Quotient Group R/Q

>>8322721
Bump. First thread on /sci/ with potential since the Egyptian math guy.

>>8322789
There's nothing to discuss, I just find it silly to have a non-measurable set where you couldn't possibly construct a single element (because Q is dense in R).

>>8322721
Bit rusty on abstract algebra. Doesn't an operation need to be defined first.

>>8322721
You can get individual elements of R/Q. In fact, you can construct sets of uncountably many elements of R/Q. What you cannot construct (without the axiom of choice) is a set that has a representative from every class in R/Q. You can get uncountable sets of representatives, but the trouble is getting a set with a representative from every class.

Look, nonmeasurable sets are weird. I'd guess that you're trying to apply intuitions from `nice' cases and `nice' objects. That won't really work when you deal with weird results related to the axiom of choice. You just have to get used to the fact that things are quite different.

>>8322721
Of course you can construct an element of R/Q. For example you can take the class [0] of 0 (whose preimage under the quotient map is the whole of Q).

>>8322859
The group operation is already there: it's the sum.

>>8322861
>>8322885
Ah yes, you're right. I was assuming R/Q was the same as "remainders of R when divided by Q", prolly got the notation mixed up.

So sqrt(2)/2 f.e. is in V, right (or rather could be)?

>>8322721
I always thought of it as the kind of semantic paradox you run into when you don't understand how you think, and only look at what you think. All narrative is a recursion, and you bring in all kinds of training for your inference that you then have to untrain when make up a narrative that is not justified by the universe your narrative is now occupying.
No greater place that this shows up as in set theory, where sets and subsets use the inference of deduction that says large things cannot fit in small things to say a set cannot be smaller than its subset, or object permanance that says any subset I make from the superset, I can pick at least one member of that set...
But guess what? You can't always do that. The narrative objects you are using determine what their arrangements can be as much as the perspective from which you view them determines what the objects can be. If you look at the same math as a recursion of set building, and also allow that recursion to go both ways, then these things sort themselves out.
For example: inference is felt when the gives of a deduction are taken as givens only after the conclusion, so that even nonsense premises can engender the feeling of inference. Deduction moves from conclusion to givens as much as it does givens to conclusions.
The axiom of choice brings in things from universes that the one you are working in might not support, just as the nesting of sets brings in abstractions you have learned in a spatial universe that might not be justified in the mathematical universe in which you are in.
tl;dr: sets don’t behave like sets when measuring space that doesn’t behave like space. No one likes the Axiom of Choice

>>8322894
Correct, sqrt{2}/2 is a fine representative for one of the classes in R/Q and therefore could be in V.

>>8322898
>No one likes the Axiom of Choice

That's demonstrably false. Many people not only like it, but they believe that it's a natural axiom to have and use.

Much of the rest of your post is true, though. We build intuitions about `measure' with nice cases (like intervals in R or circles, squares, etc in R^2). When you move to another measure like the Lebesgue measure, those intuitions cannot be fully trusted (especially when you throw in the AoC). Instead, you'll have to retrain your intuitions when you're working in this new "universe".

I must be totally misunderstanding things here, isn't [math]\mathbb{Q}+\sqrt{2}[/math], i.e. the set of all real numbers which are some rational plus [math]\sqrt{2}[/math], an element of the set? Where's the trickery here?

>>8323321
Never mind, I just looked up the Vitali set. I thought OP was saying that elements of [math]\mathbb{R/Q}[/math] couldn't be constructed.

>>8322898
But it is without Choice that extremely weird shit comes out (infinite Dedekind-finite sets, fucked up cardinals, surjective functions might not have right inverses etc.). Moreover, in Gödel's constructible universe, which is a pretty intuitive way to construct a model of ZF, has Choice true in it.

>>8323337
Also, without Choice, an infinite product of empty sets may be nonempty. In fact, Choice is equivalent to the proposition that such product is empty.

>>8323603
>Also, without Choice, an infinite product of nonempty sets may be empty.