denumerable

The axiom of choice asserts that this holds true even when the number of sets is uncountable (non-denumerable).

It asserts that you can pick a countable(denumerable using the terminology in the post) sequence from any infinite set.

We have designed algorithms which barely work on denumerable sets, but are unsuitable in more general spaces.

This notation is absolutely fine for someone who can infer that the support of the distribution of X is some denumerable set {x_i}, equipped with p.m.f.

In other words, since the leftmost path is an infinite string of zeros, it is clearly denumerable and can have one-to-one correspondence to natural numbers (ordinals rather than cardinals though).

At the same time, programming itself, like mathematics, is not a world of numbers or any of this, and is in fact the expression of complexity: the complexity of the sign as written or drafted by the mathematician, itself expressed in a non-denumerable set.

I've got to disagree on the intuitiveness of Cantor's diagonal theory, though; that was one of the most beautiful and intuitive proofs in set theory for me: make a list (denumerable) of decimally expanded reals; change the digit on the diagonal of each; you've now created a number not on the list: QED.

Neither here nor there, but FYI in English "software" is typically partitive and not denumerable, i.e.

It's just the way this proof is often presented that leads people to believe that it only proves that at least one is irrational; but once you accept that there is one irrational number, you can "index" it and add it to your denumerable list (pretend it's rational, say) and then run the diagonlization argument again, yielding yet another irrational number.