- #1

Fredrik

Staff Emeritus

Science Advisor

Gold Member

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## Main Question or Discussion Point

Does the axiom of choice make the class of all sets bigger or smaller? Does it perhaps bring new sets into the universe and kick others out of it at the same time? (Maybe in a way that ensures that the first question doesn't make sense?)

The AC gives us permission to construct certain sets from other sets. This suggests that it makes the set-theoretic "universe"

On the other hand, it's equivalent to Zorn's lemma, which implies things like "every vector space has a basis", so it seems that the AC is what ensures that certain annoying sets (like vector spaces that don't have a basis) are

The AC gives us permission to construct certain sets from other sets. This suggests that it makes the set-theoretic "universe"

*bigger*.On the other hand, it's equivalent to Zorn's lemma, which implies things like "every vector space has a basis", so it seems that the AC is what ensures that certain annoying sets (like vector spaces that don't have a basis) are

*not*included in the set-theoretic universe. This seems to suggest that the AC makes the set-theoretic universe*smaller*, not bigger.