TITLE: Dedekind-finite arithmetic vs natural numbers arithmetic
QUESTION [8 upvotes]: 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 $|A|^2=2|B|^2$?

REPLY [4 votes]: A joint paper by David Blair, Paul Howard, and me, "Divisibility of Dedekind-Finite Sets" (available at http://www.math.lsa.umich.edu/~ablass/ddiv.pdf and published in J. Math. Logic 5 (2005) 49-85) contains information of the sort you want about some rather specific linear diophantine equations.