 Uncountability obstruction to enumerating infinite Cartesian products

An infinite Cartesian product of finite sets need not admit any complete list enumeration. For the repeated binary factor, suppose the output lists all sequences as $z_0,z_1,\dots$. Define a sequence by $b_i=1-z_i(i)$. It is a valid binary sequence but differs from the $i$-th output at coordinate $i$, a contradiction. This is Cantor's diagonal argument. Consequently fair binary products can recursively enumerate the choices from finitely many possibly infinite lists, but cannot solve the unrestricted infinite-family specification. A finite-family algorithm's restriction must be explicit.