[thaumasiotes]

The rationals are a totally ordered set. There definitely is a natural, canonical ordering to the rationals. It's the same numeric-magnitude metric we use all the time. 1/3 is less than 2/3.

That ordering doesn't have the property that all sets of rationals contain a least element, or that any rational has a successor rational. (That would make them "well ordered".) But it's a natural ordering.

>> The argument as to why the irrational numbers are uncountable and the rationals are countable is more involved than what you've made out. But very simply you can think of it as you need an infinite string of digits to describe each irrational number, but each rational number can be written as two finite strings of digits (in the form A/B, where A and B are integers). So to write our the irrationals you have an infinite number of strings, where each string is also infinitely long, while with the rationals you have an infinite number of strings, but each string is finite.

This argument doesn't actually work. If there were only a countable number of irrational numbers, you could specify them all fully by doing no more than a countable amount of work, even stipulating that describing a single irrational number requires listing a countably infinite number of digits.