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!
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!