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Topic: § 534 Finis
Replies: 30   Last Post: Feb 22, 2015 8:14 AM

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 Tucsondrew@me.com Posts: 1,161 Registered: 5/24/13
Re: § 534 Finis
Posted: Aug 7, 2014 3:31 PM

On Thursday, August 7, 2014 9:17:15 AM UTC-7, muec...@rz.fh-augsburg.de wrote:
> On Thursday, 7 August 2014 17:40:44 UTC+2, Zeit Geist wrote:
>

> > > I think the following: For all natural numbers I have shown that uncounted rationals remain.
> > All Natural Numbers are Finite, too.
> > Is N Finite?

> If infinity can be finished, then N is ""all natural numbers. I have assumed that and have shown that then rationals remain uncounted.

Only for All Finite Sets do Any Rationals Remain Uncounted.

> But since it is impossible to identify remaining rationals, the original assumption has been contradicted.

The Remaining Rationals are Identified by the Naturals that Reamin in Any of your Cases, even though No Naturals Remain in Every Case. No Contradiction.

(It was the only assumption used in my proof that has not been well established as reliable from hundreds of years of mathematics.) Therefore finished infinity has been contradicted. "All natural numbers" is a meaningless notion. So we must say: N contains every natural number. Since there is not a largest one, N is without end, we call it endless or infinite or unfinished.
>

However, N is the Set of All Naturals Numbers, as Defined by PA.
It is Infinite.

> > Using the Same Step in your False Argument can Show it is.

> No, it is not. But try to show it.

Proof by Contradiction that N is Finite.

Assume to the Contrary that N is Infinite.
For Every n e N, Define M_n = { m e N | 0 <= m =< n } and L_n = N \ M_n.
Now, for Every L_n there are Infinitely Many Natural Numbers Remaining, so Every M_n is Finite.
Since All n e N were Used in Forming the M_n's and L_n's Nothing Remains to Make N Infinite.

qed.

Nice Logic WM!

> > > Therefore all natural numbers cannot count all rational numbers.

> > Therefore All Natural Numbers com Not count All the Natural Numbers.

> Correct, since there are not "all natural numbers".

You can Not have it both ways.
As I said you can a System of Only Finite Sets.
However, you or Nobody Else has Ever Shown an Inconsistency with Assuming they Exist.

> Regards, WM

ZG