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

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 LudovicoVan Posts: 4,025 From: London Registered: 2/8/08
Re: Matheology § 288
Posted: Jun 18, 2013 2:38 PM

"Julio Di Egidio" <julio@diegidio.name> wrote in message
news:kppqpg\$div\$1@dont-email.me...
> "apoorv" <skjshr@gmail.com> wrote in message
>

>> Suppose we draw up triangles
>> ...............................l..
>> ...........................1....1
>> ........................1....1....1
>> .....................1....1....1....1
>> ....................................................
>> ...........................................................
>> 1....1....1....1....1....1....1....1....1....1....1....1
>>
>> Each of the diagonal is of the order type {1,2,3....w}

>
> That would be w+1, the order type of N* := {1,2,3,...} U {w}.
>

>> The question is what is the order type of the bottom horizontal line?
>
> Consider this:
>
> 1-> 1
> 2-> 12
> 3-> 123
> ...
> n-> 123...n
> ...
> ___
> w-> 123...n...___w
>
> The order type of the w-th entry is again w+1.
>
> So, the "triangle" is "equilateral", at every step as in the limit.
>
> There just seems to be a sort of dissymmetry, so that the n-th entry has
> order type n but the w-th entry has order type w+1. But I guess this is
> simply because we are overloading the symbol w. To sort it out, maybe
> something looking like the following would work?
>
> Let w* := w+1 be the order type of N* := N U {w*}.
>
> Then we would rather write the bottom line line as:
>
> w*-> 123...n...___w*
>
> and the w*-th entry would indeed have order type w*.
>
> Otherwise, how to make head or tails of that "dissymmetry"?

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

Date Subject Author
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 LudovicoVan
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6/16/13 apoorv
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6/18/13 LudovicoVan
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6/20/13 LudovicoVan
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6/15/13 mueckenh@rz.fh-augsburg.de
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6/16/13 mueckenh@rz.fh-augsburg.de
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6/16/13 Virgil
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6/16/13 Virgil
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