be ridiculed, not admired. No
different really from great achievements of the 13th century, like
Aquinas' elegantly reasoned demonstration that we won't have to crap in
the afterlife.

*** That's simply not true, for very important practical and economic
reasons. Having "bugs" in computer programs so that they do strange things
(like go into infinite loops) is a huge economic problem. Computer languages
are Turing Machines; the theory of computability tells us exactly how far we
can in automated program checking, and how to structure computer languages
and systems to minimise this cost.


The reality is that even computable functions are not computable.
Computability is not determined by a definition on a piece of paper.
It's determined by the ability to mobilize physical resources in the
real world.

*** Well, yes if you are interested only in the real world.


david petry

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Jul 1, 2006, 3:33:49 PM7/1/06
to

Ralf Bader wrote:

> Restricting mathematics to the computable realm, for
> example to computable real numbers as Wildberger suggests, because
> anything non-computable is kind of intractable, makes it impossible to
> characterize and clarify the notion of computability, because there is
> no longer any "background" one could view computability in contrast to.
> Computability would become a notion written in white letters on a white
> sheet of paper (or screen).

Wouldn't that argument also suggest that scientists shouldn't restrict
themselves to reality (the observable universe) ? You seem to be
suggesting that science needs a "background" of fantasy to make a
contrast with reality in order to clarify the notion of reality.

.

Ralf Bader

unread,
Jul 1, 2006, 4:08:00 PM7/1/06
to
Kevin Karn wrote:

> Ralf Bader wrote:

>> However, other parts of Wildberger's paper are, plainly spoken, just
>> dull. What he is saying about category theory reveals just one thing:
>> Besides a handful of basic vocabulary, Wildberger knows nothing about
>> the subject. His explanation "Why real numbers are a joke" is a joke,
>> as are his remarks about mathematics and computer science. There has
>> been substantial work -in fact, this is one of the great mathematical
>> achievements of the 20th century- to clarify the notion of
>> computability. Restricting mathematics to the computable realm, for
>> example to computable real numbers as Wildberger suggests, because
>> anything non-computable is kind of intractable, makes it impossible to
>> characterize and clarify the notion of computability, because there is
>> no longer any "background" one could view computability in contrast to.
>> Computability would become a notion written in white letters on a white
>> sheet of paper (or screen).
>
> The theory of computability begins with a false premise, i.e. that
> there is such a thing as an infinite tape, or (equivalently) a tape to
> which you can always add one more cell. There is no such machine, and
> no such tape. In short, Turing Machines (as defined) do not exist.

Do you think you are telling me anything new? Or that those who developed
that theory didn't know that infinite machines can't be actually built? The
sense and purpose of that theory is not what you assume (whatever that may
be), and this in such an obvious way that I can only wonder about your
state of mind.


Ralf

ken.q...@excite.com

unread,
Jul 1, 2006, 4:43:27 PM7/1/06
to
Peter Webb wrote:
> The theory of computability begins with a false premise, i.e. that
> there is such a thing as an infinite tape, or (equivalently) a tape to
> which you can always add one more cell. There is no such machine, and
> no such tape. In short, Turing Machines (as defined) do not exist.
>
> *** Not true in one essential regard. All problems that can be solved -
> produce a valid output - do so after a finite number of steps, and hence
> only use a finite amount of tape. The problems you can't solve are those
> that require an infinite number of steps to compute - ie they never halt.
> Infinite tape is only needed for problems which can't be solved, and that is
> basically why they can't be. You can't write 1/3 in base 10, because a
> Turing Machine which pumped out 0.333... would require an infintely long
> tape for the answer. Therefore it is impossible. No Turing machine that
> solves a problem could actually ever use an infintely long tape, because it
> would never halt. The infintely long tape is only needed for things you
> can't do.
>

I would think an infinite tape is required because, for
any tape with finite length L, there is going to be a solvable problem
that requires a tape with length > L.

So it doesn't seem to me to be [only] the unsolvable problems that
require an infinite tape.

I suppose one way around this is, if you have a machine with tape
length L and the problem barfs because the tape isn't long enuf,
you keep adding to the tape until its long enuf.

Thanks.

Ken

Kevin Karn

unread,
Jul 3, 2006, 4:30:51 AM7/3/06
to

I would argue that the Turing-undecidability of the halting problem is
a completely useless fact with regard to real-world (physical)
computers. My reasons:

1) All physical computers are finite automata (FA), and the halting
problem is decidable for FAs. We don't need to define Turing machines
to know that fact. This completely settles the issue for all physical
computers, and we don't need to concern ourselves with non-physical
computers because there are no such things. Non-physical computers are
fictions which can't actually compute anything.

2) The halting problem for physical computers is easily settled: All
programs halt. There is no such thing as a computer which loops
forever, because a computer which did so would violate physical law.
It would either be a perpetual motion machine, or consume an infinite
amount of energy.

3) The real-world solution to loops is simply to terminate the program
with the operating system (or a reboot) and rewrite the program to
eliminate the loop. I don't believe the people who do this have any
need to know or rely on the theory of Turing machines/infinite tapes.

Maybe you have a good counterargument, or an example where the theory
of Turing machines is genuinely indispensable to the solution of some
practical, real-world problem. I'd love to hear it if you do. I'm very
interested in more rigorous approaches to the question: "What exact
practical purpose does mathematical construct X serve?"

Patricia Shanahan

unread,
Jul 3, 2006, 9:08:53 AM7/3/06
to

True, but the known solutions to the FA halting problem involve analysis
of each state.

For a general purpose computer the number of states is 8^n, where n is
the computer's total storage in bytes. For this purpose, the disk drives
dominate. If you have 100 GB of disk space the disk has more than
8^10^11 distinct states.

In practice, no brute force method is remotely practical. Any method
that is practical for the number of states our desktop FAs have is also
likely to work for unbounded memory.

> 2) The halting problem for physical computers is easily settled: All
> programs halt. There is no such thing as a computer which loops
> forever, because a computer which did so would violate physical law.
> It would either be a perpetual motion machine, or consume an infinite
> amount of energy.
>
> 3) The real-world solution to loops is simply to terminate the program
> with the operating system (or a reboot) and rewrite the program to
> eliminate the loop. I don't believe the people who do this have any
> need to know or rely on the theory of Turing machines/infinite tapes.

Agreed. But you are forgetting what happens after a program has been
killed for failure to finish within a reasonable time. I've often had to
debug when that is the only symptom. The first stage is to try to find
out whether it is a performance problem or a functional problem. Would
the program have finished, but was taking too long? Or did it get into a
perpetual loop?

Absent the proof of the halting problem I might have wasted time on
ideas for a program to resolve the issue. As it is, I KNOW that any
program that can answer that question, for a general program, must
somehow depend on the finiteness of its number of states. I don't have
any practical ideas that meet that condition.

Moreover, an absence of infinite loops would be a useful compile time
check, if we knew how to implement it.

Patricia Shanahan

unread,
Jul 3, 2006, 1:32:23 PM7/3/06
to
Kevin Karn wrote:
...

> Maybe you have a good counterargument, or an example where the theory
> of Turing machines is genuinely indispensable to the solution of some
> practical, real-world problem. I'd love to hear it if you do. I'm very
> interested in more rigorous approaches to the question: "What exact
> practical purpose does mathematical construct X serve?"

I've already commented on the halting problem issue.

Turing machines are also very important in some aspects of algorithm
performance analysis.

For example, it is useful to know if a problem is NP-complete or not. An
NP-complete problem has a polynomial time solution if, and only if, all
the NP-complete problems do.

The NP-complete problems have the general form "Is there a solution at
least this good to this problem", where there is a polynomial-time check
for the assertion that a string is a solution and is at least "this good".

At this point, nobody knows of a polynomial time solution to any
NP-complete problem.

Given a novel problem with the same general flavor as the NP-complete
problems one of the first steps is to try to prove it is NP-complete. If
it is, then barring a really major breakthrough, any polynomial-time
solution will be approximate, or probabilistic, or limited in some other
way.

Evaluating whether a problem is NP-complete or not gives some guidance
was what sort of approach to take.

Of course, the space demands for an algorithm are also important, but
can be considered separately.

What constitutes "practical, real-world" depends partly on your point of
view. From my point of view, it includes picking an algorithm to solve a
problem.

Patricia

george

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Jul 3, 2006, 2:13:06 PM7/3/06
to


> > Rupert says...
> >
> > >Norman Wildberger, an Associate Professor at my university

> > One might as well declare that: There is an all-seeing Leprechaun!
> > or There is an unstoppable mouse!

> Daryl McCullough wrote:
> > What a ridiculous comment. First of all, the axiom of infinity
> > doesn't actually say that there exists an infinite set.

Well, yes, it does, actually.
It says that there is a set containing all the naturals.
Since there are an infinite number of those, this set MUST be infinite.

Rupert wrote:
> In the presence of the other axioms, it is equivalent to the
> proposition "There exists an infinite set."

That is LESS clear since the other axioms do NOT include a
DEFINITION of "infinite". If one of the other axioms is Choice
then that will not matter (several intuitive definitions of
infinity are available, adequate, and, again, IN the presence of
Choice, EQUIVALENT(importantly)). But if it is not -- and one would
not expect it to be, if one is attacking the "basic" theory....

> In some presentations of
> the axioms (the one at the start of Jech's textbook "Set Theory", for
> example), the axiom is given as "There exists an infinite set." My
> guess is that Wildberger read one of these presentations of the axioms.

Hmmm. If one is going to pontificate about the relevance of the field
in its entirety, then one is obligated to seek a BROAD perspective
first. You have to read more than 1 treatment of the area, not just
assume that the one that YOUR undergrad instructor taught YOU is
globally shared.

> I think that what he needs to do is acquaint himself with the work that
> has already been done on alternative foundations and try and situate
> his views within the context of that work, explaining how they are
> similar and how they differ.

I want to respond in two different ways to this.
Way 1: WHY??? WHY are you asking him to perform
some sort of intellectual investigation here? IF he is going
to perform that then he needs to understand that he is performing
it as PENANCE! He has SINNED here! What he ACTUALLY needs
is to cash the relevant REALITY check, which reads, "EVEN as random
a collection of amateurs as sci.logic can tell that what YOU just wrote
is CRAP. It's EMBARRASSING. TO YOU." He needs to internalize that
this is what has happened. Once he understands that, he will not need
anyone's advice about how to react to it or cure it. If he never
understands
that, then NO path of continuation can get him anywhere relevant.

Way 2: He's perfectly entitled to hate set theory if he wants.
Sometimes you get a gestalt; you get an overall reaction to
the overall picture; you just KNOW that there is something
FUNDAMENTALLY WRONG with it. If he knows this then the
proper use of alternative foundations FOR HIM is not so much to
enable him to "situate" his work in comparison to it, as it is to
provide examples of how & why set theory gets it wrong, to provide
grist for his mill, to help him avoid re-inventing the wheel. But in
order
to do this properly, you need to understand what's ACTUALLY going on
in set theory, as opposed to thinking that your simplistic mistaken
gloss is already shared by everybody.

george

unread,
Jul 3, 2006, 2:20:06 PM7/3/06
to

Rupert wrote:
> I did actually take one of his courses once, back in my undergrad days.
> It was in combinatorics. It was a good course.

But this is telling.
Since you now have a Ph.D. yourself, this was
at least 8 years ago. And after all this time, he
is still Associate and NOT FULL Professor.
This does have implications.
But his Ph.D. is from Yale and he taught briefly at Stanford.
So I would say that the problem is not so much that he is
a crank as perhaps that he is "a practicing mathematician"
who has never needed to take the PHILOSOPHY seriously,
before. If he is (late in life) FINALLY coming around to what
the actually INTERESTING stuff is, then maybe we shouldn't
be so hard on him.

Rupert

unread,
Jul 3, 2006, 7:00:16 PM7/3/06
to

george wrote:
> Rupert wrote:
> > I did actually take one of his courses once, back in my undergrad days.
> > It was in combinatorics. It was a good course.
>
> But this is telling.
> Since you now have a Ph.D. yourself, this was
> at least 8 years ago.

I've almost handed in my Ph.D. thesis. I'm handing it in on July 24.

I think the combinatorics course was in my Honours year, in 2002.

I can't remember exactly when he joined us but I think it was fairly
early in my undergrad degree, which was quite a while ago.

> And after all this time, he
> is still Associate and NOT FULL Professor.
> This does have implications.
> But his Ph.D. is from Yale and he taught briefly at Stanford.
> So I would say that the problem is not so much that he is
> a crank as perhaps that he is "a practicing mathematician"
> who has never needed to take the PHILOSOPHY seriously,
> before. If he is (late in life) FINALLY coming around to what
> the actually INTERESTING stuff is, then maybe we shouldn't
> be so hard on him.

I agree. I think he is a good mathematician and he's just starting to
take an interest in foundations. I've been encouraging him to learn
more mathematical logic so he can situate his views with respect to
those of others.

Rupert

unread,
Jul 3, 2006, 7:05:55 PM7/3/06
to

george wrote:
> > > Rupert says...
> > >
> > > >Norman Wildberger, an Associate Professor at my university
> > > One might as well declare that: There is an all-seeing Leprechaun!
> > > or There is an unstoppable mouse!
>
> > Daryl McCullough wrote:
> > > What a ridiculous comment. First of all, the axiom of infinity
> > > doesn't actually say that there exists an infinite set.
>
> Well, yes, it does, actually.
> It says that there is a set containing all the naturals.
> Since there are an infinite number of those, this set MUST be infinite.
>
> Rupert wrote:
> > In the presence of the other axioms, it is equivalent to the
> > proposition "There exists an infinite set."
>
> That is LESS clear since the other axioms do NOT include a
> DEFINITION of "infinite". If one of the other axioms is Choice
> then that will not matter (several intuitive definitions of
> infinity are available, adequate, and, again, IN the presence of
> Choice, EQUIVALENT(importantly)). But if it is not -- and one would
> not expect it to be, if one is attacking the "basic" theory....
>

It's true that one needs choice to prove that a set is Dedekind
infinite if and only it is Tarski infinite, however one does not need
choice to prove that there exists a Dedekind infinite set if and only
if there exists a Tarski infinite set.

Norman's a good guy. He helped me out during my Honours year with
understanding isometries of hyperbolic space, and he takes the
postgrads out to dinner every year. Of course I'm not going to rubbish
his work.

I think, in general, it's a good thing when someone becomes interested
in foundations and finds out what a rich and interesting subject it is.
Norman seems to be developing an interest in foundations, so I'm
encouraging him to take it further.

george

unread,
Jul 3, 2006, 9:31:23 PM7/3/06
to

MoeBlee wrote:
> That paper by Wildberger just reads like a rant about foundations by
> someone who just seems not to be very interested in foundations.

He's new. Rupert knows him better than we do so I say
we let Rupert read him. Suppose you were strong and
co-ordinated enough to be a good swimmer, but had
been born in Kansas. Suppose you'd never been IN the
water, and then one day you got thrown in. For the first
few hours, it would NOT be obvious that you would (soon
enough) be a good swimmer.

> Whether he os genuinely interested in foundations, I can't say; but
> that paper sure doesn't reflect such an interest, even as a critique of
> standard foundations.

One of the reasons I decided to take Computer Science instead of
Math as an undergrad was that I was appalled at the extent to which
mathematicians were stumbling along without getting serious
about "data structures", as we did in Computer Science.
I think NW is *on* to something here. He just needs to be less
contemptuous of SOME of what has gone before.

albs...@gmx.de

unread,
Jul 4, 2006, 11:21:48 AM7/4/06
to

Daryl McCullough wrote:
> Rupert says...
>
> >Norman Wildberger, an Associate Professor at my university (the
> >University of New South Wales), has written a discussion of the
> >foundations of mathematics called "Set Theory: Should You Believe"
> >which is current available on his website at
> >http://web.maths.unsw.edu.au/~norman/
>
> Frankly, I think his paper is crap. He sounds like a less loony
> version of David Petry. Most of his objections to set theory are
> without any content whatsoever, and seem either uninformed or
> willfully dishonest. For example, he writes, about the axioms
> of ZFC:
>
> However even to a mathematician it should be obvious that these
> statements are awash with difficulties. What is a property?
> What is a parameter? What is a function? What is a family of sets?
>
> These comments make it seem that the axioms of set theory are
> dependent on a number of concepts, "property", "parameter",
> "function", etc. that are left undefined or vaguely defined in set
> theory. That's completely false. Those terms don't *appear* in the axioms
> of set theory, they only appear in his particular *paraphrase* of
> those axioms. They were used under the assumption that people
> already *have* a notion of what those statements mean.
>
> Another silly and content-free comment: He writes about the axiom of
> infinity:
>
> 6. Axiom of infinity: There exists an infinite set.
>
> ...

>
> One might as well declare that: There is an all-seeing Leprechaun!
> or There is an unstoppable mouse!
>
> What a ridiculous comment. First of all, the axiom of infinity
> doesn't actually say that there exists an infinite set. It says
> that there exists a set containing all natural numbers. It's called
> the axiom of infinity because earlier definitions of "infinite set"
> (for example, by Dedekind) could be seen to imply that the set of
> naturals was an infinite set.

That's total nonsense what you say here.
The axiom of infinity declares the existence of infinite sets since you
can't have them without declaring their existence.
And doing like this, you achieve a lot of unlogical consequences.
Your comment about this facts debunk you as total layman in this
concern.


Best regards
Albrecht S. Storz


>
> Wildberger's talk about modern mathematics as religion, with priests
> and secret knowledge, etc. is just stupid. It's a remark that a
> smart-aleck teenager would make, but has no serious point.


>
> I certainly agree with Professor Wildberger that the axioms of ZFC

> are not intuitively obvious to beginners, but that really wasn't
> their point---they weren't invented as teaching tools, but as a way
> of investigating foundations.
>
> Maybe Wildberger is right, that a different foundation other
> than set theory might be more intuitive and useful for students.
> If he has such an alternative in mind, great. But this particular
> paper trashing set theory makes no contribution along those lines.
> The paper is complete crap. (In my opinion)

george

unread,
Jul 4, 2006, 1:06:31 PM7/4/06
to

albs...@gmx.de wrote:
> That's total nonsense what you say here.
> The axiom of infinity declares the existence of infinite sets since you
> can't have them without declaring their existence.

Of course you can. It is a general feature of first-order models
in general that they CAN contain, that you CAN have, elements
that are never "declared" or mentioned in the language.
The canonical example is a non-standard model of Peano Arithmetic,
which has tons of infinite numbers that the language cannot refer to
(but that cause existential and universal generalizations to come up
with different truth values from the ones they would have if we
limited our attention to things "declared" to exist, i.e., things that
are named by terms in the language, which refer only to the natural
numbers of the standard model).

> And doing like this, you achieve a lot of unlogical consequences.

Here, you're simply being ridiculous; he IS NOT "doing like this";
he's just commenting on the perfectly standard axiomatization of
ZFC and its perfectly standard axiom of infinity. Which, just for the
record, DOES NOT include ANY definition of what "infinite" means.
Any logical consequences are going to be logical consequences of
the axioms of ZFC, AND NOT of "doing like this".

> Your comment about this facts debunk you as total layman in this
> concern.

At least he was commenting in English.
In your case, we'll attribute it to translation error.

Ralf Bader

unread,
Jul 4, 2006, 3:47:52 PM7/4/06
to
david petry wrote:

I think there is a lot of fantasy (aka hypotheses) in science.


Ralf

John Jones

unread,
Jul 4, 2006, 4:06:44 PM7/4/06
to
The fact that mathematicians are unable and unwilling to translate
their tasks into common discourse leads one to believe either that the
foundations of mathematics are suspect, or that 'foundations' in
mathematics are not originary, simple and conceivable acts and
properties but fix-its enmeshed in the body of the mathematical
integrated text.

Even non-mathematicians can glean enough from the cracks between the
extensive symbology of mathematics to recognise conceptual
inconsistency. Set theory is rich with inconsistency which the
mathematician, high up in the branches, fails to see. It is doubtful
whether the mathematician has the skills to deal with mathematics at
the very deepest foundational levels, such skills generally being the
province of the philosopher.

But the problem is not confined to mathematics: The integral texts of
mathematics (the vast integrated, linked structure built from
formalism, author and argument in which mathematics is stored as a body
of knowledge) can, when the discipline is in crisis, be mandated upon
argument and theory, stifling authentic debate and creating divides and
allegiances. The phenomena occurs in any scholastic text.

quote source if used

albs...@gmx.de

unread,
Jul 5, 2006, 9:38:08 AM7/5/06
to

george wrote:
> albs...@gmx.de wrote:
> > That's total nonsense what you say here.
> > The axiom of infinity declares the existence of infinite sets since you
> > can't have them without declaring their existence.
>
> Of course you can. It is a general feature of first-order models
> in general that they CAN contain, that you CAN have, elements
> that are never "declared" or mentioned in the language.

Okay. Than you also have unicorns, wolkenkuckucksheime, the wizard of
oz, kindergartens and rucksacks and all what you can imagine in ZFC
since you can't proove that they ar not in.
I'm right?
If yes: How should we deal with this objects?

> The canonical example is a non-standard model of Peano Arithmetic,
> which has tons of infinite numbers that the language cannot refer to
> (but that cause existential and universal generalizations to come up
> with different truth values from the ones they would have if we
> limited our attention to things "declared" to exist, i.e., things that
> are named by terms in the language, which refer only to the natural
> numbers of the standard model).
>
> > And doing like this, you achieve a lot of unlogical consequences.
>
> Here, you're simply being ridiculous; he IS NOT "doing like this";
> he's just commenting on the perfectly standard axiomatization of
> ZFC and its perfectly standard axiom of infinity. Which, just for the
> record, DOES NOT include ANY definition of what "infinite" means.
> Any logical consequences are going to be logical consequences of
> the axioms of ZFC, AND NOT of "doing like this".


The problem, I've tried to point on, is, that a few people think, the
set of the natural numbers exists in a platonic way and the Axiom of
Infinity just does state this fact. But it isn't like this.
We can say without problems:
- natural numbers exists
- there are infinite many natural numbers
- there are sets of natural numbers
but we can't conclude: the set of all natural numbers exists.

The assumption of infinite sets leads to logical contradictions.


>
> > Your comment about this facts debunk you as total layman in this
> > concern.
>
> At least he was commenting in English.
> In your case, we'll attribute it to translation error.

Nice. :-)

george

unread,
Jul 5, 2006, 9:59:17 AM7/5/06
to


> > >Daryl McCullough wrote:
> > >> What a ridiculous comment. First of all, the axiom of infinity
> > >> doesn't actually say that there exists an infinite set. It says
> > >> that there exists a set containing all natural numbers. It's called
> > >> the axiom of infinity because earlier definitions of "infinite set"
> > >> (for example, by Dedekind) could be seen to imply that the set of
> > >> naturals was an infinite set.

Kevin Karn wrote:
> > >You're a mush-mouth, and you're not fooling anybody.

You're being ridiculous.
He has taken more courses in this stuff than
you have. Locally he's one of the better teachers/explainers.
He knows AND YOU DON'T. The question is not whether he
is or isn't fooling anybody. The question IS, why are YOU making
a fool of YOURself in public? Don't you realize this stuff will
basically
exist in perpetuity? Do you really want all the world to know for
all time Just How Much of a Total Fucking Moron YOU were in 2006?

> Why don't you start over again and say
> that in predicate calculus so we can check the logic.

Who "we", white man?
Most of the people who have taken EVEN ONE INTRODUCTORY
course in set theory ALREADY KNOW what the logical translations
of everything he was saying were. When he actually provided you
with some logical definitions, your response was to say that
they were not logical. That does NOT prove that you know logic
better than he does. It DOES prove that you don't know logic
when you see it.

> > Okay:
> >
> > Definition 1: A set S is Dedekind-infinite if there exists a
> > function f from S to S such that (1) f is one-to-one, but
> > (2) f is not a surjection.
> >
> > Do you want definitions of "function", "one-to-one", "surjection",
> > or can I assume that you know some mathematics already?

No, Darryl, obviously, you canNOT assume that,
as is revealed by KK's reply, which was:

> Sorry, that's not predicate calculus. That's English.

Sorry. That's not predicate calculus.
It's not even witty repartee.
It's childish stupidity.

george

unread,
Jul 5, 2006, 10:13:40 AM7/5/06
to
Rupert wrote:
> I think he is a good mathematician and he's just starting to
> take an interest in foundations. I've been encouraging him to learn
> more mathematical logic so he can situate his views with respect to
> those of others.

I went back to his home page and could not find
"Set Theory: Should You Believe", but I *did* find
"A new look at multisets", with this abstract:

> For more than a century, mathematicians have
> been hypnotized by the allure of set theory.
> Unfortunately, the theory has at least two crucial failings.
> First of all, infinite set theory doesn't make proper logical sense.

This is sort of indisputably obviously true;
Skolem's paradox and dueling semanticses
for 2nd-order logic make this very abundantly clear.

> Secondly, the fundamental data structures in
> mathematics ought to be the same ones that
> are the most important in computer science,
> science and ordinary life---namely the multiset and the list.

And this, arguably, is even MORE obviously true.
EVERYbody uses lists (i.e. ordered n-tuples) a LOT
more than they use un-ordered sets. In writing the
stuff down, you HAVE to impose an order whether you like
it or not. And as for Computer Science, the pure unordered
set simply CANNOT be represented; you have to impose
an order first and then try to throw it away later.
The value of set theory is that lists and other things can
ALL be ENCODED as sets; you get a valuable sort of
homogeneity in a comprehensive neutral framework.
But it is not QUITE neutral enough in that {{a},{a,b}}
REALLY is UNnecessarily COMPLICATED compared to
something that knows that ordered things are primitive.
Finally, the fact that multisets are more useful than just
plain sets is, again, obvious.

This is basically little more than a plea for people to write
math in assembly language instead of brute 1s and 0s.
I don't see how it can be controversial. He also talks about
notational innovations. Again, it would be hard to see how
anybody could accuse him of making the situation worse.
But maybe he needs to read Kuhn. The scientific revolution
doesn't happen until the status quo becomes embarrassingly
intolerable as a result of new discoveries. You can't move
people to a new paradigm/notation MERELY because that
would be BETTER.

george

unread,
Jul 5, 2006, 10:29:33 AM7/5/06
to

Rupert wrote:
> I think, in general, it's a good thing when someone becomes interested
> in foundations and finds out what a rich and interesting subject it is.
> Norman seems to be developing an interest in foundations, so I'm
> encouraging him to take it further.

There is a foundations-of-math discussion list that
is archived at NYU. It is moderated and sort of hard
to get admitted to (it's got giants like Feferman and he
definitely does not want to go into it with the attitude that
he ALREADY knows more than these people). But it will
have some discussions that might help him see that people
who have been studying this for a long time share some of
his concerns. One thread where he could begin is


http://www.cs.nyu.edu/pipermail/fom/2005-October/thread.html#9141

Patricia Shanahan

unread,
Jul 5, 2006, 10:49:26 AM7/5/06
to
albs...@gmx.de wrote:
> george wrote:
>> albs...@gmx.de wrote:
>>> That's total nonsense what you say here.
>>> The axiom of infinity declares the existence of infinite sets since you
>>> can't have them without declaring their existence.
>> Of course you can. It is a general feature of first-order models
>> in general that they CAN contain, that you CAN have, elements
>> that are never "declared" or mentioned in the language.
>
> Okay. Than you also have unicorns, wolkenkuckucksheime, the wizard of
> oz, kindergartens and rucksacks and all what you can imagine in ZFC
> since you can't proove that they ar not in.
> I'm right?
> If yes: How should we deal with this objects?

ZFC doesn't have any axioms about unicorns or rucksacks, so you cannot
prove that a unicorn is equal to itself, or not equal to a rucksack.

To reason effectively about things other than sets it would be necessary
to add axioms relating to them, and it would no longer be ZFC.

It does have axioms of the form "for any set", so it is possible to
prove theorems about any set that is not prohibited by its axioms.

Patricia

albs...@gmx.de

unread,
Jul 5, 2006, 11:18:28 AM7/5/06
to

Are unicorns and rucksacks not sets? Can you know it and are you shure?
:-)

But serious:
I had discussed abut this issues a long time and as I had understanded,
the majority was sure about the fact that ZFC contains only this, what
ZFC declares to exist.

Does it make sense to think about objects in ZFC which aren't defined
in it?
At the end: what is the consequence? You don't have infinite sets
without the Axiom of Infinity.
If not so, why should ZFC has this axiom?

Best regartds
Albrecht S. Storz

Jack Campin - bogus address

unread,
Jul 5, 2006, 11:37:27 AM7/5/06
to
> I had discussed abut this issues a long time and as I had understanded,
> the majority was sure about the fact that ZFC contains only this, what
> ZFC declares to exist.
> Does it make sense to think about objects in ZFC which aren't defined
> in it?

You can define lots of things in the language of ZFC which it can't
prove the existence of. Large cardinals, for example - since these
have been the main area of research into set theory for the last 50
years they aren't exactly obscure.

Look at Rudy Rucker's "Infinity and the Mind" for a basic idea (his
writing style is horribly gee-whiz, but he does convey the point of
the project).

============== j-c ====== @ ====== purr . demon . co . uk ==============
Jack Campin: 11 Third St, Newtongrange EH22 4PU, Scotland | tel 0131 660 4760
<http://www.purr.demon.co.uk/jack/> for CD-ROMs and free | fax 0870 0554 975
stuff: Scottish music, food intolerance, & Mac logic fonts | mob 07800 739 557

R. Srinivasan

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Jul 5, 2006, 12:14:52 PM7/5/06
to

This FOM thread is about questioning Platonism as a foundation for
modern mathematics and set theory.

Here is a quote from the end of Pg.10 of Norman Wildberger's paper "Set
theory: should you believe?" at:

<http://web.maths.unsw.edu.au/~norman/views2.htm>

"But here is a very important point: we are not obliged, in modern
mathematics, to actually have a rule or algorithm that specifies the
sequence r1, r2, r3, · · · . In other words, 'arbitrary'
sequences are allowed, as long as they have the Cauchy convergence
property. This removes the obligation to specify concretely the objects
which you are talking about. Sequences generated by algorithms can be
specified by those algorithms, but what possibly could it mean to
discuss a 'sequence' which is not generated by such a finite rule?
Such an object would contain an 'infinite amount' of information,
and there are no concrete examples of such things in the known
universe. This is metaphysics masquerading as mathematics."

I fully agree with this quote which NW makes in the context of
questioning classical real analysis. I would suggest to NW to look up
my work on NAFL, in particular the paper available at
<http://arxiv.org/abs/math.LO/0506475>, if he would like to know how to
avoid Platonism in mathematics and theoretical science in general (in
particular physics, comp. science). "Arbitrary" infinite sequences are
not allowed in the presentation of real analysis in this paper; I claim
that *all* other existing versions of real analysis in any other
existing logic is subject to the same objections made by NW. I would
also suggest to NW not to waste time burying himself too deeply in the
abstruse stuff discussed in the FOM newsgroup; most of the people there
are already committed to this or that viewpoint and will not accept any
questioning of the status quo. In other words, these people are
anything but objective in analysing and questioning the sorry state of
the foundations of logic and mathematics as exists today. However I
will not deny that there are some interesting discussions on that list
from time to time. But if at all anyone successfully questions the
existing foundations and proposes an alternative, it will have to be an
amateur with no stakes in the status quo and with no obligation to
his/her colleagues to justify the status quo; the FOM list does not
meet this criterion.

Here is another quote from NW's paper (in the section "Does mathematics
require axioms?" on pg. 8,) with which I disagree violently:

"Mathematics does not require 'Axioms'. The job of a pure
mathematician is not to build some elaborate castle in the sky, and to
proclaim that it stands up on the strength of some arbitrarily chosen
assumptions. The job is to investigate the mathematical reality of the
world in which we live. For this, no assumptions are necessary. Careful
observation is necessary, clear definitions are necessary, and correct
use of language and logic are necessary. But at no point does one need
to start invoking the existence of objects or procedures that we cannot
see, specify, or implement."

This position seems to contradict his own objections to Platonism in
the previous quote; there is no such thing as the "mathematical reality
of the world in which we live". When one states "rules" which may be
finite, but which refer to infinitely many objects, one *has*
postulated the existence of infinitely many objects (a point which NW
does not seem to accept). No truths about infinitely many objects can
be taken as "self evident" or "definitions"; they are in fact
*postulates*, and you cannot establish these truths by "observation",
as NW seems to think. Peano Arithmetic is a bunch of postulates, it is
not "self-evident" that there must exist infintely many natural numbers
in "the world in which we live". Mathemaitcs is about producing proofs
and proofs do require axioms as starting points. To say that
mathematics does not require axioms is the same is saying that
mathematics does not require formal systems or even logic itself. But
that will quickly lead us into contradictions and just plain confusion
as well. The fact is that if he wants to reject Platonism and infinite
sets, most of the mathematics that he swears by will have to go down
the drain; they constitute infinitary reasoning. Welcome to the world
of NAFL, which I believe, could easily co-exist with the status quo and
at least lead to a completely new perspective. And maybe new results as
well (especially in theoretical physics, quantum computing, theoretical
comp. science).

Regards, RS

george

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Jul 5, 2006, 12:18:41 PM7/5/06
to

Rupert wrote:
> I think, in general, it's a good thing when someone becomes interested
> in foundations and finds out what a rich and interesting subject it is.
> Norman seems to be developing an interest in foundations, so I'm
> encouraging him to take it further.

Here is a message from the thread I mentioned earlier,
one that I think NW could profitably engage (even if it
is a year old by now):

http://www.cs.nyu.edu/pipermail/fom/2005-October/009130.html

It intriguingly mentions
an earlier article by Boolos (#8 in "Logic, Logic, and Logic"),
"Must We Believe in Set Theory?"

george

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Jul 5, 2006, 12:34:20 PM7/5/06
to

> george wrote:
> > It is a general feature of first-order models
> > in general that they CAN contain, that you CAN have, elements
> > that are never "declared" or mentioned in the language.

albs...@gmx.de wrote:
> Okay. Than you also have unicorns, wolkenkuckucksheime, the wizard of
> oz, kindergartens and rucksacks and all what you can imagine in ZFC
> since you can't proove that they ar not in.
> I'm right?

You can have them IN SOME MODEL of ZFC,
IF they exist TO BEGIN with. If you don't know whether
they exist or not, then you don't know whether they can
or cannot be in any model of ZFC.

> If yes: How should we deal with this objects?

The usual way is by adding extra axioms that mention them,
or that define them in terms of the language.
That is in fact exactly what the axiom of infinity does.
If you remove the axiom of infinity from ZFC, all the
models of ZFC that have infinite sets WILL CONTINUE
to exist and will continue to be models of the theory without
the axiom.

> The problem, I've tried to point on, is, that a few people think, the
> set of the natural numbers exists in a platonic way

Are you sure??
It cannot be proven to exist, not in first-order logic anyway.

> and the Axiom of Infinity just does state this fact.

NO, IT DOESN'T. JEEzus!
Will people PLEASE STOP making BASIC TECHNICAL errors!
What ZFC's axiom of infinity DOES say is that there exists
a set containing every natural number. But it does NOT say
that this set contains all AND ONLY the natural numbers.
THAT is something that CANNOT be said in first-order logic AT ALL!
Not by ZFC OR BY *ANY* recursive axiom-set!

> But it isn't like this.
> We can say without problems:
> - natural numbers exists

No, you CAN'T say THAT, EITHER, not in
in first-order logic anyway. You can say that 0 exists.
You can say that 1 exists. You can say that 2 exists.
For every natural number n, you can say that n exists.
But you CAN*NOT* say that "natural numbers" generically exist.

> - there are infinite many natural numbers

Obviously, since you cannot even say what a natural
number is in the first place (not until you ascend to
2nd-order logic), you canNOT say that you have infinitely
many of them. Moreover, until AFTER you have the
axiom of choice, you cannot even say what "infinite" means
(there are at least 5 *different* definitions).

> - there are sets of natural numbers

Finite ones, yes. But any set that includes
infinitely many natural numbers may, also, for all you know,
include things bigger than any natural number. FOL is not
expressively powerful enough to exclude supernaturals.

> but we can't conclude: the set of all natural numbers exists.

Right.


> The assumption of infinite sets leads to logical contradictions.

Wrong.
Obviously, IF it did, that would be PROVABLE, so the burden of proof
IS ON you. We are NOT holding our breath.
As I said before, there are 5 different definitions of infinity, and
there are 5 corresponding THEOREMS of ZFC, to the effect that
the-set-required-to-exist-by-the-axiom-of-infinity satisfies each
of those definitions. I don't know for sure but I strongly suspect
that even after you remove choice, you will still get 5 different
theorems showing how the set satisfies the 5 different definitions.
You do NOT get any LOGICAL contradictions.
What you do get is philosophical qualms that have more
to do with the nature of first-order logic&languages (BOTH of which
in fact PREsuppose the existence of the set of natural numbers
in a very DEEP and strong way) than with set theory.

george

unread,
Jul 5, 2006, 12:45:55 PM7/5/06
to

> > One thread where he could begin is
> >
> >
> > http://www.cs.nyu.edu/pipermail/fom/2005-October/thread.html#9141


R. Srinivasan wrote:
> This FOM thread is about questioning Platonism as a foundation for
> modern mathematics and set theory.

Please quit the ignorant paraphrases.
The issue is not what the thread in general is about; there is
a lot going on at one time in October-2005 on that list.
The issue is, many messages in the thread are relevant to NW's issues.


> Here is a quote from the end of Pg.10 of Norman Wildberger's paper "Set
> theory: should you believe?" at:
>
> <http://web.maths.unsw.edu.au/~norman/views2.htm>
>
> "But here is a very important point: we are not obliged, in modern
> mathematics, to actually have a rule or algorithm that specifies the
> sequence r1, r2, r3, · · · . In other words, 'arbitrary'
> sequences are allowed, as long as they have the Cauchy convergence
> property. This removes the obligation to specify concretely the objects
> which you are talking about.

This is an incredibly ignorant comment.
IF the sequence has the Cauchy convergence property, THEN THERE MUST
exist an algorithmically-specifiable sequence that will get THE SAME
job done.
There must exist an algorithmic approximation, and the approximation
can be as close as you like. So this is just a stupid objection.

> Sequences generated by algorithms can be
> specified by those algorithms, but what possibly could it mean to
> discuss a 'sequence' which is not generated by such a finite rule?

It could mean that you don't CARE, as OPPOSED to that you don't know,
what the missing bits are.

> Such an object would contain an 'infinite amount' of information,

Not necessarily; again, see "don't-know vs. don't-care
non-determinism".
NW has not borrowed as much of the Computer Science perspective as
he would like to think he has.

> and there are no concrete examples of such things in the known
> universe.

This is just idiotic bullshit. All of space exists and all of it is
continuous, as far as he knows. If all elements of some infinite
set exist, then all subclasses of those elements, INCLUDING THE ONES
THAT ARE NOT RECURSIVE, exist too.
He might as well have claimed that there are no concrete examples of
infinity
in the known universe, that its mass, its number of electrons,
neutrinos,
quarks, and photons, are ALL finite. Appeal to the "concrete"
universe
in this context is just BULLSHIT. Maybe he IS a crank after all.

> This is metaphysics masquerading as mathematics.

Every discipline, INCLUDING math, NEEDS a take on metaphysics.

> Here is another quote from NW's paper (in the section "Does mathematics
> require axioms?" on pg. 8,) with which I disagree violently:
>
> "Mathematics does not require 'Axioms'. The job of a pure
> mathematician is not to build some elaborate castle in the sky, and to
> proclaim that it stands up on the strength of some arbitrarily chosen
> assumptions. The job is to investigate the mathematical reality of the
> world in which we live.

OK, he's a crank.

> For this, no assumptions are necessary.

RUPERT: YOUR PROF IS A FUCKING IDIOT!

I know he has a Ph.D. from Yale, but seriously, he
simply doesn't know what he's talking about here.
I am not an applied psychologist so I do not know how
to cause the relevant evolution in his perspective.
But he really would've been much better off sticking to
improving the usual choice of foundational data structures.
What he CALLS "the mathematical reality of the world in
which we live" IS A THING BASED on axioms and assumptions.
You do not GET to have ANY mathematical ANYthings until
AFTER you make some assumptions. If he thinks that
there was a mathematical world BEFORE there were these
assumptions then he was simply living Socrates' unexamined life.

her...@cox.net

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Jul 5, 2006, 3:17:08 PM7/5/06
to


Infinity is a tough nut to crack.

--
hz

John Jones

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Jul 5, 2006, 4:15:43 PM7/5/06
to

Number is defined by a limit, infinity has no limit and cannot be a
number.

david petry

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Jul 5, 2006, 6:06:19 PM7/5/06
to

george wrote:

> Sometimes you get a gestalt; you get an overall reaction to
> the overall picture; you just KNOW that there is something

> FUNDAMENTALLY WRONG with it. [...] But in order


> to do this properly, you need to understand what's ACTUALLY going on
> in set theory, as opposed to thinking that your simplistic mistaken
> gloss is already shared by everybody.

That's not really true. What Wildberger is saying, and what so many
Usenet "cranks" have been saying, is that there is a reality underlying
mathematics, which is the world of computation. Almost anybody (except
maybe pure mathematicians and logicians and philosophers) can see that
set theory goes beyond that reality and creates a fantasy world.
Wildberger has no obligation to understand the details of set theory to
earn the right to be a critic of set theory.

george

unread,
Jul 5, 2006, 7:33:42 PM7/5/06
to

david petry wrote:
> george wrote:
>
> > Sometimes you get a gestalt; you get an overall reaction to
> > the overall picture; you just KNOW that there is something
> > FUNDAMENTALLY WRONG with it. [...] But in order
> > to do this properly, you need to understand what's ACTUALLY going on
> > in set theory, as opposed to thinking that your simplistic mistaken
> > gloss is already shared by everybody.
>
> That's not really true.

Of course it is.
I have read enough of him to see that he is getting a gestalt.

> What Wildberger is saying, and what so many
> Usenet "cranks" have been saying, is that there is a reality underlying
> mathematics, which is the world of computation.

Conputation is clearly and certainly real, completely irrespective
of whether it does or doesn't underlie anything.

> Almost anybody (except
> maybe pure mathematicians and logicians and philosophers) can see that
> set theory goes beyond that reality and creates a fantasy world.

You are a crank.
NW is maybe not.

> Wildberger has no obligation to understand the details of set theory to
> earn the right to be a critic of set theory.

Now you're just being an idiot.

Ross A. Finlayson

unread,
Jul 5, 2006, 10:28:04 PM7/5/06
to

William of Ockham wrote:

> Rupert wrote:
> > Norman Wildberger, an Associate Professor at my university (the
> > University of New South Wales), has written a discussion of the
> > foundations of mathematics called "Set Theory: Should You Believe"
> > which is current available on his website at
> > http://web.maths.unsw.edu.au/~norman/
> >
> > I am sure he would appreciate any feedback. He can be reached at
> > n.wild...@unsw.edu.au
>
> Naturally I agree 100% with the sentiments of the paper. But the
> further you go on, the flimsier the arguments get. In particular, he
> seems to give the axiom of infinity as "there exists an infinite set"
> and nothing more. I say "seems" because both my browsers had a problem
> with his page. All the standard versions of the axiom, even the bad
> ones, have more than this. Did anyone else have the same difficulty?
> Zermelo's original 1908 axiom says ""There exists in the domain at
> least one set Z that contains the null set as an element and is so
> constituted that to each of its elements a there corresponds a further
> element of the form {a}, in other words, that with each of its elements
> a it also contains the corresponding set {a} as element" i.e. it does
> not claim there is an "infinite" set, only that there is a set with the
> aforesaid properties. The Von Neumann is obviously different, but,
> again, is explicity. Where is the author getting this version of the
> axiom from? If it is not my browser - a lot of the formulae were not
> coming out.
>
> On the objection "what is a property" historically Zermelo's original
> formulation had a flaw that was spotted by Russell. Grattan Guiness
> has a good history of this.
>
> As I say I am 100% with the man, but it is one thing to agree with a
> conclusion, another thing to agree whether the conclusion follows from
> the stated assumptions.
>
> I didn't understand the objections above to his remarks about
> computabilty. What exactly is the problem with what he says? Not that
> he says much.

Well, obviously, someone should inform Wildberger that has statements
are being discussed here on sci.logic so it is certain that he knows
about it and has a chance to address people who disagree for various
reasons.

I'm not bothered. I can validate some of Wildberger's statements you
find disagreeable, were I to care.

With the luxury of there being only one null axiom theory, and that
being the only theory, I'm not much concerned about foundations of
mathematics, and my varieties of statements about them. They stand for
themselves.

There is no universe in ZF. If you admit paradoxes in ZF, that does
not agree with then making claims of its consistency, for a statement
and its negation both being true. In the null axiom theory, all the
statements are negated at once.

Ross

albs...@gmx.de

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Jul 6, 2006, 3:59:13 AM7/6/06
to


OH, I SEE, YOU UNDERSTAND ALL.

Thanks for the contact.

Kevin Karn

unread,
Jul 6, 2006, 7:03:53 AM7/6/06
to
R. Srinivasan wrote:

<snip>

> I would
> also suggest to NW not to waste time burying himself too deeply in the
> abstruse stuff discussed in the FOM newsgroup; most of the people there
> are already committed to this or that viewpoint and will not accept any
> questioning of the status quo. In other words, these people are
> anything but objective in analysing and questioning the sorry state of
> the foundations of logic and mathematics as exists today. However I
> will not deny that there are some interesting discussions on that list
> from time to time. But if at all anyone successfully questions the
> existing foundations and proposes an alternative, it will have to be an
> amateur with no stakes in the status quo and with no obligation to
> his/her colleagues to justify the status quo; the FOM list does not
> meet this criterion.

This is a question from the sociological angle, but I'm curious to know
what you think. Why, in your opinion, is the orthodoxy in set theory
etc. so entrenched? Why is the idea of questioning/rejecting infinity
so threatening? What exactly is the stake which people have in the
status quo? What do they have to lose if the status quo is upset? Why
is the resistance so fierce? (Sorry for so many questions. :-) I think
you can see what I'm driving at.)

> Mathemaitcs is about producing proofs
> and proofs do require axioms as starting points. To say that
> mathematics does not require axioms is the same is saying that
> mathematics does not require formal systems or even logic itself. But
> that will quickly lead us into contradictions and just plain confusion
> as well.

When NW said that "You don't need axioms", I understood him to be
saying something more nuanced -- i.e. that "You don't need set theory,
or the axioms of set theory, to do mathematics."

As he said:
" Whenever discussions about the foundations of mathematics arise, we
pay lip service to the `Axioms' of Zermelo-Fraenkel, but do we every
use them? Hardly ever. With the notable exception of the `Axiom of
Choice', I bet that fewer than 5% of mathematicians have ever employed
even one of these `Axioms' explicitly in their published work. The
average mathematician probably can't even remember the `Axioms'. I
think I am typical---in two weeks time I'll have retired them to their
usual spot in some distant ballpark of my memory, mostly beyond recall.
"

It's very clear that you don't need set theory or the axioms of set
theory to do mathematics. After all, virtually the entire body of
pre-20th century mathematics was developed without set theory, or
axioms of set theory. Even today, mathematicians like NW do productive
work without even knowing set theory, or the axioms of set theory, and
this clearly shows that the whole field is an unnecessary appendix.

R. Srinivasan

unread,
Jul 6, 2006, 10:38:27 AM7/6/06
to
Kevin Karn wrote:
[...]

>
> This is a question from the sociological angle, but I'm curious to know
> what you think. Why, in your opinion, is the orthodoxy in set theory
> etc. so entrenched? Why is the idea of questioning/rejecting infinity
> so threatening? What exactly is the stake which people have in the
> status quo? What do they have to lose if the status quo is upset? Why
> is the resistance so fierce? (Sorry for so many questions. :-) I think
> you can see what I'm driving at.)
>
Well, your questions have confounded me too, and I don't have clear
answers. I guess it is partly human nature to be intolerant of dissent,
but that doesn't explain everything. I think it is a big mistake made
by the academicians to plunge headlong into set theory and infinitary
mathematics, ignoring the misgivings of geniuses like Poincare, Weyl,
Brouwer and many others in the early part of the twentieth century. I
guess the dominance of great mathematicians like Hilbert ensured that
these objections were ignored. The institutionalization of research
produced many professionals who had no option but to churn out one
paper after another in the mainstream research areas, with virtually
zero support for any significant pursuit of alternative foundations.
All I can say is that this is just plain wrong. Whoever it was that
decided the funding priorities over the last several decades (ever
since the early 1900's) should have had the vision to support and
encourage alternative viewpoints that may have flatly contradicted the
status quo. Why can't these dissenters co-exist with the mainstream
guys and get funding? That would have led to a healthy balance that is
clearly absent today. It is amazing that today research into
foundations is a very low priority for mathematicians and even
logicians. I would recommend that you look at a recent paper by Lee
Smolin on these issues in the context of theoretical physics (I forget
the title of the paper, but you can search his website and get it).

[..]


>
> When NW said that "You don't need axioms", I understood him to be
> saying something more nuanced -- i.e. that "You don't need set theory,
> or the axioms of set theory, to do mathematics."
>

In the passage that I quoted in my previous post, what NW said was that
mathematics did not need axioms at all, period (whether set theoretical
or otherwise). That viewpoint presumes a certain 'reality' that
mathematcians have to take for granted and simply make definitions to
access that reality, in their proofs. This is not right -- there is no
way for us humans to access any such reality (certainly about
infinitely many objects, such as, natural numbers). This alleged
'reality' is just a figment of our imaginations -- which is what I
would call axiomatic declarations in the human mind. i.e., these
"truths" have their (temporary) existence in the human mind as axioms.
Using the rules of inference in some system of logic that the human
mind accepts, one can deduce theorems from these axioms. Ultimately all
the theorems of such a theory are therefore just declarations made by
the human mind. This is the position that I take in my work on the
logic NAFL, and you can show why infinite sets are not acceptable in
NAFL theories (but infinite proper classes, which are not mathematical
objects (i.e., sets) are acceptable in NAFL theories).

> As he said:
> " Whenever discussions about the foundations of mathematics arise, we
> pay lip service to the `Axioms' of Zermelo-Fraenkel, but do we every
> use them? Hardly ever. With the notable exception of the `Axiom of
> Choice', I bet that fewer than 5% of mathematicians have ever employed
> even one of these `Axioms' explicitly in their published work. The
> average mathematician probably can't even remember the `Axioms'. I
> think I am typical---in two weeks time I'll have retired them to their
> usual spot in some distant ballpark of my memory, mostly beyond recall.
> "
>
> It's very clear that you don't need set theory or the axioms of set
> theory to do mathematics. After all, virtually the entire body of
> pre-20th century mathematics was developed without set theory, or

> axioms of set theory. Even today, mathematicians like Wildberger do productive


> work without even knowing set theory, or the axioms of set theory, and
> this clearly shows that the whole field is an unnecessary appendix.

How would you define real numbers, for example, without set theory?
Wildberger seems to assert in his paper that as long as we have a
finitely stated rule for generating the n'th term in a Cauchy sequence
of rationals, there is no need to consider a real number as an infinite
object. I don't agree with this. For starters, you need to talk about
an "arbitrary" natural number n to generate this rule -- what is this
n? It only makes sense to consider n as a variable that can take on
*any* of infinitely many possible values; in my view this presumes the
existence of an infinite class of natural numbers. Similarly the range
of the sequence is another infinite class of rationals described by the
rule for generating r_n, the n'th rational in the list. Wildberger
rejects this and says that a function does not need to be considered as
having an infinite domain and an infinite range, as long as there is a
finite rule to generate the function. My own view is that the existence
of these infinite classes themselves is not the problem; it is
quantifying over these classes, i.e., formally referring to infinitely
many such infinite classes in a formula, that constitutes infinitary
reasoning, tacit in set theory or almost any modern mathematical
theory. This is what my logic NAFL avoids and one can still do real
analysis in NAFL as I pointed out in my previous post. Wildberger's
objection to an *arbitrary* real number x (see my previous post) can
now be rationalized as an objection to quantifying over all the
possible real values that x can take, since each real is an infinite
object. Otherwise what precisely is Wildberger's objection to an
arbitrary real number, since he already accepts arbitrary natural
numbers?

Regards, RS

george

unread,
Jul 6, 2006, 1:50:43 PM7/6/06
to

Kevin Karn wrote:
> This is a question from the sociological angle, but I'm curious to know
> what you think. Why, in your opinion, is the orthodoxy in set theory
> etc. so entrenched?

Because it's adequate to encode most things.
It's a comprehensive neutral framework.
It makes a good model-construction language.
But it is not what "practicing mathematicians"
actually use, in any case, so from that standpoint, it is
NOT entrenched. It is entrenched as a foundation and
AS A TOOL FOR INVESTIGATING MATHEMATICAL *LOGIC*.
It is NOT entrenched in other ways.

> Why is the idea of questioning/rejecting infinity so threatening?

It's not threatening. It's just stupid. It's so stupid that EVEN
after
people (like NW and others) prove that we could actually, in an
Occam's Razor sense, get along without it, people keep it ANYway.
Because pretending that infinity did not exist WOULD JUST BE STUPID.
Infinity OBVIOUSLY exists. Infinity OBSERVABLY exists, once you
think you know what a natural number is, once you notice that EVERY
natural number has the property that you can add 1 to it and get
something
differing from ALL SMALLER natural numbers. Suppose you TRIED
to confine yourself to considering ONLY FINITE things. Well, guess
what:
there are AN INFINITE NUMBER OF finite natural numbers. So you have
ALREADY COMMITTED yourself to an infinity of things. So pretending
that you haven't, is, I repeat, JUST OBVIOUSLY STUPID. So that's
why people who (futilely) try to do it get dismissed as idiots.

> > Mathemaitcs is about producing proofs
> > and proofs do require axioms as starting points. To say that
> > mathematics does not require axioms is the same is saying that
> > mathematics does not require formal systems or even logic itself. But
> > that will quickly lead us into contradictions and just plain confusion
> > as well.
>
> When NW said that "You don't need axioms", I understood him to be
> saying something more nuanced -- i.e. that "You don't need set theory,
> or the axioms of set theory, to do mathematics."

Well, you're even stupider than he is.
He really WAS saying "you don't need axioms"
and he really was proving about himself, thereby, that
he is too stupid to realize that HE needs axioms and
is in fact relying on them constantly.

>
> As he said:
> " Whenever discussions about the foundations of mathematics arise, we
> pay lip service to the `Axioms' of Zermelo-Fraenkel, but do we every
> use them? Hardly ever. With the notable exception of the `Axiom of
> Choice', I bet that fewer than 5% of mathematicians have ever employed
> even one of these `Axioms' explicitly in their published work. The
> average mathematician probably can't even remember the `Axioms'. I
> think I am typical---in two weeks time I'll have retired them to their
> usual spot in some distant ballpark of my memory, mostly beyond recall.
> "

This is just silly.
Seriously, despite the fact that all of this is true, he marks himself
as a crank by having dared to say it. OF COURSE he doesn't use
the axioms OF SET theory because HE is NOT a SET THEORIST!
HE uses the axioms of WHATEVER FIELD HE SPECIALIZES IN!

The value and relevance of set theory is that the axioms of THAT
field AND of ALL THE OTHERS can ALL be SIMULTANEOUSLY
ENCODED in this ONE dialect (ZFC) -- that is why IT gets to BE
the FOUNDATION. SOME of these concepts (like category theory)
are also equally comprehensive and could serve as alternative
foundations, and are in fact invited to, sometimes, especially when
their take on the universe gets hard to translate into set theory
(which in the case of category theory, it is).

> It's very clear that you don't need set theory or the axioms of set
> theory to do mathematics.

Well, of course, any old axioms will do.
It goes withOUT saying that once you pick ANY consistent
axiom-set and investigate what follows from it, YOU ARE "DOING
mathematics". That is not the point. Set theory is special because
it is a comprehensive neutral framework, because it is one size that
can fit all. What is equally clear (and what you are overlooking,
because
you're just STUPID like that) is that you DO need something AS RICH
or AS POWERFUL as set theory in order to be able to do ALL of
mathematics
WITHIN THIS ONE framework.

> After all, virtually the entire body of
> pre-20th century mathematics was developed without set theory,
> or axioms of set theory.

As I said, any old simpler body of axioms will get you
a little something. The mathematical endeavor that motivated
set theory was describing the sets of points where infinite series
failed to converge. So yes, a whole bunch of calculus and geometry
all got developed first before anybody ever knew they NEEDED to
care about infinite set theory. But the point is, to the extent that
it got developed RIGHT, it got developed axiomatically.

> Even today, mathematicians like NW do productive
> work without even knowing set theory, or the axioms of set theory, and
> this clearly shows that the whole field is an unnecessary appendix.

Dipshit: you can't tell the difference between an appendix
and a foundation. What you have said makes about as much
sense as saying: "Game developers write massively complicated
pieces of graphics software without knowing anything about integrated
circuits or registers and accumulators; that just proves that the
whole field of microcode is an unnecessary appendix."
It proves nothing of the kind. It proves that as long as other
people are maintaining the integrity of the lower levels, YOU
are free to focus on the HIGHER ones. The fact that you write
your computer code in HTML instead of machine language does
not mean that machine language is irrelevant. It is in fact
what your code gets translated into and executed as.

Russell Easterly

unread,
Jul 6, 2006, 3:13:37 PM7/6/06
to

"george" <gre...@cs.unc.edu> wrote in message
news:1152208239.6...@75g2000cwc.googlegroups.com...

>
> Kevin Karn wrote:
>> This is a question from the sociological angle, but I'm curious to know
>> what you think. Why, in your opinion, is the orthodoxy in set theory
>> etc. so entrenched?
>
> Because it's adequate to encode most things.
> It's a comprehensive neutral framework.
> It makes a good model-construction language.
> But it is not what "practicing mathematicians"
> actually use, in any case, so from that standpoint, it is
> NOT entrenched. It is entrenched as a foundation and
> AS A TOOL FOR INVESTIGATING MATHEMATICAL *LOGIC*.
> It is NOT entrenched in other ways.
>
>> Why is the idea of questioning/rejecting infinity so threatening?
>
> It's not threatening. It's just stupid. It's so stupid that EVEN
> after
> people (like NW and others) prove that we could actually, in an
> Occam's Razor sense, get along without it, people keep it ANYway.
> Because pretending that infinity did not exist WOULD JUST BE STUPID.

Why is it stupid to think every natural number is finite?

> Infinity OBVIOUSLY exists.

There is nothing obvious about infinity.

> Infinity OBSERVABLY exists, once you
> think you know what a natural number is, once you notice that EVERY
> natural number has the property that you can add 1 to it and get
> something
> differing from ALL SMALLER natural numbers. Suppose you TRIED
> to confine yourself to considering ONLY FINITE things.

Why can't we limit ourselves to the natural numbers.
What reason do we have to assume infinity exists?

> Well, guess what:
> there are AN INFINITE NUMBER OF finite natural numbers.

You assume there is an "infinite" number of natural numbers.

> So you have
> ALREADY COMMITTED yourself to an infinity of things.

You are only commited to infinity if you insist it exists.

> So pretending
> that you haven't, is, I repeat, JUST OBVIOUSLY STUPID. So that's
> why people who (futilely) try to do it get dismissed as idiots.

Anyone who disagree's with your assumptions is a heretic.

How can you agree with what he says and then
call him a fool for saying it?
He is a fool for pointing out the emperor is naked?

Certainly, his statements won't advance his career
as a mathematician. That is a sad statement about
about mathematics as a field of study.

> OF COURSE he doesn't use
> the axioms OF SET theory because HE is NOT a SET THEORIST!

And only set theorist can create theories about mathematics.

> HE uses the axioms of WHATEVER FIELD HE SPECIALIZES IN!
>
> The value and relevance of set theory is that the axioms of THAT
> field AND of ALL THE OTHERS can ALL be SIMULTANEOUSLY
> ENCODED in this ONE dialect (ZFC)

Total crap.
The statement "each and every natural number is finite" is provably
false in ZF. How can you say ZF encodes all of mathematics?

> -- that is why IT gets to BE
> the FOUNDATION.

It is a foundation that denies one of the most basic truths
of natural numbers.

> SOME of these concepts (like category theory)
> are also equally comprehensive and could serve as alternative
> foundations, and are in fact invited to, sometimes, especially when
> their take on the universe gets hard to translate into set theory
> (which in the case of category theory, it is).
>
>> It's very clear that you don't need set theory or the axioms of set
>> theory to do mathematics.
>
> Well, of course, any old axioms will do.
> It goes withOUT saying that once you pick ANY consistent
> axiom-set and investigate what follows from it, YOU ARE "DOING
> mathematics". That is not the point. Set theory is special because
> it is a comprehensive neutral framework,

How is ZF neutral when it requires one to accept that
infinte sets exist before ZF can even define natural number.
How is ZF neutral when ZF requires the existence of
transfinite inductive sets to define "number"?

> because it is one size that
> can fit all. What is equally clear (and what you are overlooking,
> because
> you're just STUPID like that) is that you DO need something AS RICH
> or AS POWERFUL as set theory in order to be able to do ALL of
> mathematics

All except finite mathematics.


Russell
- 2 many 2 count


Barb Knox

unread,
Jul 6, 2006, 5:45:44 PM7/6/06
to
In article <9aKdnc7HQJRt_TDZ...@comcast.com>,
"Russell Easterly" <logi...@comcast.net> wrote:

> "george" <gre...@cs.unc.edu> wrote in message
> news:1152208239.6...@75g2000cwc.googlegroups.com...

[SNIP]

> > Well, guess what:
> > there are AN INFINITE NUMBER OF finite natural numbers.
>
> You assume there is an "infinite" number of natural numbers.

No, he knows how to DERIVE that fact from commonly-accepted facts about
natural numbers and definitions of "infinite". (Note that just DEFINING
something does not thereby call it into existence; e.g. one can give a
definition for Pegasus or Santa Clause or "the largest natural number").

An easily understood definition of a "finite" set is one that has some
1-1 correspondence with an initial segment of the natural numbers. If a
set so corresponds with {0, 2, ..., n} for some natural number n, then
the set is finite. Then "infinite" is simply "not finite" in that sense
(i.e. can NOT be put into 1-1 correspondence....)

So, if one assumed that the set of all natural numbers is finite (in the
above sense), there must be some n such that {0, 1, ...} = {0, 1, ...,
n}. But note that for any n, the set of all naturals can be written {0,
1, ..., n, n+1, ...}, which clearly does NOT correspond with {0, 1, ...,
n}. Therefore the assumption that the naturals form a FINITE set is
FALSE. Thus we have derived that the naturals form an INFINITE set.

Comments?

[SNIP]

Russell Easterly

unread,
Jul 6, 2006, 6:48:48 PM7/6/06
to

"Barb Knox" <s...@sig.below> wrote in message
news:see-8F2115.0...@lust.ihug.co.nz...

> In article <9aKdnc7HQJRt_TDZ...@comcast.com>,
> "Russell Easterly" <logi...@comcast.net> wrote:
>
>> "george" <gre...@cs.unc.edu> wrote in message
>> news:1152208239.6...@75g2000cwc.googlegroups.com...
>
> [SNIP]
>
>> > Well, guess what:
>> > there are AN INFINITE NUMBER OF finite natural numbers.
>>
>> You assume there is an "infinite" number of natural numbers.
>
> No, he knows how to DERIVE that fact from commonly-accepted facts about
> natural numbers and definitions of "infinite".

Definitions that assume infinite sets exist.

? (Note that just DEFINING


> something does not thereby call it into existence; e.g. one can give a
> definition for Pegasus or Santa Clause or "the largest natural number").

Not if the definition requires us to accept infinite sets exist.

> An easily understood definition of a "finite" set is one that has some
> 1-1 correspondence with an initial segment of the natural numbers.

"Initial segment of the natural numbers"?
You mean the smallest inductive set formed by taking
the intersection of all transfinite inductive sets?

Why do I have to assume transfinite inductive sets exist
before I can define "finite"?

What makes this definition "easily understood"?

> If a
> set so corresponds with {0, 2, ..., n} for some natural number n, then
> the set is finite. Then "infinite" is simply "not finite" in that sense
> (i.e. can NOT be put into 1-1 correspondence....)

"Infinite" is part of set theory's definition of natural number.
Anyone who doesn't accept this definition is a heretic.

> So, if one assumed that the set of all natural numbers is finite (in the
> above sense),

"Above sense" meaning we have already assumed infinte sets exist.

> there must be some n such that {0, 1, ...} = {0, 1, ...,
> n}. But note that for any n, the set of all naturals can be written {0,
> 1, ..., n, n+1, ...}, which clearly does NOT correspond with {0, 1, ...,
> n}. Therefore the assumption that the naturals form a FINITE set is
> FALSE.

Because you assumed the set was infinite to begin with.

> Thus we have derived that the naturals form an INFINITE set.

You derive what you have already assumed is true.


Russell
- Integers are an illusion

george

unread,
Jul 6, 2006, 7:12:09 PM7/6/06
to

> >> Why is the idea of questioning/rejecting infinity so threatening?

I rebutted,


> > It's not threatening. It's just stupid. It's so stupid that EVEN after
> > people (like NW and others) prove that we could actually, in an
> > Occam's Razor sense, get along without it, people keep it ANYway.
> > Because pretending that infinity did not exist WOULD JUST BE STUPID.

The incredibly moronic Russell Easterly
THEN had the IDIOCY to inquire,

> Why is it stupid to think every natural number is finite?

This is a lot worse than stupid. It is malicious and outrageous.
NOBODY EVER ALLEGED that natural numbers weren't finite.
Unless you are going to impute that to a model of PA+~Con(PA)
or something. And the rest of us will not AGREE with it that that
it says/means that, EVEN if IT thinks it does.

>
> > Infinity OBVIOUSLY exists.
>
> There is nothing obvious about infinity.

Of course there is. It is obvious that no natural number is
the number OF natural numbers. That REALLY IS obvious.
DEAL with it.

> > Infinity OBSERVABLY exists, once you
> > think you know what a natural number is, once you notice that EVERY
> > natural number has the property that you can add 1 to it and get
> > something differing from ALL SMALLER natural numbers.
> > Suppose you TRIED
> > to confine yourself to considering ONLY FINITE things.
>
> Why can't we limit ourselves to the natural numbers.

Because they are arbitrarily big and because there are infinitely
many OF them, dipshit. IF you commit yourself to ALL of them,
THEN you commit yourself TO AN INFINITE COLLECTION of things.
The ONLY way to avoid making this commitment is to try to stop at
some largest natural number. But that, OF COURSE, is inconsistent.
So you CANNOT avoid it.


> What reason do we have to assume infinity exists?

Dipshit: WE DON'T assume that infnity exists: WE JUST PROVE
that it exists simply by PROVING that you cannot have a largest
natural number! THE ONLY "assumption" we are making is that if
you have two finite strings, the result of appending them is always
still a finite string!

>
> > Well, guess what:
> > there are AN INFINITE NUMBER OF finite natural numbers.
>
> You assume there is an "infinite" number of natural numbers.

BULLSHIT! WE PROVE that!
I mean, suppose the number of natural numbers WERE finite.
Then there would be SOME natural number that WAS the number
of natural numbers. Call it M (for Maximum). Well, guess what:
M+1 *IS* a natural number! And there are clearly at least M
natural numbers 0,1,2,...M-1. So there are at least M+2
natural numbers. THIS CONTRADICTS YOUR ASSUMPTION
that were only M natural numbers! SO THAT ASSUMPTION IS FALSE.
So howEVER many natural numbers there are, if we are talking about
ALL of them, then we have just PROVED that it is MORE THAN ANY
finite number! THEREFORE IT IS INFINITE.

Jeezus.

How did you graduate from high school without learning that?

george

unread,
Jul 6, 2006, 7:15:26 PM7/6/06
to

Russell Easterly wrote:
> "Barb Knox" <s...@sig.below> wrote in message
> news:see-8F2115.0...@lust.ihug.co.nz...
> > In article <9aKdnc7HQJRt_TDZ...@comcast.com>,
> > "Russell Easterly" <logi...@comcast.net> wrote:
> >
> >> "george" <gre...@cs.unc.edu> wrote in message
> >> news:1152208239.6...@75g2000cwc.googlegroups.com...
> >
> > [SNIP]
> >
> >> > Well, guess what:
> >> > there are AN INFINITE NUMBER OF finite natural numbers.
> >>
> >> You assume there is an "infinite" number of natural numbers.
> >
> > No, he knows how to DERIVE that fact from commonly-accepted facts about
> > natural numbers and definitions of "infinite".
>
> Definitions that assume infinite sets exist.

No, really, they don't.
They don't assume ANYthing about sets.
They don't mention sets at all.
All that they assume is that for EVERY natural
number n, n+1 is also a FINITE natural number.
The definitions never mention infinity or presume anything
about infinity at all. They DON'T prove the existence of infinity
INternally. INternally all they prove is ~ExAy[y<x].
But from that and the simple fact that if x<y then ~(x=y),
it follows that there are an infnite number of things in the domain.
This is NOT ASSUMED about the domain. It is PROVED about
the domain. We LISTED our assumptions, moron.

Russell Easterly

unread,
Jul 6, 2006, 7:30:25 PM7/6/06
to

"george" <gre...@cs.unc.edu> wrote in message
news:1152227726.3...@s53g2000cws.googlegroups.com...

>
> Russell Easterly wrote:
>> "Barb Knox" <s...@sig.below> wrote in message
>> news:see-8F2115.0...@lust.ihug.co.nz...
>> > In article <9aKdnc7HQJRt_TDZ...@comcast.com>,
>> > "Russell Easterly" <logi...@comcast.net> wrote:
>> >
>> >> "george" <gre...@cs.unc.edu> wrote in message
>> >> news:1152208239.6...@75g2000cwc.googlegroups.com...
>> >
>> > [SNIP]
>> >
>> >> > Well, guess what:
>> >> > there are AN INFINITE NUMBER OF finite natural numbers.
>> >>
>> >> You assume there is an "infinite" number of natural numbers.
>> >
>> > No, he knows how to DERIVE that fact from commonly-accepted facts about
>> > natural numbers and definitions of "infinite".
>>
>> Definitions that assume infinite sets exist.
>
> No, really, they don't.
> They don't assume ANYthing about sets.
> They don't mention sets at all.
> All that they assume is that for EVERY natural
> number n, n+1 is also a FINITE natural number.
> The definitions never mention infinity or presume anything
> about infinity at all. They DON'T prove the existence of infinity
> INternally. INternally all they prove is ~ExAy[y<x].

Everyone keeps telling me the statement every natural
number is finite is undecidable in ZF-I.

> But from that and the simple fact that if x<y then ~(x=y),
> it follows that there are an infnite number of things in the domain.

Why does it follow?
Sounds like you have no idea what is in the domain.

> This is NOT ASSUMED about the domain. It is PROVED about
> the domain. We LISTED our assumptions, moron.

You prove you have no idea what the domain is.
"Infinite" is just another way of saying "who knows?".

Russell Easterly

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Jul 6, 2006, 7:52:01 PM7/6/06
to

"george" <gre...@cs.unc.edu> wrote in message
news:1152227529....@k73g2000cwa.googlegroups.com...
>

>> Why is it stupid to think every natural number is finite?
>
> This is a lot worse than stupid. It is malicious and outrageous.
> NOBODY EVER ALLEGED that natural numbers weren't finite.

So the statement "Each and every ordinal has a largest element"
is easily provable in ZF-I?

> Unless you are going to impute that to a model of PA+~Con(PA)
> or something. And the rest of us will not AGREE with it that that
> it says/means that, EVEN if IT thinks it does.
>
>>
>> > Infinity OBVIOUSLY exists.
>>
>> There is nothing obvious about infinity.
>
> Of course there is. It is obvious that no natural number is
> the number OF natural numbers. That REALLY IS obvious.
> DEAL with it.

Why can't I prove this statement in ZF-I?

>> > Infinity OBSERVABLY exists, once you
>> > think you know what a natural number is, once you notice that EVERY
>> > natural number has the property that you can add 1 to it and get
>> > something differing from ALL SMALLER natural numbers.
>> > Suppose you TRIED
>> > to confine yourself to considering ONLY FINITE things.
>>
>> Why can't we limit ourselves to the natural numbers.
>
> Because they are arbitrarily big and because there are infinitely
> many OF them, dipshit. IF you commit yourself to ALL of them,

Why would I even consider committing myself to the idea
there is an "ALL of them". We already know there is no
"ALL of them".

You just said "It is obvious that no natural number is


the number OF natural numbers."

> THEN you commit yourself TO AN INFINITE COLLECTION of things.

Maybe you do.

> The ONLY way to avoid making this commitment is to try to stop at
> some largest natural number.

Or accept there is no such thing as "ALL of them".

> But that, OF COURSE, is inconsistent.

Assuming there is an "ALL of them" is inconsistent.

> So you CANNOT avoid it.

I can. I don't assume there is an "all of them".

>
>> What reason do we have to assume infinity exists?
>
> Dipshit: WE DON'T assume that infnity exists: WE JUST PROVE
> that it exists simply by PROVING that you cannot have a largest
> natural number!

How does that prove there is an "all of them"?

> THE ONLY "assumption" we are making is that if
> you have two finite strings, the result of appending them is always
> still a finite string!

Which is exactly what my proof shows.
There are no infinite ordinals.

>>
>> > Well, guess what:
>> > there are AN INFINITE NUMBER OF finite natural numbers.

If by "infinite" you mean we have no clue how many natural numbers
there are then I agree.

>> You assume there is an "infinite" number of natural numbers.
>
> BULLSHIT! WE PROVE that!
> I mean, suppose the number of natural numbers WERE finite.

I am not saying the number of natural numbers is finite.
I am saying there is no set of all natural numbers.

> Then there would be SOME natural number that WAS the number
> of natural numbers. Call it M (for Maximum). Well, guess what:
> M+1 *IS* a natural number!

Unless we call it OMEGA and pretend its not a natural number.

> And there are clearly at least M
> natural numbers 0,1,2,...M-1. So there are at least M+2
> natural numbers. THIS CONTRADICTS YOUR ASSUMPTION
> that were only M natural numbers! SO THAT ASSUMPTION IS FALSE.
> So howEVER many natural numbers there are, if we are talking about
> ALL of them,

Why are we talking about "ALL of them"?
It should be obvious there is no "ALL of them".

> then we have just PROVED that it is MORE THAN ANY
> finite number! THEREFORE IT IS INFINITE.
>
> Jeezus.
>
> How did you graduate from high school without learning that?

Sheer stubbornness.
I have a low tolerance for BS.

Nam Nguyen

unread,
Jul 7, 2006, 1:42:55 AM7/7/06
to

Russell Easterly wrote:

>
> Why would I even consider committing myself to the idea
> there is an "ALL of them". We already know there is no
> "ALL of them".

It seems severely hard to imagine a kind of mathematics in which
the notion of "ALL of them" is a taboo! We may as well get rid
of the notion of "ONE of them", since the two must be together,
as far as FOL is concerned. Of course, if you happen to suggest
"fractional" quantification instead of, I think that would be an
interesting thing to listen to.

>
> Russell
> - 2 many 2 count
>
>

--
----------------------------------------------------
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
----------------------------------------------------

Nam Nguyen

unread,
Jul 7, 2006, 2:05:26 AM7/7/06
to

Nam Nguyen wrote:

>
>
> Russell Easterly wrote:
>
>>
>> Why would I even consider committing myself to the idea
>> there is an "ALL of them". We already know there is no
>> "ALL of them".
>
>
> It seems severely hard to imagine a kind of mathematics in which
> the notion of "ALL of them" is a taboo! We may as well get rid
> of the notion of "ONE of them", since the two must be together,
> as far as FOL is concerned. Of course, if you happen to suggest
> "fractional" quantification instead of, I think that would be an
> interesting thing to listen to.
>
>>
>> Russell
>> - 2 many 2 count

Hmm! Incidentally, "MANY" sort of sounds like "fractional"
quantification, doesn't it? I mean "there are many natural numbers"
or "there are many transcendentals" means there must be more than ONE,
but does not connote the notion of ALL! The problem is how would we
treat the negation of "there are many"? Would that be "there is none"?

Rupert

unread,
Jul 7, 2006, 3:03:48 AM7/7/06
to

No, that's not true. It's quite common to define "finite number" as
"natural number". What is true is that if ZF is consistent, then there
are models of ZF in which there are hyperfinite natural numbers. (But
here the notion of "hyperfinite" can only be defined from outside the
model, not from within the model).

Kevin Karn

unread,
Jul 7, 2006, 5:21:29 AM7/7/06
to

Your argument hinges on the idea that you can write any n, and that is
extremely dubious.

Here's a basic physical fact:
"Therefore, so long as the number of particles is finite, and
the volume of space occupied by the particles is bounded, and
their total energy is bounded, then even though (classically) the
number of point particle states is uncountably infinite, and even
though the number of possible quantum wavefunctions is also
uncountably infinite, the amount of information in the system is
finite!"
http://www.cise.ufl.edu/research/revcomp/physlim/PhysLim-CiSE/PhysLim-CiSE-5.pdf

This implies that the amount of information contained in the observable
universe is finite. Call that amount x bits. Now, consider a number n
which, due to the length and complexity of its binary representation,
requires more than x bits to express. You can't write n because there's
not enough stuff in the observable universe to write with. So your
derivation fails. Beyond a certain point, you physically can't write
the next number in the series.

abo

unread,
Jul 7, 2006, 7:17:19 AM7/7/06
to
Barb Knox wrote:

>
> An easily understood definition of a "finite" set is one that has some
> 1-1 correspondence with an initial segment of the natural numbers. If a
> set so corresponds with {0, 2, ..., n} for some natural number n, then
> the set is finite. Then "infinite" is simply "not finite" in that sense
> (i.e. can NOT be put into 1-1 correspondence....)

Maybe that's an easily understood definition, but surely it's not the
most natural (is that a pun?). It's more natural to say that a set is
"finite" if there is a natural number which numbers it. E.g. if x, y,
z are distinct and the natural number 3 exists, then {x,y,z} is a
finite set. It can be seen that this definition is not necessarily
equivalent to yours...

With the natural definition, one has to worry about "finite" viewed
externally and viewed internally. For instance, there are theories
which have models where the natural numbers are {0,1,...n} for some n,
yet internally one can prove that no natural number numbers the natural
numbers (say that 3 times fast!). There is no inconsistency, because
if the natural numbers were {0,1,..,n}, then no natural number does
number it (since the set has n+1 elements).

Generally in this discussion, it might be helpful to clarify whether
one is talking about infinity internally or externally to a theory, and
also remark that there are at least three different ways of looking at
infinity:

1/ "Infinity ad infinitum" - a series going on and on
2/ "Potential infinity" - for every natural number there exists a set
X and an element x such that X numbers n and x is not in X
3/ "Actual infinity" - there exists a set X such that no natural number
numbers X

Jack Campin - bogus address

unread,
Jul 7, 2006, 7:51:49 AM7/7/06
to
> How is ZF neutral when it requires one to accept that
> infinte sets exist before ZF can even define natural number.
> How is ZF neutral when ZF requires the existence of
> transfinite inductive sets to define "number"?

It doesn't. You're quite free to define "natural number" inductively
in ZF if you want. But there wouldn't be much point, since ZF was
designed to be a formalization of Cantorian set theory - and *that*
had already generalized the notion of "number" into the transfinite,
in order to represent the foundations of analysis. Permitting the
definition of transfinite ordinals and cardinals was an essential
goal, which ZF achieves by defining cardinals and ordinals in a
uniform way that doesn't assume finiteness where it's not needed.

It's quite simple to show that inductively defined numbers are a
proper set of the numbers ZF defines, just look at the early parts
of any text on the foundations of mathematics...

...in book form (Hatcher, Mendelson, Suppes, whatever). You'd be
better off shutting down your computer and going to a library with
a paper notebook to work out the exercises.

============== j-c ====== @ ====== purr . demon . co . uk ==============
Jack Campin: 11 Third St, Newtongrange EH22 4PU, Scotland | tel 0131 660 4760
<http://www.purr.demon.co.uk/jack/> for CD-ROMs and free | fax 0870 0554 975
stuff: Scottish music, food intolerance, & Mac logic fonts | mob 07800 739 557

Jack Campin - bogus address

unread,
Jul 7, 2006, 9:18:51 AM7/7/06
to
> It's very clear that you don't need set theory or the axioms of set
> theory to do mathematics. After all, virtually the entire body of
> pre-20th century mathematics was developed without set theory, or
> axioms of set theory.

But not measure and probability, topology or theoretical computer
science.

Jack Campin - bogus address

unread,
Jul 7, 2006, 9:22:34 AM7/7/06
to
> The institutionalization of research
> produced many professionals who had no option but to churn out one
> paper after another in the mainstream research areas, with virtually
> zero support for any significant pursuit of alternative foundations.
> All I can say is that this is just plain wrong. Whoever it was that
> decided the funding priorities over the last several decades (ever
> since the early 1900's) should have had the vision to support and
> encourage alternative viewpoints that may have flatly contradicted the
> status quo. Why can't these dissenters co-exist with the mainstream
> guys and get funding?

They *do* get the same funding as mainstream set theorists, i.e. nothing.

Are you under the impression that set theory was like the space programme,
with the US and the Soviets vying for the strongest large cardinal axiom?

george

unread,
Jul 7, 2006, 12:59:21 PM7/7/06
to

> > Well, guess what:
> > there are AN INFINITE NUMBER OF finite natural numbers.

Russell Easterly wrote:
> You assume there is an "infinite" number of natural numbers.

We DON'T JUST "assume" that, moron!
We start with some basic assumptions about natural
numbers that have NOTHING TO DO WITH SETS,
assumptions that YOU agree to, and from them we
PROVE that there is an infinite number of natural numbers.
Idiot.


> You are only commited to infinity if you insist it exists.

Again, bullshit.
You can insist that infinity does NOT exist, and you will
STILL wind up committed to infinity, because your insistence
that it does NOT exist PRODUCES A CONTRADICTION!


> Anyone who disagree's with your assumptions is a heretic.

We are NOT talking about assumptions like ZF's axiom of infinity
here! We are talking about the basic axioms of Peano Arithmetic!
NOBODY, especially NOT YOU, disagrees with those!
You DON'T NEED set theory's axiom of infinity to prove that
there are infinitely many natural numbers! It is the fact that there
were ALREADY KNOWN to be infinitely many natural numbers that
MOTIVATED set theory to DEFINE "infinite set" as "set big enough
to contain all the natural numbers".

[switching topics, after NW said that working mathematicians
never actually use the axioms of set theory]

> How can you agree with what he says and then
> call him a fool for saying it?

Because he is one, that's why; it is a stupid thing to say.

> He is a fool for pointing out the emperor is naked?

In that context, people were generally professing&pretending
that the emperor was finely clothed, so something NEEDED to
be said. Here, that is NOT happening. NOBODY IS ALLEGING
that working mathematicians usually prove things in ZFC.
Since, precisely as he said, THAT happens only 5% of the time,
EVERYBODY ALREADY KNOWS that it happens only 5% of
the time. Going around publishing to the world, like it is some
insight that we needed the likes of YOU to tell us, that THE SKY IS
BLUE,
IS *STUPID*. If you think that the fact that the sky is blue is
surprising
or noteworthy, then that marks YOU as an IDIOT compared to EVERYbody
else who ALREADY knew that that was OLD news!

> > OF COURSE he doesn't use
> > the axioms OF SET theory because HE is NOT a SET THEORIST!
>
> And only set theorist can create theories about mathematics.

Nobody HAS EVER said that.
HE didn't say it. You're an idiot for implying that he did.

> > HE uses the axioms of WHATEVER FIELD HE SPECIALIZES IN!
> >
> > The value and relevance of set theory is that the axioms of THAT
> > field AND of ALL THE OTHERS can ALL be SIMULTANEOUSLY
> > ENCODED in this ONE dialect (ZFC)
>
> Total crap.

How the fuck would YOU know??

> The statement "each and every natural number is finite" is provably
> false in ZF.

So show me a proof, YOU IGNORANT DIPSHIT.

> How can you say ZF encodes all of mathematics?

Obviously, no finite paradigm encodes all of anything; you
can ALWAYS diagonalize. But you can SAY that in ZF.

>
> > -- that is why IT gets to BE
> > the FOUNDATION.
>
> It is a foundation that denies one of the most basic truths
> of natural numbers.

NO, IT DOESN'T. I don't know WHO TOLD you that
something was "provably false in ZF", but you HAVE been
LIED to.

> How is ZF neutral when it requires one to accept that
> infinte sets exist before ZF can even define natural number.

IT DOESN'T, dumbass!
YOU DON'T *KNOW* shit about ZF!
GO READ A FUCKING BOOK, moron!

> How is ZF neutral when ZF requires the existence of
> transfinite inductive sets to define "number"?

I repeat, it doesn't.
And it DOESN'T define number, actually.
It sort of halfway-defines "ordinal".
And that's the best it can do. At first-order, anyway.

[ZF can encode a lot of math]
> All except finite mathematics.

No, actually, the finite part is the only part you CAN get
right. Once you add the axiom of infinity, dueling interpretations
become possible.

george

unread,
Jul 7, 2006, 1:04:36 PM7/7/06
to

Russell Easterly wrote:
> Everyone keeps telling me the statement every natural
> number is finite is undecidable in ZF-I.

Quote somebody, you lying moron.
"Every natural number is finite" IS NOT EVEN
*statable* in ZF. You CANNOT SAY "every natural
number" in ZF. The closest you can say is "every finite
ordinal". And you can't even say THAT properly UNTIL
you go from ZF to ZF*C*.
And it goes withOUT saying that every finite
ordinal is finite. That's just basic English Grammar.
It's also a trivial theorem of ANY theory in which it is statable
at all. It is VERY far from undecidable.


> You prove you have no idea what the domain is.

Neither do you, dumbass.
It is a fact about the first-order logic paradigm
that NOBODY knows *OR CARES* what the domain is!
The domain CAN BE ANYthing!
The only thing that matters is that it be the right SIZE and
that you interpret the functions and predicates OVER it properly.

> "Infinite" is just another way of saying "who knows?".

In ZFC, infinite has at least 5 very CLEAR equivalent
definitions. And most of us know what they all mean.
If you don't, that really does just mean you're stupid.

george

unread,
Jul 7, 2006, 1:06:02 PM7/7/06
to

Russell Easterly wrote:

> So the statement "Each and every ordinal has a largest element"
> is easily provable in ZF-I?

Of course not.
But that has nothing to do with whether NATURAL NUMBERS
are or are not finite. NOT all ordinals are natural numbers.
NOT all ordinals are finite.

george

unread,
Jul 7, 2006, 1:18:08 PM7/7/06
to

Russell Easterly wrote:
> > Of course there is. It is obvious that no natural number is
> > the number OF natural numbers. That REALLY IS obvious.
> > DEAL with it.
>
> Why can't I prove this statement in ZF-I?

Because there are too many different definitions of
infinity, that's why. YOU NEED CHOICE. You need
to be in ZF*C* ~ I. Once you go THERE, you can DEFINE
finite. Unfortunately, you still can't define "natural number".
But you CAN then define "finite ordinal". And it IS provable
that no finite ordinal is big enough to count ALL the finite
ordinals.

> >> Why can't we limit ourselves to the natural numbers.

We can. There are plenty of perfectly good finite set theories;
Peano Arithmetic in fact encodes a finite set theory.
ALL of them have an INFINITE number of finite numbers or
finite sets.

> >
> > Because they are arbitrarily big and because there are infinitely
> > many OF them, dipshit. IF you commit yourself to ALL of them,

> Why would I even consider committing myself to the idea
> there is an "ALL of them".

Because IF YOU LEFT ONE OUT, then you COULDN'T
do arithmetic! You COULDN'T agree that EVERY natural
number has a successor and a predecessor if you weren't
committted to ALL of them, dumbass.

> We already know there is no "ALL of them".

We already know that all of them exist.
Just because the class of all of them isn't ALSO ONE OF them
doesn't mean it doesn't exist. Arguably, the fact that all of its
elements exist means it MUST exist. The line YOU are taking
implies that even though trillions and trillions of cells in your body
all exist, YOU can't exist because YOU are not a cell.


> You just said "It is obvious that no natural number is
> the number OF natural numbers."

Indeed it is.

> > THEN you commit yourself TO AN INFINITE COLLECTION of things.
>
> Maybe you do.

Everybody who commits himself to the natural numbers does that.
The only way to avoid doing it is to avoid committing yourself to
infinitely many of the naturals. There is no point in talking about
natural numbers at all if you are going to leave out infinitely many
more of them than you use.

> > The ONLY way to avoid making this commitment is to try to stop at
> > some largest natural number.
>
> Or accept there is no such thing as "ALL of them".

That is NOT coherent. If you accept each one individually
then you can't justify rejecting them collectively. I repeat, you
can't
concede that all of your own cells exist and bear the right geometric,
electric, elastic, and chemical relationships to each other, BUT YOU
DON'T
exist.


> > But that, OF COURSE, is inconsistent.
>
> Assuming there is an "ALL of them" is inconsistent.

Obviously, that is NOT inconsistent, since you CANNOT PROVE
the inconsistency. Of course, we reserve the right to retract this
if someone else later proves one. But even after they do, it will
STILL not be OBVIOUS, you ignorant moron.


> > So you CANNOT avoid it.
>
> I can. I don't assume there is an "all of them".

Yes, actually, you do. About each of them, you assume
that it exists. THAT IS WHAT IT *MEANS* to assume
"there is an all of them". Even if this collection does NOT
exist as an ELEMENT of the domain of your discourse, it
still HAS to exist, in YOUR paradigm, AS your (total/overall)
domain of discourse.


> How does that prove there is an "all of them"?
>
> > THE ONLY "assumption" we are making is that if
> > you have two finite strings, the result of appending them is always
> > still a finite string!
>
> Which is exactly what my proof shows.

No it isn't.
You have no proof.
You have no axioms.
You don't have shit.

> There are no infinite ordinals.

The fact that there are no infinite ordinals AS ELEMENTS of
your domain does NOT prevent your domain ITSELF (since it
doesn't contain itself) FROM BEING an infinite ordinal.
Or from being an infinite well-ordered CLASS of ordinals
(it it were a set, that would be a contradiction, just as it would
if the collection of all finite ordinals were a finite ordinal).


> >> > Well, guess what:
> >> > there are AN INFINITE NUMBER OF finite natural numbers.
>
> If by "infinite" you mean we have no clue how many natural numbers
> there are then I agree.

That is stupid. HowEVER many it is, you can just PUT a NAME
on it; thereafter, it is THAT many. The fact that this name has
nothing
in common with previous smaller finite names/numbers IS JUST
IRRELEVANT. You might as well have said "we have no idea what
the square root of -1 is". You REALLY Are Stupid.

Karl Malbrain

unread,
Jul 7, 2006, 1:43:24 PM7/7/06
to

george wrote:

> Russell Easterly wrote:
> > >> > Well, guess what:
> > >> > there are AN INFINITE NUMBER OF finite natural numbers.
> >
> > If by "infinite" you mean we have no clue how many natural numbers
> > there are then I agree.
>
> That is stupid. HowEVER many it is, you can just PUT a NAME
> on it; thereafter, it is THAT many. The fact that this name has
> nothing
> in common with previous smaller finite names/numbers IS JUST
> IRRELEVANT. You might as well have said "we have no idea what
> the square root of -1 is". You REALLY Are Stupid.

That is not an honest assessment. The name given to anything is of the
smallest importance, but not irrelevant.

Russell already called this quantity "who knows" -- otherwise
"unknowable".

karl m

her...@cox.net

unread,
Jul 7, 2006, 4:58:16 PM7/7/06
to

John Jones wrote:
>
> her...@cox.net wrote:

[...]

> > Infinity is a tough nut to crack.
> >
> > --
> > hz
>
> Number is defined by a limit, infinity has no limit and cannot be a
> number.


Please go on.

Loosen your belt. Expand!


--
hz

david petry

unread,
Jul 7, 2006, 5:19:13 PM7/7/06
to

Jack Campin - bogus address wrote:
> > It's very clear that you don't need set theory or the axioms of set
> > theory to do mathematics. After all, virtually the entire body of
> > pre-20th century mathematics was developed without set theory, or
> > axioms of set theory.
>
> But not measure and probability, topology or theoretical computer
> science.

That's not really true, even if many pure mathematicians wish it were.
Set theory is merely unnecessary bells an whistles tacked on to those
subjects.

"those of us who work in probability theory or any other area of
applied mathematics have a right to demand that this disease [Cantorian
set theory] be quarantined and kept out of our field" (From
"Probability Theory: The Logic of Science" by E. T. Jaynes)

[...] ordinary mathematical practice does not require an enigmatic
metaphysical universe of sets. (Nik Weaver 2005)

"Set theory is based on polite lies, things we agree on even though we
know they're not true. In some ways, the foundations of mathematics has
an air of unreality." (William P. Thurston)

"[Cantor's paradise] is a paradise of fools, and besides feels more
like hell" (Doron Zeilberger 2006)

"I am convinced that the platonism which underlies Cantorian set theory
is utterly unsatisfactory as a philosophy of our subject, despite the
apparent coherence of current set-theoretical conceptions and methods
... platonism is the medieval metaphysics of mathematics; surely we can
do better" (From "Infinity in Mathematics: Is Cantor Necessary?" by
Soloman Feferman)

Peter Niessen

unread,
Jul 7, 2006, 5:53:17 PM7/7/06
to
Am 4 Jul 2006 08:21:48 -0700 schrieb albs...@gmx.de:

> That's total nonsense what you say here.
> The axiom of infinity declares the existence of infinite sets since you
> can't have them without declaring their existence.
> And doing like this, you achieve a lot of unlogical consequences.
> Your comment about this facts debunk you as total layman in this
> concern.

Non sense or Crazy:
No Axiom => No Set
Try to proof: A set will be exist!
--
Mit freundlichen Grüssen
Peter Nießen

Peter Niessen

unread,
Jul 7, 2006, 5:58:19 PM7/7/06
to
Am 5 Jul 2006 08:18:28 -0700 schrieb albs...@gmx.de:

> why should ZFC has this axiom?

That's very simpel:
No Axiom => No set will be exist.

Karl Malbrain

unread,
Jul 7, 2006, 6:39:03 PM7/7/06
to

george wrote:
> Russell Easterly wrote:
> > "Barb Knox" <s...@sig.below> wrote in message
> > news:see-8F2115.0...@lust.ihug.co.nz...
> > > In article <9aKdnc7HQJRt_TDZ...@comcast.com>,
> > > "Russell Easterly" <logi...@comcast.net> wrote:
> > >
> > >> "george" <gre...@cs.unc.edu> wrote in message
> > >> news:1152208239.6...@75g2000cwc.googlegroups.com...
> > >
> > > [SNIP]
> > >
> > >> > Well, guess what:
> > >> > there are AN INFINITE NUMBER OF finite natural numbers.
> > >>
> > >> You assume there is an "infinite" number of natural numbers.
> > >
> > > No, he knows how to DERIVE that fact from commonly-accepted facts about
> > > natural numbers and definitions of "infinite".
> >
> > Definitions that assume infinite sets exist.
>
> No, really, they don't.
> They don't assume ANYthing about sets.

You did just finish defining a commitment to ALL natural numbers in an
infinite sized "collection":

RE: Why can't we limit ourselves to the natural numbers.

GG: Because they are arbitrarily big and because there are infinitely


many OF them, dipshit. IF you commit yourself to ALL of them,

THEN you commit yourself TO AN INFINITE COLLECTION of things.

The ONLY way to avoid making this commitment is to try to stop at

some largest natural number. But that, OF COURSE, is inconsistent.


So you CANNOT avoid it.

> They don't mention sets at all

How is this "infinite collection" different from an infinite set?

karl m

Karl Malbrain

unread,
Jul 7, 2006, 6:51:06 PM7/7/06
to

Nam Nguyen wrote:
> Russell Easterly wrote:
>
> >
> > Why would I even consider committing myself to the idea
> > there is an "ALL of them". We already know there is no
> > "ALL of them".
>
> It seems severely hard to imagine a kind of mathematics in which
> the notion of "ALL of them" is a taboo! We may as well get rid
> of the notion of "ONE of them", since the two must be together,
> as far as FOL is concerned.

An excellent dialectical observation, thanks.

As far as definitions go: "ALL of them" means "each and every one of
them"

karl m

R. Srinivasan

unread,
Jul 7, 2006, 11:44:35 PM7/7/06
to

Jack Campin - bogus address wrote:
> > The institutionalization of research
> > produced many professionals who had no option but to churn out one
> > paper after another in the mainstream research areas, with virtually
> > zero support for any significant pursuit of alternative foundations.
> > All I can say is that this is just plain wrong. Whoever it was that
> > decided the funding priorities over the last several decades (ever
> > since the early 1900's) should have had the vision to support and
> > encourage alternative viewpoints that may have flatly contradicted the
> > status quo. Why can't these dissenters co-exist with the mainstream
> > guys and get funding?
>
> They *do* get the same funding as mainstream set theorists, i.e. nothing.
>
> Are you under the impression that set theory was like the space programme,
> with the US and the Soviets vying for the strongest large cardinal axiom?
>
Maybe the mainstream set theorists/logicians do not get too much
external funding, but at least they are reasonably certain of getting
their salaries from in-house funding. And they can find many journals
in which they can publish their stuff. On the other hand, there is
simply no incentive for anyone to take up research that questions the
mainstream viewpoint.
Regards, RS

Nam Nguyen

unread,
Jul 8, 2006, 1:07:13 AM7/8/06
to

Nam Nguyen wrote:
>
>
> Nam Nguyen wrote:
>>
>> Russell Easterly wrote:
>>
>>>
>>> Why would I even consider committing myself to the idea
>>> there is an "ALL of them". We already know there is no
>>> "ALL of them".
>>
>> It seems severely hard to imagine a kind of mathematics in which
>> the notion of "ALL of them" is a taboo! We may as well get rid
>> of the notion of "ONE of them", since the two must be together,
>> as far as FOL is concerned. Of course, if you happen to suggest
>> "fractional" quantification instead of, I think that would be an
>> interesting thing to listen to.
>>
>>>
>>> Russell
>>> - 2 many 2 count
>
> Hmm! Incidentally, "MANY" sort of sounds like "fractional"
> quantification, doesn't it? I mean "there are many natural numbers"
> or "there are many transcendentals" means there must be more than ONE,
> but does not connote the notion of ALL! The problem is how would we
> treat the negation of "there are many"? Would that be "there is none"?

I might just work!

Definition
==========

Let the Mx, and 0x denote the 2 new "Many" and "None" quantifiers,
respectively. To introduce them to FOL framework is to define some
*new* rules governing them, in relation to themselves and to the rest
of the existing FOL - particularly the Ax and Ex quantifiers.

Out of a hunch, we'd propose these two:

Rule 1.a: ~(Mx) = 0x
Rule 1.b: ~(0x) = Mx

Also, the intuition doesn't seem to mind this one:

Rule 2: Mx -> Ex

But then complication arises. Suppose F = (~Ex -> Mx) and
we envision it to be a non-trivial theorem of a T; suppose
further that ~Ex is provable. How then could we ever prove
F? Specifically, how would we get from ~Ex to Mx, where the
new rules above don't seem to suggest what kind of arrow ("->")
would lead to this new quantifier? But we might just have a luck
here! We know that ~Ex = Ax, and that the transitivity of
implication means if (A1 -> A2 -> A3) is provable then so is
(A1 -> A3). So now we'd like to have:

(~Ex -> Ax -> Mx)

which is to say we need this new rule:

Rule 3: Ax -> Mx

So there they are: 1.a, 1.b, 2, 3 as the new rules governing
the semantics and the behaviours of the Many and the None
quantifiers.

Application
===========

We could now entertain some possible applications (or impacts ?)
of Mx and 0X on FOL.

Example 1: ZF re-wording (or re-formulating).

Let the null set axiom be defined as:

Ex0y(y e x)

Example 2: Cardinality-free induction.

Let L(0,<) be a language where 0 is a constant and < is
the familiar strictly-less-than binary predicate. Let T
be the following:

T = { AxMy(x < y) }

I guess we could discuss more on these two Mx and 0x. But I just
wonder if I've gone being technically wrong, too far? Thanks for
any comments on this.

--
----------------------------------------------------
Your manuscript is both good and original. However, that which
is good is not original, and that which is original is not good.
Samuel Johnson
----------------------------------------------------


Nam Nguyen

unread,
Jul 8, 2006, 1:08:59 AM7/8/06
to

Nam Nguyen wrote:

>
>
> Nam Nguyen wrote:
>
>>
>>
>> Nam Nguyen wrote:
>>
>>>
>>> Russell Easterly wrote:
>>>
>>>>
>>>> Why would I even consider committing myself to the idea
>>>> there is an "ALL of them". We already know there is no
>>>> "ALL of them".
>>>
>>>
>>> It seems severely hard to imagine a kind of mathematics in which
>>> the notion of "ALL of them" is a taboo! We may as well get rid
>>> of the notion of "ONE of them", since the two must be together,
>>> as far as FOL is concerned. Of course, if you happen to suggest
>>> "fractional" quantification instead of, I think that would be an
>>> interesting thing to listen to.
>>>
>>>>
>>>> Russell
>>>> - 2 many 2 count
>>
>>
>> Hmm! Incidentally, "MANY" sort of sounds like "fractional"
>> quantification, doesn't it? I mean "there are many natural numbers"
>> or "there are many transcendentals" means there must be more than ONE,
>> but does not connote the notion of ALL! The problem is how would we
>> treat the negation of "there are many"? Would that be "there is none"?
>
>
> I might just work!

"It might just work!" that is! Sorry.

Nam Nguyen

unread,
Jul 8, 2006, 4:10:45 AM7/8/06
to

Nam Nguyen wrote:

>
> Nam Nguyen wrote:
>>
>> Hmm! Incidentally, "MANY" sort of sounds like "fractional"
>> quantification, doesn't it? I mean "there are many natural numbers"
>> or "there are many transcendentals" means there must be more than ONE,
>> but does not connote the notion of ALL! The problem is how would we
>> treat the negation of "there are many"? Would that be "there is none"?
>

> It might just work!


>
> Definition
> ==========
>
> Let the Mx, and 0x denote the 2 new "Many" and "None" quantifiers,
> respectively. To introduce them to FOL framework is to define some
> *new* rules governing them, in relation to themselves and to the rest
> of the existing FOL - particularly the Ax and Ex quantifiers.

The semantics of Mx, 0x are related - but not identical - to that of
Ax, Ex. So while the formulae and expressions below are not wrong,
some clarifications are needed.

> Out of a hunch, we'd propose these two:
>
> Rule 1.a: ~(Mx) = 0x
> Rule 1.b: ~(0x) = Mx

Rule 1.a: ~(Mx[phi(x)]) = 0x[phi(x)]
Rule 1.a: ~(0x[phi(x)]) = Mx[phi(x)]

Note: unlike in the case of Ax and Ex, the ~ would not appear
on the right side of "=".

> Also, the intuition doesn't seem to mind this one:
>
> Rule 2: Mx -> Ex

Rule 2. Mx[phi(x)] -> Ex[phi(x)]

> But then complication arises. Suppose F = (~Ex -> Mx) and
> we envision it to be a non-trivial theorem of a T; suppose
> further that ~Ex is provable. How then could we ever prove
> F? Specifically, how would we get from ~Ex to Mx, where the
> new rules above don't seem to suggest what kind of arrow ("->")
> would lead to this new quantifier? But we might just have a luck
> here! We know that ~Ex = Ax, and that the transitivity of
> implication means if (A1 -> A2 -> A3) is provable then so is
> (A1 -> A3). So now we'd like to have:
>
> (~Ex -> Ax -> Mx)

F was msleading: it should be F = (~Ex[phi(x)] -> Mx[~phi(x)]).
Similarly, it should have been:

(~Ex[phi(x)] -> Ax[~phi(x)] -> Mx[~phi(x)])

> which is to say we need this new rule:
>
> Rule 3: Ax -> Mx

Rule 3: Ax[phi(x)] -> Mx[phi(x)]

Kevin Karn

unread,
Jul 8, 2006, 9:07:47 AM7/8/06
to
Jack Campin - bogus address wrote:
> > The institutionalization of research
> > produced many professionals who had no option but to churn out one
> > paper after another in the mainstream research areas, with virtually
> > zero support for any significant pursuit of alternative foundations.
> > All I can say is that this is just plain wrong. Whoever it was that
> > decided the funding priorities over the last several decades (ever
> > since the early 1900's) should have had the vision to support and
> > encourage alternative viewpoints that may have flatly contradicted the
> > status quo. Why can't these dissenters co-exist with the mainstream
> > guys and get funding?
>
> They *do* get the same funding as mainstream set theorists, i.e. nothing.

Bullshit.

Hugh Woodin, a set theory bagwhan from UC at Berkeley, has sucked down
more than a million dollars in U.S. federal grants in the last 15
years. See the NSF site:

http://www.nsf.gov/awardsearch/afSearch.do?PILastName=Woodin&PIFirstName=W.+Hugh#results

Set theory is a business, just like any other business. To understand
it, you just have to follow the money.

Frederick Williams

unread,
Jul 8, 2006, 10:31:37 AM7/8/06
to
Nam Nguyen wrote:
>
> ... The problem is how would we

> treat the negation of "there are many"? Would that be "there is none"?

No. The negation of "there are many" is "there are few or none".

--
Remove "antispam" and ".invalid" for e-mail address.

Nam Nguyen

unread,
Jul 8, 2006, 11:13:37 AM7/8/06
to

Frederick Williams wrote:

> Nam Nguyen wrote:
>
>>... The problem is how would we
>>treat the negation of "there are many"? Would that be "there is none"?
>
>
> No. The negation of "there are many" is "there are few or none".

One could _choose_ take it that, in natural language. It's just that in
technical language, imho, it's nearly impossible to distinguish between
"few" and "many"; so I choose "none". In any rate what I said originally
(quoted above) is just an intuition and might contain ambivalence.

Nam Nguyen

unread,
Jul 8, 2006, 11:21:55 AM7/8/06
to

Nam Nguyen wrote:

>
>
> Frederick Williams wrote:
>
>> Nam Nguyen wrote:
>>
>>> ... The problem is how would we
>>> treat the negation of "there are many"? Would that be "there is none"?
>>
>>
>>
>> No. The negation of "there are many" is "there are few or none".
>
>
> One could _choose_ take it that, in natural language. It's just that in

"One could _choose_ to take it that way,..." I meant.

> technical language, imho, it's nearly impossible to distinguish between
> "few" and "many"; so I choose "none". In any rate what I said originally
> (quoted above) is just an intuition and might contain ambivalence.

I think in the end the new rules we have chosen would be the ones
that would determine how we _technically_ mean by the new concepts.

george

unread,
Jul 8, 2006, 2:38:37 PM7/8/06
to

Daryl McCullough wrote:
> Another silly and content-free comment: He writes about the axiom of
> infinity:
>
> 6. Axiom of infinity: There exists an infinite set.
>
> ...
>
> One might as well declare that: There is an all-seeing Leprechaun!
> or There is an unstoppable mouse!
>
> What a ridiculous comment. First of all, the axiom of infinity
> doesn't actually say that there exists an infinite set. It says
> that there exists a set containing all natural numbers. It's called
> the axiom of infinity because earlier definitions of "infinite set"
> (for example, by Dedekind) could be seen to imply that the set of
> naturals was an infinite set.

Exactly. He reveals further confusion about infinity,
albeit in a more constructive way, in *this* quote:

>> Do modern texts on set theory bend over
>> backwards to say precisely what is and
>> what is not an infinite set? Check it out for
>> yourself---I cannot say that I have found much
>> evidence of such an attitude, and I have looked.

This is probably the single most embarrassing thing
this man has ever said in public. There really are VERY
clear definitions of infinite set -- in fact there are 6 of them --
and they are not THAT hard to find. Here is a relatively
recent contribution of them to THIS arena (sci.logic/sci.math
newsgroups). I think one of the more helpful things Rupert
could do is make sure that THIS old article from Arthur L. Rubin
back in 1998, gets forwarded to NW:

If X and Y are sets, then we define X <= Y
if either of the equivalent definitions hold:

(1) There is a one-to-one function from X into Y.
(2) There is a one-to-one function from a subset of X onto Y.

If X and Y are sets, write X == Y (actually written with a single pair
of wavy lines, and read X is equipollent to Y) if either of the two
equivalent conditions below hold:

(1) X <= Y and Y <= X
(2) There is a 1-1 function from X onto Y.

If X and Y are sets, then write X < Y if X <= Y and not X == Y.

If X is a set, define "X+1" to be any set Y containing X such that
Y\X is a singleton. (As used, it won't matter which set is used, and
the axiom of choice will not be required in any of the proofs, but
this is NOT obvious. There is, however, a formal definition that
will work, but I don't need to go into that much detail at this time.)

If X is a non-empty set, define "X-1" to be any subset of X with only
one element removed. (Again, the axiom of choice will not be required
in any of the proofs, but, this time, there is no suitable formal
definition.)

Define X <=_* Y if any of the equivalent definitions below hold:

(1) X is empty, or there is a function from Y onto X.
(2) There is a function from a subset of Y onto X.
(3) There is a function from Y+1 onto X+1.

If X is a set, define P(X) (usually written with a script P) to be the
power set of X, the collection of all subsets of X.

"omega" will be the first infinite ordinal. (Ordinal numbers are
defined in such a way that "n" = {0, 1, 2, ..., n-1}. I don't think
I need to go into the details of the defintion.)

X is n-finite (n for normal) if any of the following equivalent
definitions hold:

(1) X is equipollent to a finite ordinal.
(2) X < omega
(3) P(X) is PD-finite
(4) P(X) is D*-finite
(5) P(P(X)) is D-finite

X is A-finite if for any subset Y of X, Y is n-finite or X\Y is
n-finite. (A stands for amorphous.)

X is PD-infinite if P(X) is D-infinite (defined below).

X is D*-infinite if any of the following equivalent conditions hold:

(1) There is a proper subset Y of X such that X <=* Y
(2) X <=* X-1
(3) X+1 <=* X
(4) omega <=* X (oops, that's not quite the same but 1-3 imply 4)

X is D-infinte (Dedekind-infinite) if any of the following equivalent
conditions hold:

(1) There is a proper subset Y of X such that X <= Y
(2) There is a proper subset Y of X such that X == Y
(3) X+1 <= X
(4) X+1 == X
(5) X <= X-1
(6) X == X-1
(7) omega <= X

It can be seen that, for any set X, with the following definitions,
(1) -> (2) -> (3) -> (4) -> (5) -> (6)
(1) X is D-infinite
(2) X is D*-infinite
(3) X is D*-infinite (definition 4, but I don't have a separate
name)
(4) X is PD-infinite
(5) X is A-infinite
(6) X is n-infinite

(BTW, some of the lemmas used are:

(A) If X <=_* Y, then P(X) <= P(Y).
(B) If X is n-infinite, then P(X) is D*-infinite
(C) If X is D*-infinite, then P(X) is D-infinite.)

With more effort and a lot of knowledge of set theory, one can show
that
none of these implications is reversable.

For some background, including the definitions of <=, and <=_*, I
recommend _Set Theory for the Mathematician_, by Jean E. Rubin. I have
a vague memory of a paper by one or more Rubin's about relationships
between definitions of finiteness, but I can't find a reference at the
moment.

--

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