= Uncountability obstruction to enumerating infinite Cartesian products
{title2=$\{0,1\}^{\mathbb N}\text{ is uncountable}$}
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 $z_0,z_1,\ldots$. 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.
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