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Topic: Matheology § 285
Replies: 84   Last Post: Jun 15, 2013 6:05 PM

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 Tucsondrew@me.com Posts: 1,161 Registered: 5/24/13
Re: Matheology § 285
Posted: Jun 12, 2013 11:59 AM

On Wednesday, June 12, 2013 4:50:23 AM UTC-7, muec...@rz.fh-augsburg.de wrote:
> On Tuesday, 11 June 2013 21:22:40 UTC+2, Zeit Geist wrote:
>

> > On Tuesday, June 11, 2013 11:49:59 AM UTC-7, muec...@rz.fh-augsburg.de wrote: > On Tuesday, 11 June 2013 20:23:17 UTC+2
>
>
>

> > > > > It is obvious: If we show in set theory a proposition P(n) for the first n elements of a well-ordered set (where n is an arbitrarily large natural number), then we do show it for all elements of the set.
>
>
>

> > This is not obvious. Indeed, it is false. If we show in set theory a proposition P(n) for the first n elements of a well-ordered set (where n is an arbitrarily large natural number), then we do show it for EACH element of the set.
>
>
>
> *All elements* of |N are finite. The *set of all* elements is not (but only because the axiom of infinity says so).
>

The "collection" of all natural numbers is Infinite.
The Axiom of Infinity makes it a set.

Let N be the set of all natural numbers.
Suppose N is finite.
Any set X is finite iff |X| = n, for some n e N.
Hence |N| = m, a natural number.
Now, m e N, so Y = { y | 0 <= y <= m } c N.
Therefore, |N| < |Y|.
But since N is finite and YcN, this is impossible.
Thus, N is infinite.

>
> > The validity P may not carry over to the limit point of Omega.
>
>
>
> It need not, since the axiom of infinity exists and can be applied.
>

What does it mean for any axiom to "exist"?

Neither the AoI or Induction over the Naturals implies
the following

For all n e N, if phi(n) then phi(N).

First, we need to show phi(N) is even defined.
Second, we need to show that
If Z is set of ordinals, where for all z e Z, phi(z);
then phi(Z) must hold.

That is, sort of, Transfinite Induction, Induction beyond the finite.

It doesn't all always works.

For all n e N, 2 * n e N.
But 2 * N ~e N.

It fails here.
>
> By axiom of infinity *there are all elements*. We have to accept that. Now goning on: If not all elements of |Q could be well-ordered, then there must be a first configuration of elements
>
>
>
> configuration n: {q_1, q_2, ..., q_n}
>
>
>
> that cannot be well-ordered by magnitude or size. But there is no such configuration. Every configuration can be well-ordered by size. This implies there is no q that stays outside the well order.
>

Every finite one can, but not the entire set of Q.

Any finite set of real numbers can be well-ordered according to magnitude.
Infinite ones, not necessarily.

>
> Regards, WM

ZG

Date Subject Author
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Virgil
6/11/13 JT
6/11/13 Tucsondrew@me.com
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Virgil
6/11/13 Tucsondrew@me.com
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Tucsondrew@me.com
6/11/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 Virgil
6/11/13 Tucsondrew@me.com
6/11/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Tucsondrew@me.com
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 Tucsondrew@me.com
6/14/13 mueckenh@rz.fh-augsburg.de
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.
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Virgil
6/15/13 Virgil
6/14/13 Virgil
6/13/13 Virgil
6/12/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/12/13 Virgil
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Virgil
6/11/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Scott Berg
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Virgil
6/12/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/11/13 LudovicoVan
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 LudovicoVan
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Tucsondrew@me.com
6/11/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/12/13 Virgil
6/12/13 Virgil
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Virgil
6/12/13 Virgil
6/11/13 LudovicoVan
6/11/13 Virgil
6/11/13 Tanu R.
6/11/13 Tucsondrew@me.com
6/11/13 Tucsondrew@me.com
6/11/13 Tucsondrew@me.com
6/11/13 Virgil
6/11/13 Tanu R.
6/11/13 Virgil
6/11/13 Tanu R.