[vessenes]

I don’t know the answers to all of your questions - but I believe you’d benefit from some mathematical history books around the formalization of the real analysis; I’m not the best person to give you that history.

A couple comments, though - first, all mathematics is linguistics and arguably it is all sleight of hand - that said the word “gaps” that you’ve rightly pointed out is vague is a journalists word standing in for a variety of concepts at different times.

The existence of the irrationals themselves were a secret in ancient greece - and hence known for thousands of years, but the structure of the irrationals has not been well understood until quite recently.

To talk precisely about these gaps, if you’re not a mathematical historian, you have to borrow terminology from the tools that were used to describe and formalize the irrationals -> if former concepts about the lines sound hand-wavy to you, it is because they WERE handwavy. And this handwaviness is about infinity as well, the two are intimately connected. In modern terms, the measure of the rationals across any subset of the (real) number line is zero - that is the meaning of the “gaps”. There is, between any two rationals, a great unending sea where if you were to choose a point completely at random, the odds of that point being another rational is zero.

EDIT: for a light but engaging read about topics like this, David Foster Wallace’s Everything and More is excellent.

[bhk]

Thanks for the references.

I think you will agree that the bulk of your comment employs a post-set-theory nomenclature.

Regarding "if you were to choose a point completely at random, the odds of that point being another rational is zero", I ponder the question of how one might casually "choose" a value with infinite entropy.