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Virgil
Posts:
8,833
Registered:
1/6/11


Re: ? 417 An implication of actual infinity
Posted:
Jan 15, 2014 2:57 AM


In article <c637c9bf9ba84aaf887c2d5499c746cf@googlegroups.com>, mueckenh@rz.fhaugsburg.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 



