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I don't find the AC intuitive at all. It assumes that a choice function exists for any set of non-empty sets, including infinite ones. How can anyone have an in

I don't find the AC intuitive at all. It assumes that a choice function exists for any set of non-empty sets, including infinite ones. How can anyone have an intuition for such things? Except of course when the intuition is built upon taking finite cases to some limit. But in those cases the AC is reduced to a theorem, making it completely unnecessary.

v64 9y ago
I once had a homework assignment to attempt to construct the choice function for the set made up of all non-empty subsets of the reals. A lot of intuitive ideas come to mind and it's a good exercise to show why they don't work. You should feel weird about the axiom of choice after working through a few.
cgmg 9y ago
AC is equivalent to the statement: The Cartesian product of any family of nonempty sets is nonempty. This is very intuitive.
- cuspycode 9y ago
  So how does anyone form an intuition for even defining the Cartesian product of infinite sets in general, without assuming the AC in the first place?
  - xamuel 9y ago
    Consider something like the Cartesian product of countably infinitely many copies of the naturals: this is nothing more than the set of all countably-infinitary tuples (n1,n2,...). It's obviously nonempty, e.g. because it contains (0,0,...). It seems intuitively obvious that similar reasoning should make any arbitrary Cartesian product of nonempty sets be empty.
    - danharaj 9y ago
      That reasoning applies only to products of an arbitrary family of the same set, provided you know an element of that set. That is an exceptionally simple special case, don't you think?
    - cuspycode 9y ago
      Yes, that particular example is intuitive. But I don't see how a similar reasoning should apply to arbitrary infinite sets however. That's a huge intuitive leap in my opinion, and it's one that I can't follow.
      - xamuel 9y ago
        Ok, you're presenting to an audience consisting of lots of tables, each with at least one person at it. You say: "Will every table please have a representative stand up." To question the AC is to suggest that for some configuration of (sufficiently many) tables, it's impossible for the audience to fulfill your request, on account of there not existing a function f(table)=representative. This sounds absurd, because obviously each table can independently choose a representative, and it isn't like they need to coordinate with the other tables in order to fulfill your request.
        cuspycode 9y ago
        "Lots of tables" is the finite case, for which the AC is a theorem. An infinite set of tables is qualitatively different. An arbitrary f(table) function will have to contain an infinite amount of information, which is beyond anything my intuition can grasp. For f(table) functions that contain finite information, the intuition is easy and the AC is reduced to a theorem.
- danharaj 9y ago
  It's only intuitive if you take cartesian products infinite families for granted. I think it's only considered intuitive because when people think "cartesian product" their model is finite families.
nerdponx 9y ago
Do you have an example of a union without a known choice function?

v64 9y ago

I once had a homework assignment to attempt to construct the choice function for the set made up of all non-empty subsets of the reals. A lot of intuitive ideas come to mind and it's a good exercise to show why they don't work. You should feel weird about the axiom of choice after working through a few.

cgmg 9y ago

AC is equivalent to the statement: The Cartesian product of any family of nonempty sets is nonempty. This is very intuitive.

cuspycode 9y ago
So how does anyone form an intuition for even defining the Cartesian product of infinite sets in general, without assuming the AC in the first place?
- xamuel 9y ago
  Consider something like the Cartesian product of countably infinitely many copies of the naturals: this is nothing more than the set of all countably-infinitary tuples (n1,n2,...). It's obviously nonempty, e.g. because it contains (0,0,...). It seems intuitively obvious that similar reasoning should make any arbitrary Cartesian product of nonempty sets be empty.
  - danharaj 9y ago
    That reasoning applies only to products of an arbitrary family of the same set, provided you know an element of that set. That is an exceptionally simple special case, don't you think?
  - cuspycode 9y ago
    Yes, that particular example is intuitive. But I don't see how a similar reasoning should apply to arbitrary infinite sets however. That's a huge intuitive leap in my opinion, and it's one that I can't follow.
    - xamuel 9y ago
      Ok, you're presenting to an audience consisting of lots of tables, each with at least one person at it. You say: "Will every table please have a representative stand up." To question the AC is to suggest that for some configuration of (sufficiently many) tables, it's impossible for the audience to fulfill your request, on account of there not existing a function f(table)=representative. This sounds absurd, because obviously each table can independently choose a representative, and it isn't like they need to coordinate with the other tables in order to fulfill your request.
      - cuspycode 9y ago
        "Lots of tables" is the finite case, for which the AC is a theorem. An infinite set of tables is qualitatively different. An arbitrary f(table) function will have to contain an infinite amount of information, which is beyond anything my intuition can grasp. For f(table) functions that contain finite information, the intuition is easy and the AC is reduced to a theorem.
danharaj 9y ago
It's only intuitive if you take cartesian products infinite families for granted. I think it's only considered intuitive because when people think "cartesian product" their model is finite families.

nerdponx 9y ago

Do you have an example of a union without a known choice function?