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

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 fom Posts: 1,968 Registered: 12/4/12
Re: Matheology § 288
Posted: Jun 18, 2013 9:35 PM

On 6/18/2013 1:38 PM, Julio Di Egidio wrote:
> "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.

The omega-th entry is the union of its predecessors
as is typical of its definition as a limit ordinal.

The successor of the omega-th entry takes omega as an
element and has order type omega+1.

Your ">>___" is the omega-th entry.

You should consider the possibility that apoorv's
question is not well-construed.

Presumably the answer to his question will be
something like omega+n where n is ambiguous. The
bottom line is countably infinite with a tail that
is characterized as a sequence of successors.

In fact, it could be alpha+n for any countable
limit ordinal alpha and any finite n since apoorv gave
no partition rule for the bottom line. Finite
natural numbers enjoy a coincidence between ordinality
and cardinality. apoorv's question conflates the
two concepts (in my opinion).

>> 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.
>

If you find a way to restate this, maybe I could make sense
of it and try to complement it with some thoughts. I have a
paper which observes that the difference between the Fregean
analysis of number and the Cantorian analysis of number is that
the Cantorian analysis differentiates ordinality from cardinality.
This, of course, becomes distinctly different in the transfinite
realm. This may apply to what you are thinking about.

By the way, for each ordinal, its "existence" as an object of the
theory depends upon its successor. That is,

null = {}

{} in {{}}

{{}} in {{}, {{}}}

{{}, {{}}} in {{}, {{}}, {{}, {{}}}
}

etc.

Remember, to be a set, a class must be an element of a set.

This is why your omega-th entry is in a set with order
type omega+1.

Moreover, when working with ordinal-indexed sequences in
set theory, it is fairly common practice to work with
strict order relations:

{n_beta | beta < alpha}

where, of course, one has

beta in alpha

because ordinals are well-ordered by membership.

This reflects how sequences are treated at limit
ordinals.

Naturally I am making these statements with a
specific set of constructions in mind relative
to a specific theory. Different constructions
by different authors may be different. I make
this disclaimer because the "That's not what I
am talking about" is far too common.

Anyway, I expect to see the same response you
gave to ZG. Sorry about further regurgitation.

Date Subject Author
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 LudovicoVan
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 LudovicoVan
6/14/13 Tucsondrew@me.com
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 Tucsondrew@me.com
6/14/13 Virgil
6/14/13 Tucsondrew@me.com
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Tucsondrew@me.com
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Virgil
6/15/13 Virgil
6/16/13 apoorv
6/16/13 Virgil
6/16/13 mueckenh@rz.fh-augsburg.de
6/16/13 Virgil
6/16/13 apoorv
6/16/13 Virgil
6/16/13 apoorv
6/17/13 Virgil
6/17/13 apoorv
6/17/13 Virgil
6/17/13 Tanu R.
6/17/13 Virgil
6/17/13 fom
6/16/13 FredJeffries@gmail.com
6/16/13 apoorv
6/16/13 apoorv
6/17/13 FredJeffries@gmail.com
6/17/13 Virgil
6/17/13 FredJeffries@gmail.com
6/18/13 LudovicoVan
6/18/13 mueckenh@rz.fh-augsburg.de
6/18/13 Virgil
6/18/13 LudovicoVan
6/18/13 Tucsondrew@me.com
6/18/13 LudovicoVan
6/18/13 Tucsondrew@me.com
6/18/13 mueckenh@rz.fh-augsburg.de
6/18/13 Tucsondrew@me.com
6/19/13 mueckenh@rz.fh-augsburg.de
6/19/13 Tucsondrew@me.com
6/19/13 mueckenh@rz.fh-augsburg.de
6/19/13 Tucsondrew@me.com
6/20/13 mueckenh@rz.fh-augsburg.de
6/20/13 LudovicoVan
6/20/13 mueckenh@rz.fh-augsburg.de
6/20/13 LudovicoVan
6/20/13 mueckenh@rz.fh-augsburg.de
6/20/13 LudovicoVan
6/20/13 mueckenh@rz.fh-augsburg.de
6/20/13 LudovicoVan
6/20/13 Virgil
6/20/13 Virgil
6/20/13 Virgil
6/20/13 Virgil
6/20/13 Virgil
6/19/13 Virgil
6/19/13 Virgil
6/18/13 Virgil
6/18/13 fom
6/18/13 fom
6/18/13 LudovicoVan
6/18/13 fom
6/19/13 LudovicoVan
6/19/13 mueckenh@rz.fh-augsburg.de
6/19/13 Virgil
6/19/13 LudovicoVan
6/19/13 fom
6/20/13 mueckenh@rz.fh-augsburg.de
6/20/13 LudovicoVan
6/20/13 mueckenh@rz.fh-augsburg.de
6/20/13 LudovicoVan
6/20/13 mueckenh@rz.fh-augsburg.de
6/20/13 LudovicoVan
6/20/13 mueckenh@rz.fh-augsburg.de
6/20/13 LudovicoVan
6/20/13 Virgil
6/20/13 Virgil
6/20/13 FredJeffries@gmail.com
6/20/13 LudovicoVan
6/21/13 LudovicoVan
6/21/13 LudovicoVan
6/20/13 FredJeffries@gmail.com
6/20/13 Virgil
6/20/13 Virgil
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Virgil
6/14/13 Virgil
6/14/13 Virgil
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 Virgil
6/14/13 Tucsondrew@me.com
6/14/13 Virgil
6/14/13 Tucsondrew@me.com
6/14/13 Virgil
6/14/13 Tucsondrew@me.com
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Tucsondrew@me.com
6/15/13 Virgil
6/15/13 Tanu R.
6/14/13 Tucsondrew@me.com
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 LudovicoVan
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 LudovicoVan
6/14/13 Tucsondrew@me.com
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 Virgil
6/14/13 Tucsondrew@me.com
6/14/13 Virgil
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Virgil
6/15/13 Tanu R.
6/14/13 Virgil
6/14/13 Virgil
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 Bergholt Stuttley Johnson
6/14/13 Virgil
6/14/13 Tucsondrew@me.com
6/14/13 Virgil
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Virgil
6/15/13 Tanu R.
6/15/13 Bergholt Stuttley Johnson
6/14/13 Virgil
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 Virgil
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Virgil
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Virgil
6/16/13 mueckenh@rz.fh-augsburg.de
6/16/13 Virgil
6/16/13 mueckenh@rz.fh-augsburg.de
6/16/13 fom
6/16/13 mueckenh@rz.fh-augsburg.de
6/16/13 fom
6/16/13 Virgil
6/17/13 mueckenh@rz.fh-augsburg.de
6/17/13 Virgil
6/17/13 fom
6/16/13 mueckenh@rz.fh-augsburg.de
6/16/13 fom
6/16/13 mueckenh@rz.fh-augsburg.de
6/16/13 Virgil
6/16/13 fom
6/16/13 Virgil
6/16/13 fom
6/16/13 Virgil
6/16/13 mueckenh@rz.fh-augsburg.de
6/16/13 Virgil
6/17/13 mueckenh@rz.fh-augsburg.de
6/17/13 Virgil
6/16/13 Virgil
6/16/13 mueckenh@rz.fh-augsburg.de
6/16/13 Virgil
6/15/13 Tanu R.