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Let $A$ denote a particular (fixed) algorithm to encode ordinals as real numbers. The exact technical description of $A$ is irrelevant for this question: it can be any algorithm that is mathematically ...

Let $\Sigma$ be a finite set or "alphabet", $\Sigma^*$ the free monoid on $\Sigma$ or set of "words". If $w,w'\in \Sigma^*$, write $w\leq w'$ when $w$ is a "subword" of $w'$, i.e., can be obtained by ...