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.
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!