Do you "belive" in the axiom of choice?
I think we can get an ice cream with infinite flavors. xD
>>7782374
No, the axiom of V=L is the only logical one.
>>7782374
You're basically forced to believe in it if you want the WOP to be true, and if you aren't pants-on-head retarded you should intuitively expect that it is
why would you ever have to believe in AoC?
you accept it as valid or you don't. belief doesnt enter the picture
>Axiom of Choide
You misspelled "Axiom of Chode". Hope that helps. Have a nice day^H^H^Hsage.
>>7782374
No but I believe that every vector space has a basis.
>>7782393
>intuitively expect
Translation: I really wish it were true, and so it is.
>>7782450
Doesn't axiom of choice only give you that every basis has the same dimension (i.e. without AoC you get that it's possible to give some vector spaces multiple basis with different dimension)?
Does it bother you that a unique dimension is obviously not the case with topological basis for topologies?
>>7782555
More than that, it lets you declare that every vector space actually has a basis. In cases such as R considered as a vector space over Q, this is the only way of saying that there's a basis at all.
>>7782561
What is an example of such a basis?
>>7782568
Here's one
http://mathworld.wolfram.com/HamelBasis.html
>>7782568
You can give any Hilbert space an ONB. Also I'm pretty sure you don't need AoC to say that R is an infinite dimensional vector space over Q.
>>7782574
The construction I learned relies on Zorn's lemma which is equivalent to AoC.
>>7782580
Field extensions are vector spaces. It's easy to give an infinite independent set of Q-vectors in R (roots of primes, powers of pi...) but you can't give an actual base without the AoC of course.
>>7782374
Can't even use induction without axiom of choice.
>>7782393
>WOP
but it is obviously wrong, even intuitively.
>>7782820
Wtf. Of course you can
>>7782390
But that implies choice...
I only believe in the Univalence axiom.
>>7783073
>we may consider them elements of some universe type U
maybe we can, maybe we can't. This claim needs a proof.
>>7782393
I'm not trolling when I say that WOP is fucking retarded and anyone that believes in the Axiom of Choice is saying "Banach-Tarski makes more sense to me than the idea that we can't just assume a choice function exists for every set just because that would be useful."
>>7782374
>my pumpkin simply has chest pains
10/10
>do you believe in axiom X
I don't care desu.
All it's conclusions seem to work, so I guess it's true.
>>7783087
this desu senpai
the stronger the assumption, the stronger the result, but what's the point?
What bothers me more is how sets are defined, and the definition is abused.
>>7783096
The definition isn't abused. Pretty much all the stupid paradoxical shit in mathematics is either based on the Axiom of Choice or comes from naive set theory and the idea that all collections of elements are sets.
>>7783096
>What bothers me more is how sets are defined, and the definition is abused.
sets are not defined in set theory. but then, set theory is formalizes what people who formalize desire to do with sets. you can invent your won set theory if you do not share their visions of manipulations of sets.
>>7783112
>you can invent your won set theory if you do not share their visions of manipulations of sets.
when I do, people insult me
>le Banach Tarski "paradox" is too unintuitive
Holy fuck, kill yourselves if this is all you can spout.
>>7783124
then you don't know what you're doing
>>7783081
All types are elements of some universe.
That's basically the definition of a type.
See: p33 of HoTT.
>>7783171
But can we consider them? Which lemma number is that?
>>7782374
I think Axiom of Regularity is a much more interesting thing. After all, you can prove that sets don't contain themselves with it, which is much more "natural" result than some of the implications of the axiom of choice (though well ordering is a really nice thing to have)
>>7782587
im just wondering
is there an easy proof that roots of primes are independent over Q?
>>7783280
Basically, if they weren't you would have square roots of different primes generating the same extension of Q. You could then argue (say by the use of discriminants) that this isn't possible.
>>7783217
What I like about Regularity is that even if a model satisfies it, there may still be an infinite descending chain [math] x_0 \ni x_1 \ni x_2 \ni \ldots [/math] -- it is only that such a chain cannot be definable.
bump for interesting thread
>>7782852
I could be wrong but, doesn't axiom of choice get invoked when proving the well ordering theorem?
I mean we had this thread about 6 weeks ago with lots of answers - I'm not gonna rewrite my same answer here again (too bad the archive is down).
>>7783073
>>7783171
Do you guys have any feel for what a computation/constuctive interpretation for a term of the Univalence axiom type (I think that's what they are looking for, if I understand correction) could even look like?
>>7783918
>induction schema
http://math.stackexchange.com/questions/432293/well-ordering-and-mathematical-induction
and so is well-ordering.
>>7783937
>Sugar packet is the volume of the coffee cup.
Yea, ok.
>>7783125
Lmao at that picture
