For a non-empty set $A$ let $A'$ denote the Cartesian product of $A$ with itself taken denumerably many times.
Now given a set $S$ whose cardinality is strictly greater than the cardinality of continuum (i.e., $\Bbb R$), in which cases do we have ${\rm card}(S')={\rm card}(S)$ and in which cases do we have ${\rm card}(S')\ne{\rm card}(S)$?
Are you reading Lang's algebra?On Page 893 of GTM211, Exercise 13.Lang remarked that the grapevine communicates to him that according to Solovay,the answer is "card S does not always equal to card S'."Maybe you can refer to Solovay's research to find the answer. I hope this is helpful to you.