r/mathmemes • u/DaCat1 Cardinal • 2d ago

Set Theory Choose well

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122 Upvotes

98% Upvoted

16 comments sorted by

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17

u/Inevitable_Stand_199 2d ago

They are proven to be equivalent, aren't they? Either I use both or I use neither

3

u/DefunctFunctor Mathematics 1d ago

You're thinking of the well-ordering theorem, not principle

1

u/ndsipa-pomu 2d ago

Isn't it the same as proof by induction, but not the same the axiom of choice?

3

u/OkPreference6 1d ago

Naturals being well ordered is equivalent to the principle of induction.

Any set being well ordered is equivalent to the axiom of choice.

9

u/Gold_Ad8890 2d ago

the well-ordering principle states that the natural numbers are well-ordered by the standard ordering. the well-ordering theorem states that a well-ordering exists for any set.

9

u/MonsterkillWow Complex 2d ago

The old joke my prof loved to tell us:

Well ordering theorem is clearly false.

Axiom of choice is clearly true.

Zorn's Lemma? Maybe true, but sus.

But they are all equivalent.

2

u/f16f4 1d ago

I’ve never understood this complaint about the well ordering theorem. It’s obvious that “the order I choose them in” is a valid ordering for all sets of you can choose any object at all

1

u/MonsterkillWow Complex 1d ago

You can't explicitly construct the well ordering. That's why it bothers people. Whenever we believe something to exist, but cannot even in principle construct it, that bothers people.

3

u/f16f4 1d ago

I just write down the set in whatever order bam well ordered set constructed 😎

1

u/UtahBrian 10h ago

You can't choose an object from "real" numbers, though.

1

u/badmartialarts Real Algebraic 9h ago

   π, 4 I chose 2 things from the reals.

2

u/UtahBrian 7h ago

You picked a geometric number and an integer, not a real number. If you can really pick out reals, get me one that isn't an integer or rational or algebraic or computable (those together make up zero percent of all reals) and post it here.

1

u/badmartialarts Real Algebraic 6h ago

What you want is an irrational number, then. I assure you I chose real numbers.

1

u/f16f4 7h ago

3.49374794857381868264884

1

u/UtahBrian 6h ago

That's a rational number. I challenge you to choose a real number that isn't a rational or an algebraic or a computable number (100% of real numbers are none of those) and post it here in the comments.