[MO] Is there a tetration for infinite cardinalities? (Question in MO)

[tommy1729]

What I wrote is in perfect harmony with sup and beth numbers.

Maybe I should apologize for my tricky question.

Tricky because trying to answer might be a trap.

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] with noninteger a.

SO exp^[1/2] does not exist in set theory.

SO set theory is not suitable for functions since only the power function 2^ is a function that exists and makes a "difference".

( Polynomials do not make a "difference" , they do not change cardinality )

Considering that infinite set theory is not suitable for functions , it makes more sense to be skeptical about its use in other branches of math such as number theory , algebra , calculus and dynamical systems.

Also notice that even substraction is not well defined (for ordinals) since w-1 does not even exist.


That should shed some light on my skeptisism.
( it is JUST the tip of the iceberg , there is way more reason to be skeptical )

Since half-iterates (functions) do usually not exist in set theory , I do not see how one can continue to combine set theory and dynamics.

regards

tommy1729

" Truth is what does not go away when you stop believing in it "