In 'Foundations of Constructive Analysis', in the notes at the end of the first chapter, Bishop poses an apparent paradox as an exercise for the reader:

Since every sequence of rational numbers can presumably be described by a phrase in the English language, and since the phrases in the English language can be sequentially ordered, the regular sequences of rational numbers can be sequentially ordered, in contradiction to Theorem 1.

'Theorem 1,' for those of you without the book open in front of you, is essentially Cantor's diagonal proof applied to Bishop's construction of the reals.

Is the exit here that the notion that phrases are 1-1 with sequences is misleading? Aren't some phrases schemata for infinite collections of sequences? In another direction, can't there be Chaitin-esque sequences that are legitimate but, in fact, not expressible as phrases in English? If you generate a sequence with the help of a physical source of entropy (and scale the values so that they meet the definition of a regular sequence) is it admissible in Bishop's definition or not?

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