Rationality is not an inscribed property but rather a natural consequence of observing the order in place.
For example, the table of elements expresses the properties of integers as inherent to reality. As such, it is rational and rationally-ordered.
This is not so much “philosophy” as it is “logic”, it is the conclusion of philosophy and mathematics and the subsequent origin of the computational reality.
If you believe logic and rationality exist, and you believe that you are part of the universe, then the universe is rational, because you are the universe being rational.
> Oh, and some of them (either nearly or exactly half) are negative.
Well, it doesn't quite work that way. Yes, you can put the negative integers in 1:1 correspondence with the positive integers, but you can also put them in 1:1 correspondence with the positive integers that are divisible by 2, or 3, or 3,000... or, for that matter, the negative integers that are divisible by three thousand. All of those sets have the same cardinality (often called "countably infinite").
There are, however, provably more real numbers than there are integers.
It is unknown whether there are any sets whose cardinality lies between the cardinality of the set of integers and the cardinality of the set of real numbers.
"Counting stuff" can be generalised to infinite sets in different ways: cardinality is one (arguably the most common) of them, but natural density or measure are others.
For example, the table of elements expresses the properties of integers as inherent to reality. As such, it is rational and rationally-ordered.
This is not so much “philosophy” as it is “logic”, it is the conclusion of philosophy and mathematics and the subsequent origin of the computational reality.