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Topic: Matheology § 288
Replies: 160   Last Post: Jun 21, 2013 8:42 AM

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 Tucsondrew@me.com Posts: 1,161 Registered: 5/24/13
Re: Matheology § 288
Posted: Jun 18, 2013 3:39 PM

On Tuesday, June 18, 2013 11:38:18 AM UTC-7, Julio Di Egidio wrote:

>
> In my ever far from solid understanding of these matters, that asymmetry may
>
> reflect the fact (I won't repeat things already said, here I just hint at a
>
> connection) that a theory of infinite sets should have an extended domain
>
> since inception, i.e.. that in the infinitary there cannot be any such thing
>
> as an "unfinished" set (the finitely-inductive set N is not an infinite set
>
> proper). The notion of countability should then itself be extended since
>
> inception, where the counting set would be an extended set as well. In
>
> fact, I would here contend that there can be no such thing as the order type
>
> of N either: w, the first limit ordinal, should correspond to the order type
>
> of N* := N U {w}, while the only usage of N (the finitely-inductive set)
>
> would be as a limit set for use within the finite.
>
>
>
> I am surely an advocate of a strict separation between the finite and the
>
> infinite. Anyway, I wonder if what I am saying makes sense, then if there
>
> exists already a theory of ordinals with characteristics similar to what I
>
> am describing. Feedback appreciated.
>

Julio,

In ZFC, w is a "Limit Ordinal". It is not the same as the idea of "Limit"
used in The Calculus, but it has some similarities.
We define w to be the set of all finite ordinals.
Hence, for all a e w, there exist a b e w, such a < b.

Also, w + 1, as in order type, is defined as w U {w}.
So, if you want w + 1, you must first have w.

I understand the difficulties in "comprehending" w.
When I "imagine" w, I "see" a line of the natural
numbers extending off "forever". I can "jump ahead"
in the line to a larger finite natural, but still can not
"see" the "end". It appears to have the "potential"
of extending to "infinity".

However, in ZFC, we know each finite ordinal
exists. We know 0 = { } exists, and each other
one can be shown to exist via induction, which
is provable in ZFC. Now, using the Axiom of
Infinity, and other axioms, we can show that the
set of all finite ordinals exists. That is, the "collection"
of all finite ordinals is a set.

Next, we DEFINE w to be the order-type of the set
of all finite ordinals. We can now easily prove that w is
NOT a finite ordinal. Better yet, we define w to be
least infinite order-type, and then show that the set of all
finite ordinals has that order-type.

That's basically, not exactly, how that's shown in ZFC.

Also, the list L = { L_n = " n 1's " | n e N } has order-type w.
Where as L U " aleph-0 1's " has order-type w + 1.

This was very general. Ask for further clarification.
I hope this helps a little.

>
> Julio

ZG

Date Subject Author
6/14/13 mueckenh@rz.fh-augsburg.de
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