It is known that the Dedekind-finite cardinals are closed under addition and multiplication, so one may do arithmetic in them, as opposed to only natural numbers.

How much can those two arithmetics be different? For example, can there be a Diophantine equation which is not solvable in the natural numbers but solvable in the Dedekind-finite cardinals? Can there be two nonempty Dedekind finite sets $A,B$ such that $|A|^2=2|B|^2$?

verydifferent. Sageev proved that assuming an inaccessible cardinal exists, there is an extension of the universe where all Dedekind-finite cardinals are comparable (in a nontrivial way, of course) and the axiom of choice for families of finite sets hold. Then, by a theorem of Ellentuck, it follows that the Dedekind-finite cardinals are a model of true arithmetic, if I remember correctly. $\endgroup$ – Asaf Karagila Apr 7 '15 at 9:57injectivesequences of a Dedekind-finite set, is also Dedekind-finite, so there is some weak exponentiation to be done here. But it seems to be closer to factorial than to exponentiation. So in short, I have no idea.) $\endgroup$ – Asaf Karagila Apr 7 '15 at 10:20