why is the assertion $\aleph_1=2^{\aleph_0}$ referred to as a hypothesis, whereas $$\forall \alpha( S_\alpha \ne \varnothing) \Rightarrow \prod_\alpha S_\alpha \ne \varnothing$$ is called an axiom? is this merely a historical 'accident' or does it indicate a difference in the way these statements are interpreted?
2026-04-04 02:25:28.1775269528
informal semantics regarding CH and AC
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The axiom of choice was something people had used without always noting that there is an assumption to be made in order to justify making infinitely many choices at once.
On the other hand, Cantor felt that there shouldn't be intermediate cardinals between the naturals and the reals, so he hypothesized that this is the case and spent a considerable amount of energy trying to prove that hypothesis.
You could also ask why Zorn's Lemma is not "Zorn's Axiom", but again, this is just a matter of naming something according to how it was born. If it was born as an assumption, it is an axiom or a postulate. If it was born as a theorem (namely a statement that was proved) it will be a lemma or theorem or whatever. If it was born as a question which was open for quite some time, it will be an hypothesis or a conjecture.