An intutive counterexample to the axiom of choice

Re:An intutive counterexample to the axiom of choice by boole1854 on Monday January 23, 2023 @10:58AM (#34490392)

Remember that I am not here trying to argue for the truth of the AC; I'm arguing against the author's argument. The author's argument, as I read it, relies on an intuition that "choosing" an element that is not-finitely-describable is problematic. However, the mathematical meaning of "choosing", in the context of the AC, is a statement about the mathematical existence of a certain function. To respond to the author's intuition requires only showing that not-finitely-describable domains are not a "problem" for the existence of functions. Maybe it is difficult to "choose", in some natural language sense, an input to f(x) = x + 1 such that the output x + 1 is a real number (e.g., you can't write down such an input one digit at a time on a piece of paper, even an infinitely long piece of paper). But this does not mean it is problematic for the function to exist in the mathematical sense.

[ Parent | on HN ]

Re:An intutive counterexample to the axiom of choice by lisper on Monday January 23, 2023 @01:05PM (#34492525)

FYI, I am the author.

> Remember that I am not here trying to argue for the truth of the AC; I'm arguing against the author's argument.

Yep, I get that.

> The author's argument, as I read it, relies on an intuition that "choosing" an element that is not-finitely-describable is problematic.

Not quite. Choosing an element from a set where all of the members are not-finitely-describable is problematic.

> However, the mathematical meaning of "choosing", in the context of the AC, is a statement about the mathematical existence of a certain function.

Yep, I get that.

> To respond to the author's intuition requires only showing that not-finitely-describable domains are not a "problem" for the existence of functions.

No, you would have to show that defining a function on this particular set is not a problem. Note that this is not the same thing as defining a function on elements of this set. That is obviously not a problem. But defining a (non-trivial) function on the set itself is.

[ Parent | on HN ]

Re:An intutive counterexample to the axiom of choice by riversflow on Monday January 23, 2023 @03:03PM (#34494186)

I just don’t see what describability, or as jiggawatts calls it, naming, has to do with AC. AC is built upon an indexed set[1]. That way you can just reference the index. It seems extremely contrived to me to argue that being finitely nameable has any bearing on underlying mathematical principle, nothing about any form of infinity is relatable to the physical confines of this universe, of which language is a member, but the concept of infinite is very useful.

[1]https://en.wikipedia.org/wiki/Axiom_of_choice

[ Parent | on HN ]

Re:An intutive counterexample to the axiom of choice by lisper on Tuesday January 24, 2023 @10:29AM (#34504765)

> I just don’t see what describability, or as jiggawatts calls it, naming, has to do with AC.

It doesn't have anything to do with AoC per se, it has to do (I claim) with people's intuitions about AoC. Specifically, an infinite set of arbitrary subsets of the set of indescribable reals is something for which a reasonable person might be a tad suspicious about postulating the existence of a choice function.

[ Parent | on HN ]