Is mathematics mostly chaos or mostly order?

If I had a semester or two of free time I'd love to hit this subject again. I once told my math prof (logician) who made a comment about transfinite cardinals: careful it's powerful but it's power from the devil. I half regret that comment in retrospect.

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!