Re: Testing something

r norman <r_s_norman@xxxxxxxxxxx> wrote:

On Tue, 09 Jun 2009 11:14:51 -0400, Walter Bushell <proto@xxxxxxxxx>
wrote:

In article
<ee971f08-d340-40ef-b189-7c4abca65f6b@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"rnorman@xxxxxxxxx" <rnorman@xxxxxxxxx> wrote:

On Jun 8, 7:46 pm, Walter Bushell <pr...@xxxxxxxxx> wrote:
In article
<e676dc0e-fdf7-49c0-9973-c6b8709a1...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,





"rnor...@xxxxxxxxx" <rnor...@xxxxxxxxx> wrote:
On Jun 8, 4:18 pm, "Stephen" <ssan...@xxxxxxxxxxxxx> wrote:
raven1 wrote:
On Mon, 8 Jun 2009 05:31:19 -0700 (PDT), "rnor...@xxxxxxxxx"
<rnor...@xxxxxxxxx> wrote:

On Jun 8, 1:27 am, j...@xxxxxxxxxxxxx (John S. Wilkins) wrote:
1, 2, 3...

Let's see, now....

It's not the Fibonacci series because that should start with two
1's. It's not the primes becase that should not start with 1.
It's not the rationals because although the well ordering theorem
and axiom of choice says there has to be a first element, I can't
really figure out what that first element should be.

OK, I give up, What is the answer?

Positive integers works.

In ascending sequence, beginning with the lowest positive integer. It
would be a lot more interesting to begin with the highest & list in
descending sequence...

I am having trouble understanding exactly what both of you are talking
about. Just what is this set called 'integer'? Could you please give
me a simple, precise, concise definition that is not circular in that
it does not refer to specific representations of number or other
categories of number. Also tell me exactly how you specify which of
these so-called integers are 'positive' and what you mean by 'lowest'.

{} -> 1
{{}} ->2
and so forth.

where the successor of any number n AKA n+1 is represented by {the
representation of n}

all the results of this are the positive integers and there are no other
positive integers.

A little informal, but the filling in the blanks is left to the reader.

The whole definition in these terms of going from set theory to a model
of the real numbers is a one semester course at the senior level of
undergraduate work or the first year of graduate study.

Damn! There's always some wise guy who actually knows the answer.

My Bernays and Fraenkel 'Axiomatic Set Theory' (yes, senior year
taking required 1st year grad level courses) has your definitions as
ordinals, although Bernays develops them differently from your
description which I believe was due to Zermelo to von Neumann to
Fraenkels. Then "a natural number is an ordinal such that itself and
every element of it is either 0 is a successor". Definitely precise
and concise but hardly simple.

But my definition just provides a model for the natural numbers. It
doesn't as my math professors told me, tell us what they are, as there
are many possible models.

My recollection of "Foundations of Mathematics" is about exhausted
now. Does it really matter that there are many models if they are all
homeomorphic (or whatever phrase of equivilance you choose)?

We have absolutely no idea what charge or mass are, let alone spin or
charm or strangeness. That doesn't stop the physicists. And those
things are supposed to be "real"! Mathematics always deals in
abstract ideas. I also don't remember the outcome of the question
whether you discover new mathematics or whether you create it.

The answer to such questions is always "yes".
--
John S. Wilkins, Philosophy, University of Sydney
http://evolvinthoughts.net
But al be that he was a philosophre,
Yet hadde he but litel gold in cofre

.

Relevant Pages

  • Re: Testing something
    ... it does not refer to specific representations of number or other ... all the results of this are the positive integers and there are no other ... The whole definition in these terms of going from set theory to a model ... My recollection of "Foundations of Mathematics" is about exhausted ...
    (talk.origins)
  • Re: Testing something
    ... really figure out what that first element should be. ... it does not refer to specific representations of number or other ... all the results of this are the positive integers and there are no other ... The whole definition in these terms of going from set theory to a model ...
    (talk.origins)
  • Re: Testing something
    ... really figure out what that first element should be. ... it does not refer to specific representations of number or other ... all the results of this are the positive integers and there are no other ... The whole definition in these terms of going from set theory to a model ...
    (talk.origins)
  • Re: Earth 8??
    ... The problems you're bringing up come from the fact that you're using common English words that have common English meanings that don't apply to transfinite numbers. ... By definition of _infinity_. ... lay speech, philosophy, and pretty much anywhere but pure mathematics. ... This defined mapping puts the positive integers and the even positive integers into a 1:1 correspondence. ...
    (rec.arts.comics.dc.universe)
  • Re: Find out if I/10^i = J/2^j (decadic - dyadic pairing)
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