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!

Maybe you'd prefer a purely set-theoretic one:

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Let R be a set. Let S be a set of all subsets of R.

We want to prove that |S| > |R|, by proving that a bijective function from R to S cannot exist. We will do that by assuming that it can, and then deriving a contradiction.

Assume there is a bijective function f : R -> S. Define D = { r ∈ R | r ∉ f(r) }. Since f is a bijection, there exists some r₀ ∈ R such that f(r₀) = D.

However, by the definition of D, we have:

- If r₀ ∈ D, then r₀ ∉ f(r₀) = D, which is a contradiction.

- If r₀ ∉ D, then r₀ ∈ f(r₀) = D, which is also a contradiction.

Therefore, our assumption that there exists a bijective function f : R -> S must be false.

Therefore, |S| > |R|.