I'd like to point out this is at odds with most of the mathematical community's interpretation. Caveat: mathematics itself does not have concerns about "interpretation" and your interpretation is completely valid of course.
My interpretation is that powerful axioms often prevent theories which asks for "too much" because the power of the axioms then can form contradictions. For example asking for everything to be a set => Russel's paradox and asking for all sets to be measurable => Banach tarski type contradiction.
Side note for mathematically trained peeps: Of course assuming all sets to be measurable and the measure to be translation and rotationally invariant leads to much easier contradictions with AOC. Banach-Tarski gives a finite decomposition contradiction (vs say countable). Also not a analyst myself, so take everything here with a grain of salt. But honestly analysis is basically completely not possible without excluded middle so I think analysts would have opinions more on classical side than myself.
My interpretation is that powerful axioms often prevent theories which asks for "too much" because the power of the axioms then can form contradictions. For example asking for everything to be a set => Russel's paradox and asking for all sets to be measurable => Banach tarski type contradiction.
Side note for mathematically trained peeps: Of course assuming all sets to be measurable and the measure to be translation and rotationally invariant leads to much easier contradictions with AOC. Banach-Tarski gives a finite decomposition contradiction (vs say countable). Also not a analyst myself, so take everything here with a grain of salt. But honestly analysis is basically completely not possible without excluded middle so I think analysts would have opinions more on classical side than myself.