Thread 16700006

>The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?
This is called a joke but it's actually true. Since all three are equivalent, we should reject all three as an outrage against common sense.

>>16700006 (OP)
>we should reject all three
And then what? Suddenly all your proofs in the very basics of any discipline become unreasonably complicated just for some “philosophical” reason
>no Hamel basis guaranteed for every vector space
>some free modules are suddenly not free anymore, have to come up with tedious criteria to see what went wrong
>basically any result about lattices (topologies on a set, group lattices, ring ideals, manifold atlases, etc etc) is thrown out the window and you have to develop a whole framework for each to decide if a maximal element exists
>the only area not affected is analysis because Schauder bases of TVS can rely on Dependent Choice instead

>>16700025
i think the retard in OP would consider the things that are possible with choice to be false anyway, those retards tend to be that way

>>16700025
>"if something's hard, it's not worth doing!
you are the drunkard searching for his keys under the streetlight when he dropped them in the park, because "I can see better over here"

>>16700073
now that's projection

>>16700025
>shit that does not matter
>shit that does not matter
>shit that does not matter
>shit that does not matter, but can use a weaker false axiom

>>16700073
What do I actually gain from this? I’ll tell you, fuck all. It’s like saying you can handwrite blindfolded. Who gives a shit?

>>16700172
you eliminate an obvious falsehood from mathematics

>>16700178
Obvious to whom? Gödel’s incompleteness theorems show that your statement is nonsensical. You absolutely can work in ZF without C, but ZFC is just as valid. And I can choose to work in either as per Gödel. And I choose to work in ZFC because I have shit to do. If you want to make your complicated for no reason, be my guest. But then good fucking luck getting any useful results beyond high school mathematics. You cannot even prove Stokes’ theorem without AC, so bye-bye multivariate calculus.

>>16700006 (OP)
They're only equivalent over ZF, if you start having finer foundations you can start discriminating them. Try without excluded middle to start with.

>>16700006 (OP)
i always laugh at mathematicians that unironically invoke the axiom of cope

>>16700025
>And then what? Suddenly all your proofs in the very basics of any discipline become unreasonably complicated just for some “philosophical” reason
you mean the introduction epsilon-delta proofs to obfuscate the method of calculus that was invented before such quackery ever existed?

>>16700413
Epsilon-delta proofs have nothing to do with AC. People inhabiting these threads are unironic drooling retards. Epsilon-delta proofs are constructive by definition. You literally need to construct an explicit delta for each epsilon.

>>16700424
>Epsilon-delta proofs have nothing to do with AC
didn't claim they do, retard