Re: An uncountable countable set

Virgil wrote:
In article <1156000079.633380.83620@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

*** T. Winter schrieb:
Oh. I must have missed something, because I have not seen a proof.
Given an injection f: N -> P(N), why does the set
M(f) = {n in N | n !in f(n)}
not exist?
Excuse me, the non-existing set is the triple {f, n, M_f(n)}

Regards, WM

What does "M_f(n)" mean? As far as I can see, M_f or M(f) is set, not a function so "M_f(n)" is not even defined, much less {f,n,M_f(n)).

In any case, there are lots of injections, f, from N to P(N).

Consider f(n) = {n}, for example. In which case one easily sees that M(f) = {}.

Which exists quite nicely, thank you very much!

But Monsieur, what about the injection from P(N) into N, via the bit strings which denote set membership, each of which also corresponds to a binary natural? Tsk, tsk. Mustn't forget that one! Remember, the only set which doesn't map is the entire set, and that maps to the largest natural, that is, ...1111 with all bits in finite positions.

TO
.

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