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Is it impossible to prove the continuum hypothesis using only axioms that are "intuitively true"?

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"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

That's what your intuition can give you.
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I'm not an expert in this area, but this general idea that anything in math can be described as "intuitively true" is problematic IMO. It's not well defined, it's insanely subjective, and every math expert and enthusiast can come up with a ton of examples of things that seem intuitively true but reality is the opposite.

But if you just want *any* axiom that can prove CH, axiom of determinacy (which is incompatible with axiom of choice) can prove CH (but not GCH).
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What rules it out is the fact that 'completed infinities' produce contradictions, such as Banach-Tarski. This has always been the case, but mathematicians chose to overlook this fact from around 1907 on.

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