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!
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!