ZFC was specifically designed to be immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox.
You are arguing that the ground moves to perfectly fit the shape of a puddle.
Zermelo was one of the first to reference "Cantor's theorem" in his papers.
These paradoxes do not occur in higher-order logic. You don't need ZFC or any first-order set theory for that. (Also, your comment doesn't address the sentence I quoted.)
I don't understand what you mean. Higher-order logic does have classical negation. First-order PA tries to approximate the (second-order) induction axiom by replacing it with an infinite axiom schema, but that doesn't rule out non-standard numbers.
Cantor's proofs showing that Z and Q are countable and R is uncountable.
Cantor's "diagonalization proof" showed that.
Turing extended to the computable numbers K, which can be conceptualized as a number where you can write a f(n) that returns the nth digit in a number.
The reals numbers are un-computable almost everywhere, this property holds for all real numbers in a set except a subset of measure zero, the computable reals K which is Aleph Zero, a countable infinity.
The set of computable reals is only as big as N, and can be mapped to N.
It is not 'non-standard numbers' that are inaccessible, it is most of the real line is inaccessible to any algorithm.
You are arguing that the ground moves to perfectly fit the shape of a puddle.
Zermelo was one of the first to reference "Cantor's theorem" in his papers.