Is mathematics mostly chaos or mostly order?

I've never made peace with Cantor's diagonaliztion argument because listing real numbers on the right side (natural number lhs for the mapping) is giving a real number including transedentals that pre-bakes in a kind of undefined infinite.

Maybe it's the idea of a completed infinity that's my problem; maybe it's the fact I don't understand how to define (or forgot cauchy sequences in detail) an arbitrary real.

In short, if reals are a confusing you can only tie yourself up in knots using confusing.

Sigh - wish I could do better!

As someone who also has never fully made his peace with the diagonality argument, but just chosen to accept it as true, as a given, this kind of bumps up against an interesting implication of different cardinalities of infinity.

To precisely define an arbitrary real you'd need some kind of finite string that uniquely identifies that real number. Finite strings can be mapped, 1 to 1, to natural numbers. Therefore there can't be a finite string for any real number that uniquely identifies it. Otherwise we'd have a mapping between natural numbers and real numbers.

In fact, the set of uniquely identifiable real numbers is a countable subset of real numbers. [1]

Somehow, this realization has helped me make peace with the uncountability of real numbers.

[1] Sorry if use words like "unique", "identify", "define" in not quite the right way. I hope the meaning I'm going for comes across.

But what about where you don't look? Either you take the orthodox axiomatic view that Real numbers are there too, or you take the constuctivist or finitist (or perhaps quantum mechanical?) view that nothing is there until you look, because the act of looking is the same as the act of creation.