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Re: Cantor's first proof,
Posted:
Nov 15, 2012 6:41 AM


Uirgil <uirgil@uirgil.ur> writes:
> In article > <00fb105b99054f2ca7316b001f3980c6@j10g2000yqc.googlegroups.com>, > William Hughes <wpihughes@gmail.com> wrote: > >> On Nov 14, 5:22 pm, Ben Bacarisse <ben.use...@bsb.me.uk> wrote: >> >> > Is there a bijection from N to N? >> >> >> Note that in http://arxiv.org/pdf/math.GM/0305310 >> WM states explicitly that there is no bijection from N to N > > How does WM justify claiming that the identity function on N is NOT > bijective?
Ah, simple. N is an "impossible set" (not a term he defines but why get bogged down in detail like that). The argument is simple: if you try to enumerate N you can construct a diagonal that is not in N; indeed it's not even a natural number. By WMlogic, this means that you have to include this nonnumber in N if you are to claim and enumeration of N. Thus you can't have an enumeration of *only* the elements of N.
It's one of the silliest things I've read for some time, and it's making me revise my option. I thought WM was "having a laugh" by devising arguments he knew to be false, just to see how far he push it, but this is so silly it's hard to see it in that light. Maybe he really thinks like this.
> Does he also claim that identity functions on finite sets can fail to be > bijective?
No.
 Ben.



