Löwenheim-Skolem gives you a countable elementarily equivalent submodel (assuming you're working in a theory in a countable language, otherwise it gives you an elementary substructure of the same cardinality of the language at best), but plenty of interesting properties of familiar mathematical objects cannot be captured by a first-order theory and are not preserved by elementary equivalence, completeness of the reals being the standard example
Yet the very notion of countability in ZFC, which is itself a first-order theory, is rendered completely relative by Löwenheim-Skolem. ZFC itself has a countable model.
If "plenty of interesting properties of familiar mathematical objects cannot be captured by a first-order theory" then that also undermines ZFC, which is a first-order theory.
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.
