In sci.math Tony Orlow <tony@lightlink.com> wrote:

Lester Zick wrote:

On Mon, 2 Apr 2007 16:12:46 +0000 (UTC),

It is not true that the set of consecutive naturals starting at 1 with
cardinality x has largest element x. A set of consecutive naturals
starting at 1 need not have a largest element at all.

To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define
"size" such that set of consecutive naturals starting at 1 with size x has a
largest element x, he can, but an immediate consequence of that definition
is that N does not have a size.

Is that true?

~v~~

Yes, Lester, Stephen is exactly right. I am very happy to see this
response. It follows from the assumptions. Axioms have merit, but
deserve periodic review.

01oo

Everything follows from the assumptions and definitions.

On Apr 12, 12:30 pm, Tony Orlow <t...@lightlink.com> wrote:

Yes, Lester, Stephen is exactly right. I am very happy to see this
response. It follows from the assumptions. Axioms have merit, but
deserve periodic review.

YOU can't REview something you've never VIEWED.

On Thu, 12 Apr 2007 15:30:57 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Lester Zick wrote:

On Mon, 2 Apr 2007 16:12:46 +0000 (UTC),

It is not true that the set of consecutive naturals starting at 1 with
cardinality x has largest element x. A set of consecutive naturals
starting at 1 need not have a largest element at all.

To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define
"size" such that set of consecutive naturals starting at 1 with size x has a
largest element x, he can, but an immediate consequence of that definition
is that N does not have a size.

Is that true?

~v~~

Yes, Lester, Stephen is exactly right. I am very happy to see this
response. It follows from the assumptions. Axioms have merit, but
deserve periodic review.

What follows from the assumptions, Tony? Truth?

were true and could be demonstrated they wouldn't have to be assumed
to begin with.

the most rigorous, exhaustive mechanics in extrapolating theorems and
experimental methods from foundational assumptions. But the minute the
same requirements of rigorous mechanics are laid on them and their own
axioms and foundational assumptions they cry foul and claim no one can
prove their assumptions and that even their definitions are completely
arbitrary and can be considered neither true nor false.

~v~~

a) A consecutive set of naturals starting with 1 with size X can not
have any maximum other than X.
b) A consecutive set of naturals starting with 1 with size X has
maximum X.

On 12 Apr, 19:54, Tony Orlow <t...@lightlink.com> wrote:

Mike Kelly wrote:

On 2 Apr, 15:49, Tony Orlow <t...@lightlink.com> wrote:

Mike Kelly wrote:

On 1 Apr, 04:44, Tony Orlow <t...@lightlink.com> wrote:

cbr...@cbrownsystems.com wrote:

On Mar 31, 5:45 pm, Tony Orlow <t...@lightlink.com> wrote:>

Yes, NeN, as Ross says. I understand what he means, but you don't.

What I don't understand is what name you would like to give to the set
{n : n e N and n <> N}. M?
Cheers - Chas

N-1? Why do I need to define that uselessness? I don't want to give a
size to the set of finite naturals because defining the size of that set
is inherently self-contradictory,

So.. you accept that the set of naturals exists? But you don't accept
that it can have a "size". Is it acceptable for it to have a
"bijectibility class"? Or is that taboo in your mind, too? If nobody
ever refered to cardinality as "size" but always said "bijectibility
class" (or just "cardinality"..) would all your objections disappear?

Yes, but my desire for a good way of measuring infinite sets wouldn't go
away.

You seem to be implying that the existence and acceptance of
cardinality as one way of measuring infinite sets precludes the
invention of any other. This is patently false. There is an entire
branch of mathematics called "measure theory" which, roughly speaking,
examines various ways to measure and compare infinite sets. Measure
theory builds upon set theory. Set theory doesn't preclude mesure
theory.

No response to this bit, of course. You're chronically incapable of
acknowledging this.

Of course, if *your* ideas were to be formalised then first of all
you'd have to pull your head out of.. the sand, accept that you've
made numerous egregiously erroneous statements about standard
mathematics, learn how to communicate mathematically and learn how to
formalise mathematical ideas precisely. Look at NSA and the Surreal
numbers if you need evidence that non-standard ideas can be expressed
clearly and coherently within an existing framework of mathematical
expression.
You may be a lost cause though. You've spent, what, three years
blathering on Usenet and your mathematical understanding and maturity
hasn't improved a jot. It seems like you genuinely don't want to
learn. Is ranting incoherently just your way of blowing off steam?

You're not entirely wrong, Mike.

Natch.

I mean, you've been a jerk through all of this, but you have a point.

Whereas you..

In order to supplant what is currently
the overly axiomatic bent of the field of mathematics and return to a
balance between the deductive and the inductive sides of logic requires
that one delve into the very foundations of logic itself, and form a new
basis for determining what constitutes evidence in the field of
mathematics, and how this evidence should be fed as deductively derived
input into the inductive process of choosing axioms from which to build
theorems. I am working on how to balance this, but it's not easy, and
life's not easy, and I have other things to do. But, I'm doing this,
too. I wish I had more time to research, but when I find the time to do
this here, it's occasional. I promise to try harder in the future.

There, there Tony. Aren't you just the cutest little mathematical
pioneer?

My threads do get attention, because I raise some valid issues.

Guess again.

You want a solution? Help me out. :)

Solution to what? All I'm trying to do here is to get you to realise
that you are wasting your time trying to develop new foundations for
mathematics or logic or whatever when you are utterly incapable of
understanding existing ideas, following proofs, writing proofs,
communicating mathematically...

It'd be nice if some day you learned some math above high school
level. Seems a rather remote possibility though because you are
WILLFULLY ignorant.

given the fact that its size must be equal to the largest element,

That isn't a fact. It's true that the size of a set of naturals of the
form {1,2,3,...,n} is n. But N isn't a set of that form. Is it?

It's true that the set of consecutive naturals starting at 1 with size x has largest element x.

No. This is not true if the set is not finite (if it does not have a
largest element).

Prove it, formally, please, from your axioms.

I don't have a formal definition of "size". You understand this point,
yes?

It's very easily provable that if "size" means "cardinality" that N
has "size" aleph_0 but no largest element. You aren't actually
questioning this, are you?

It's rather disingenuous to ask me for a formal proof of something
that is couched in your informal terms.

It is true that the set of consecutive naturals starting at 1 with
largest element x has cardinality x.

Forget cardinality. Can a set of naturals starting with 1 and with size
X possibly have any other maximum value besides X? This is inductively
impossible.

(Just to be clear, we're talking very informally here. It's quite
obviously the only way to talk to you.)

Tony, can you discern a difference between the following two
statements?

a) A consecutive set of naturals starting with 1 with size X can not
have any maximum other than X.
b) A consecutive set of naturals starting with 1 with size X has
maximum X.

Seriously, do you comprehend that they are saying different things?
This is important.