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To the guy from many years ago who said the infinite product of nonempty sets is nonempty, yes you can restate the AC that way. Doesn’t make it obvious.

Personally I despise even countable choice. Look at the proof of finite choice. You have a choice sequence of cardinality 1. Now if we are assuming (n-1)-choice is true, we can form the n-choice sequence by invoking the existence of (n-1)-choice to get an (n-1)-choice sequence for the first (n-1) sets, and then choose 1 more to get an n-choice sequence. So its an inductive proof of n-choice.

If we are to believe that the AC is obvious, then we are supposed to believe that the above proof was a complete waste of time. It doesn’t feel that way, it feels like the above proof was a full proper proof of a full proper theorem. So I absolutely refuse to accept as obvious a *generalisation* of it!!

]]>Unfortunate Mathematical Names | The Everything Seminar

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