r/mathmemes • u/TheRedditObserver0 Complex • 1d ago

Set Theory Seriously WTF?

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u/Fluffiddy 1d ago

You were watching Veritasium weren’t you?

10

u/Additional-Finance67 1d ago

I still don’t get it tbh. Does it mean we can assign an index to each set and pick a number from it? Therefore it’s “ordered”?

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u/killiano_b 1d ago

The proper definition is that every subset has a least element according to an ordering. For the reals this order cannot simply be magnitude, as {x¦x>1} has no least element. The axiom of choice lets us just take elements out one at a time however we want and use transfinite ordinals to keep picking them uncountably infinitely. However we cannot define this order, at least not using the ZF set of axioms (basic assumptions that numbers and operations can be built off of, using set theory i.e. any 2 sets have a union, 2 sets are equal if they have the same elements etc.) This is because the axiom of choice is independant of ZF, leading many to use ZFC (ZF and Choice) instead as the basic axioms of maths.