On 21 Dez., 14:17, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > In article <2fac8bb1-4c90-4421-b559-1ea7f0301...@e27g2000yqd.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes: > > On 18 Dez., 15:12, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > ... > > > Why need I to think about a last one (which there isn't) to be able to > > > think about a set that contains all natural numbers? Apparently you > > > have some knowledge about how my mind works that I do not have. > > > > Yes. A very convincing and often required proof of completenes of a > > linear set is to know the last element. > > Oh, is it often required?
Except in matheology it is always required. > > > T talk about all in case there > > is no last is silly. > > And I think it is silly to require there being a last to be able to talk > about all.
That's why you love matheology. > > > > > > > > Right, but there is no finite initial segment that contains them > > > > > > > all. > ... > > > > > Sorry, I have no knowledge of the bible. But live without that axiom > > > > > when you can't stomach it. And do not attack mathematicians who > > > > > live with that axiom. > > > > > > > > To live with that axiom does not create uncountability. See the proof > > > > here: > > > >http://groups.google.com/group/sci.logic/browse_frm/thread/46fa18c8bb= > > ... > > > > > > Where is the proof there? I see only you writing a bit of nonsense and > > > two rebuttals. > > > > One of the rebuttals has meanwhile been changed. Peter Webb > > recognized: It is true that you cannot show pi as a finite decimal, > > but you can't show 1/3 as a finite decimal either. > > So what? That is not contested and it does not show in *any* way that the > axiom of infinity does not create uncountability. So no proof at all.
It may create what you like. Either 1/3 can be identified at a finite digit or 1/3 cannot be identified at a finite digit.
Even a matheologian should understand that: If there is no digit at a finite place up to that the sequence 0.333... identifies the number 1/3, then there is no digit at a finite place up to that the number 1/3 can be identified. > > > Just what I said. > > And just wat I said: see the quote above: > > > > > > > Right, but there is no finite initial segment that contains them > > > > > > > all. > > which you contested.
I did not contest it. I said, if there is a sequence that identifies 1/3, then the identifying digits must be at finite places. But we know that for every finite place d_n, there is a sequence d_1, ..., d_n that is not 1/3 but is identical to the sequence of 1/3. Therefore we can conclude that there is no sequence identifying the number 1/3 by means of digits at finite places only. > > > > > > The infinite paths because you stated a priori that your tree did > > > > > not contain infinite paths. So it is impossible to construct in > > > > > your tree infinite paths by the axiom of infinity. > > > > > > > > The axiom of infinity establishes the set N from finite numbers. > > > > > > It establishes the *existence* of a set N of finite numbers. > > > > What else should be established? > > Does not matter. The axiom of infinity does *not* construct infinite paths > in your tree, beacuse you stated that your tree did not contain infinite > paths a priori.
The union of finite ínitial segments cannot ield an infinite initial segment? Does the sequence of 1/3 not consist of a union of all finite initial segments?