Thanks for this great answer. I have a couple of follow-up questions that anybody should feel free to jump in and answer.
The underlying reason is that if you imagine a Platonic realm where all abstractions allegedly exist, the problem is that there are actually multiple abstractions [“models”] compatible with ZF, but different from each other in many important ways. In some of them, Axiom of Choice is true. In others, it is false. This is what it means that Axiom of Choice can be neither proved nor disproved in ZF.
There are models where the AC is specifically false? I’ve been told that AC can be formulated as “the cartesian product of any collection of sets (even an infinite collection) is non-empty”, but I’m having trouble picturing something a collection of things I call “sets” failing to satisfy this property. Are the models referred to here ones “sets” are replaced by totally unrelated mathematical objects that just happen to satisfy the ZF axioms?
If I understand it correctly, (you believe) you have an intuition of “sets”, and you are going to judge any axiom depending on whether it is compatible with your intuition or not. ZFC is compatible with your intuition, negation of AC is not.
And maybe your intuition is also underspecified—that is, there can be multiple models that are different only in things too abstract for you to have an opinion on them—but still, AC has to be true in any of those models that are acceptable to you.
Fair enough. I believe this is not different in principle from what professional mathematicians do. Except they carefully separate what they believe intuitively to be true, from what they can prove from the axioms.
The question is, whether your intuition is consistent. That is, do you believe in AC only because the proponents of AC got to you first, and gave you the best arguments in their favor? (Like the one about the Cartesian product of infinitely many nonempty sets.) What would have happened if you heard about Banach-Tarski paradox first? Would you be like “yeah, that’s cool for me; you take an orange, cut it and rotate the pieces, and you get two oranges of the same size; my intuition is okay with that”? What is your intuitive opinion on whether the set of real numbers can be well-ordered?
If your intuition is consistent with these things, good for you! Then yes, the other models are the ones where “sets” refers to mathematical objects that are not sets according to your intuition, and only happen to satisfy the ZF axioms. (It still might be useful to talk about such objects, similarly how talking about geometries that do not satisfy Euclid’s fifth postulate got us some interesting results.) But some other people’s intuitions object against things that are equivalent to the axiom of choice, therefore they find it easier to reject it. And it’s not like you can prove them wrong.
If I understand it correctly, (you believe) you have an intuition of “sets”, and you are going to judge any axiom depending on whether it is compatible with your intuition or not. ZFC is compatible with your intuition, negation of AC is not.
And maybe your intuition is also underspecified—that is, there can be multiple models that are different only in things too abstract for you to have an opinion on them—but still, AC has to be true in any of those models that are acceptable to you.
I think this is a good assessment of my current thoughts. It’s funny you ask about the Banach-Tarski paradox, because I think I actually did hear about that first, although the place I heard it (VSauce) didn’t mentioned that it was a consequence of the AC.
I think I’m okay with the BT paradox, because the so-called “pieces” end up being insane collections of individual points. If it were doable with something more akin to “solid chunks”, the way an actual orange would be cut up, I might feel differently.
The well-ordering of the reals is less intuitive to me, and I think I remember being very surprised when I first read that it was true. This was in a context that linked it to the AC, and I think that was the first I heard of the controversy.
I think you’ve given me a good foundation to work on as I think this topic through even further. Thanks again!
What kind of reference you are using for your reference to sets if not the axioms? That reads to me as if “Are they just totally unrelated objects to red busses that just happen to be a bus and red?”
Some times I have seen people argue for example that the word “yellow” is grounded by the set of all yellow things. But usually that kind of definition suffers from the list being ambigious/insufficient. Like if a give a thing it either is or is not a member of that set. But listing all the members or otherwise giving some procedure to give out all the members seems like is not the most natural thing to do. Thus if you tried to take the cartesian product of yellow things and red things because you can’t exemplify a sample just from the concept you can’t build up the product from members. The collection of yellow things propbably is not a set but it has many set-like properties. By having a close inventory of sets properties they can be distinguished from confused or nearby notions.
Another possible imagination prompt would be a person faced with coordinates. Is there a real number that you can spesify that the human can point out on the x axis? No, they are always going to be off. In the same way if you present the axis and ask the human to point out their “favourite number” (that is supposed to keep stable) they will point out a slightly different real number each time they supposedly point that point out. Such a person can’t provide a choice function. It might still make sense to treat the person as being able to specify intervals, or refer to all of the points or being able to reference crossing points and others that have geometrical spesification. But in general a line is not guaranteed to have any referancale points falling within it.
Thanks for this great answer. I have a couple of follow-up questions that anybody should feel free to jump in and answer.
There are models where the AC is specifically false? I’ve been told that AC can be formulated as “the cartesian product of any collection of sets (even an infinite collection) is non-empty”, but I’m having trouble picturing something a collection of things I call “sets” failing to satisfy this property. Are the models referred to here ones “sets” are replaced by totally unrelated mathematical objects that just happen to satisfy the ZF axioms?
If I understand it correctly, (you believe) you have an intuition of “sets”, and you are going to judge any axiom depending on whether it is compatible with your intuition or not. ZFC is compatible with your intuition, negation of AC is not.
And maybe your intuition is also underspecified—that is, there can be multiple models that are different only in things too abstract for you to have an opinion on them—but still, AC has to be true in any of those models that are acceptable to you.
Fair enough. I believe this is not different in principle from what professional mathematicians do. Except they carefully separate what they believe intuitively to be true, from what they can prove from the axioms.
The question is, whether your intuition is consistent. That is, do you believe in AC only because the proponents of AC got to you first, and gave you the best arguments in their favor? (Like the one about the Cartesian product of infinitely many nonempty sets.) What would have happened if you heard about Banach-Tarski paradox first? Would you be like “yeah, that’s cool for me; you take an orange, cut it and rotate the pieces, and you get two oranges of the same size; my intuition is okay with that”? What is your intuitive opinion on whether the set of real numbers can be well-ordered?
If your intuition is consistent with these things, good for you! Then yes, the other models are the ones where “sets” refers to mathematical objects that are not sets according to your intuition, and only happen to satisfy the ZF axioms. (It still might be useful to talk about such objects, similarly how talking about geometries that do not satisfy Euclid’s fifth postulate got us some interesting results.) But some other people’s intuitions object against things that are equivalent to the axiom of choice, therefore they find it easier to reject it. And it’s not like you can prove them wrong.
I think this is a good assessment of my current thoughts. It’s funny you ask about the Banach-Tarski paradox, because I think I actually did hear about that first, although the place I heard it (VSauce) didn’t mentioned that it was a consequence of the AC.
I think I’m okay with the BT paradox, because the so-called “pieces” end up being insane collections of individual points. If it were doable with something more akin to “solid chunks”, the way an actual orange would be cut up, I might feel differently.
The well-ordering of the reals is less intuitive to me, and I think I remember being very surprised when I first read that it was true. This was in a context that linked it to the AC, and I think that was the first I heard of the controversy.
I think you’ve given me a good foundation to work on as I think this topic through even further. Thanks again!
What kind of reference you are using for your reference to sets if not the axioms? That reads to me as if “Are they just totally unrelated objects to red busses that just happen to be a bus and red?”
Some times I have seen people argue for example that the word “yellow” is grounded by the set of all yellow things. But usually that kind of definition suffers from the list being ambigious/insufficient. Like if a give a thing it either is or is not a member of that set. But listing all the members or otherwise giving some procedure to give out all the members seems like is not the most natural thing to do. Thus if you tried to take the cartesian product of yellow things and red things because you can’t exemplify a sample just from the concept you can’t build up the product from members. The collection of yellow things propbably is not a set but it has many set-like properties. By having a close inventory of sets properties they can be distinguished from confused or nearby notions.
Another possible imagination prompt would be a person faced with coordinates. Is there a real number that you can spesify that the human can point out on the x axis? No, they are always going to be off. In the same way if you present the axis and ask the human to point out their “favourite number” (that is supposed to keep stable) they will point out a slightly different real number each time they supposedly point that point out. Such a person can’t provide a choice function. It might still make sense to treat the person as being able to specify intervals, or refer to all of the points or being able to reference crossing points and others that have geometrical spesification. But in general a line is not guaranteed to have any referancale points falling within it.