I am poking on a proof of the subject theorem. Given sets $A_i$ and injections $f_i:A_i\to A_{1-i}$, $i\in \{0,1\}$, theorem defines a bijection $b$ between $A_0$ and $A_1$. $b$ uses an auxilary function
$degree (x:A_i) := \cases{ 1+degree(f_{1-i}^{-1}(x)), x\in f_{1-i}(A_{1-i}) \cr 0}$
“degree” may loop, denote this $\bot$. Then $b$ returns some value$\neq\bot$ on $\bot$ and is not constant in general. $b$ solves the halting problem. So, it is possible? (I assume that the codomain of “degree” is a flat domain.) (I have not posted this on cstheory because this is not a research-level question.)