Assuming the axiom of choice there exist subsets of reals that are not measurable...

Assuming the axiom of choice there exist subsets of reals that are not measurable. But without AC we can construct models in which every subset of reals is measurable. So why do we stick to ZFC, is knowing that every surjection in Set splits or that a ball can be cut into 5 pieces and reassembled into two balls identical as the one we started with more important that being able to measure every subset of reals?

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en.wikipedia.org/wiki/Graph_canonization
en.m.wikipedia.org/wiki/Axiom_of_choice#Equivalents
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Axiom of choice was a mistake.

Choicefags BTFO

>Assuming the axiom of choice there exist subsets of reals that are not measurable. But without AC we can construct models in which every subset of reals is measurable. So why do we stick to ZFC, is knowing that every surjection in Set splits or that a ball can be cut into 5 pieces and reassembled into two balls identical as the one we started with more important that being able to measure every subset of reals?
What are your preferred axioms?

ZF+countable choice

You obviously don't understand what an axiom is, so why do you imagine you have anything interesting to contribute to this discussion?

I do, I'm just wondering why AC is preferred over other axioms

It's preferred by people who prefer it, and not by people who don't. There's no law that says you have to use an axiom, you use the axioms that are useful for the type of math you want to do.

Its impossible to order some sets. How would select an element from them?

>How would select an element from them?
Just choose any element from the set, the set doesn't have to be orderable for you to be able to take a countable subset of it.

"It's possible to order a set"
"It's possible to find the smallest member of a set"

Doesn't this assume that every member of a set must be a number?

Yet set theory is used all the time in every part of maths.

E.g. a graph is a set of vertices with edges between some of them.
Which point is the "smallest" in a graph? there isn't such a thing. "smallest" doesn't have meaning with respect to a vertex.

Is the whole of mathematics built on shakey foundations like this?

There exists a bijection which maps the vertices and edges of a graph to a subset of natural numbers. You may then apply operations, such as sorting, on the graph in the context of this function. Furthermore, for any graph there exists a unique canonical bijection, so you don't have to rely on choosing a function randomly. Assuming a canonical form, there indeed is a smallest vertex in a graph.

ok so how do you decide which vertex is number 0?
If you could just as easily and validly decide that another vertex should be number 0 then you can't say that either vertex is smaller than the other.

en.wikipedia.org/wiki/Graph_canonization

Congrats you just used choice

>en.wikipedia.org/wiki/Graph_canonization
so you're admitting that it is arbitrary and you could just as easily label a different vertex as 0 and so find a different "smallest" vertex for the same graph.

The choice of canonization method is arbitrary, yes. However, you can't get a different smallest vertex by "reordering" the graph.
As I said, the elements are sorted in the context of a chosen function. It so happens that there exists a canonical function for this, which makes it all very useful.

All these people avoiding pointing out the obvious. It should be clear by its very name that the "Axiom of choice" is preferred.

But it's impossible to choose an element from uncountable (or bigger) family of sets

>so you're admitting that it is arbitrary
Nobody claimed it isn't arbitrary, the point is it can be ordered. There are loads of different orders on Z too.
Choice = The Cartesian product of any family of nonempty sets is nonempty.

>There exists a bijection which maps the vertices and edges of a graph to a subset of natural numbers.
[citation needed]

>But it's impossible to choose an element from uncountable (or bigger) family of sets
Why?

>Congrats you just used choice
How so?

You chose to post stupid shit.

>You chose to post stupid shit.
Can you elaborate?

>There exists a bijection which maps the vertices and edges of a graph to a subset of natural numbers.
But that's wrong.

>Congrats you just used choice
But that's wrong.

Because they're too large and it can be done in finite time. And what does AC give us in first place? Banach-Tarski paradox? Well-ordering theorem? "Hey, there is well-ordering of reals, but it's impossible to construct it, you won't know anything about it other than it exists. Every vector space has a basis. You can't construct a basis of [math]\mathbf{R}^\infty[/math] or of reals as a vector space over rationals but hey, there exist one". AC is only a source of pathological counterexamples and obstructs doing real maths

I could just as easily claim it's "more important" that every vector space have a basis, or the Cartesian product of nonempty sets is also nonempty.

en.m.wikipedia.org/wiki/Axiom_of_choice#Equivalents

>AC is only a source of pathological counterexamples and obstructs doing real maths
You're retarded. Just because it confuses (non-mathematician) brainlets like you doesn't mean it "obstructs" anything. There's a reason mathematicians use it when they want to.

But what's the point of knowing every vector space has a basis if you can't construct that basis? It's just like having a candy that you can never unwrap and eat. It's there, but it's hidden from your eyes and it can't be used for it's intended purpose, so what's the point?