09/04/2016, 08:16 PM
More on that :
Long ago , I was challenged on sci.math to answer my own question.
There cannot be a bijection from exp^[1/2](w) to w nor 2^w.
Here is why :
Let f(x) be exp^[1/2](x) + o(1).
If card f(w) = card w then
card f(f((w)) = card f( card f(w) ) = card f(w) = card(w)
But f(f(w)) = 2^w
Contradiction
If card f(w) = card 2^w then
card f(f(w)) = card f ( card f(w) ) = card f ( 2^w ) = card(2^(2^w))
But f(f(w)) = 2^w
Contradiction
Likewise for exp^[a](w).
Regards
Tommy1729
Long ago , I was challenged on sci.math to answer my own question.
There cannot be a bijection from exp^[1/2](w) to w nor 2^w.
Here is why :
Let f(x) be exp^[1/2](x) + o(1).
If card f(w) = card w then
card f(f((w)) = card f( card f(w) ) = card f(w) = card(w)
But f(f(w)) = 2^w
Contradiction
If card f(w) = card 2^w then
card f(f(w)) = card f ( card f(w) ) = card f ( 2^w ) = card(2^(2^w))
But f(f(w)) = 2^w
Contradiction
Likewise for exp^[a](w).
Regards
Tommy1729
