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Topic: ? 417 An implication of actual infinity
Replies: 4   Last Post: Jan 17, 2014 3:36 PM

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Virgil

Posts: 7,014
Registered: 1/6/11
Re: ? 417 An implication of actual infinity
Posted: Jan 15, 2014 2:57 AM
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In article <c637c9bf-9ba8-4aaf-887c-2d5499c746cf@googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:

> On Tuesday, 14 January 2014 23:14:55 UTC+1, wpih...@gmail.com wrote:
>
>

> >
> > > I am not considering the limit
> >
> >
> >
> >
> >
> > So you are not considering an infinite sum.
> >

> I am considering every natural. You say this way I never get all naturals.
> Hence your "all" is more than the set of "every".
>
> That is the trick of set theory. Every = potential inf. All = actual inf.


In properly expressed set theory, "for every x" = "for all x"
and whether that involves actual infiniteness or finiteness depends on
whether the set one is quantifying over is actually_infinite or
actually_finite. Tertium non datur.


> But if you claim that the set |N contains more than every natural number:
> What is this "more"?


Since every natural number has a successor, for every one ther eis a
more, and for all of them collectively there is, on any proper sset
theory, a more as well.

That WM has to ask about it only shows his ignorance,
>
> What differs in the infinite set |N from all FISONs with
> |{1, 2, 3, ...,n}| =< n


It is equally true that for all n in |N, |{1, 2, 3, ...,n}| >=n


> although on the RHS all n and on the LHS all FISONs appear, one after
> another? It is impossible to have more than all natural numbers, isn't it?


The set of all sets of natural numbers, 2^|N, is far "more" that the
set of all natural numbers, |N, since there are lots of injections from
|N to 2^|N, but provably none from 2^|N to |N , nor any surjections from
|N to 2^|N.

> Or is here "all" different from every?

Neither "all" nor "every" is unambiguous in WMytheology, but they both
are outside of WMytheology
--





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