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

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Matheology � 285
Posted: Jun 12, 2013 2:52 PM

mueckenh@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 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.
>
> 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.

Each of WM's well-ordered sets is also finite, so if the well-ordered
property is carried over to to the limit set, finiteness property must
be as well.

Note that every finite ordered set is also well-ordered but not every
infinite set is well-ordered, so being well-ordered can be lost when
going from a sequence of finite sets to an infinite limit set, just like
being finite can be lost.

Since finiteness implies well-orderedness, loss of finiteness in the
limit can produce loss of well-orderedness in that limit.
But there is no such

> configuration. Every configuration can be well-ordered by size.

Every finite ordered set is automatically well-ordered, but, e.g., the
set of integers is not.

Finiteness of an ordered set implies its well-orderedness.
When a sequence of finite ordered sets has an infinite limit, that
implication is lost.
--

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.