Here is a topologically accurate photograph of my crocodile:
In particular, topologically he is a solid sphere. He can be described as being shaped like himself, but that is vague. There are so many ways it can be shaped like himself!
Specifically, I am asking how many homeomorphisms there are from my solid sphere crocodile to itself. I know that, under continuum hypothesis, this quantity is less than or equal to $\aleph_2$. I don't know its exact value though.
Also, without continuum hypothesis or axiom of choice, do we still know this quantity?
Since your sphere has $|\mathbb R|$ points and contains a countable dense subset (for example, the points with rational coordinates), the number of homeomorphisms cannot be larger than $|\mathbb R|$, since any homeomorphism is completely given by its values on a dense subset, so there are at most $$|\mathbb R^{\mathbb N}| = |(2^{\mathbb N})^{\mathbb N}| = |2^{\mathbb N\times\mathbb N}| = |2^{\mathbb N}| = |\mathbb R| $$ continuous functions from the sphere to itself.
On the other hand, there are also at least continuum many homeomorphisms, for example the rigid rotations around some axis.
By Cantor-Schröder-Bernstein, then, the number of homeomorphisms is exactly $|\mathbb R|$.
(And this is independent of the continuum hypothesis, the generalized hypothesis, and the Axiom of Choice).