I don’t think ZF(C) is the problem. If whatever alternative you come up with doesn’t have equivalent results, then I think it isn’t expressive enough (or you’ve possibly even assumed away infinites, which would be empirically questionable). And whatever solution you might come up with can probably be expressed in ZF, and with less work than trying to build new foundations for math. I think it’s better to work within ZF, but with additional structure on sets or using different ethical axioms.
I confess my comment was motivated by seeing something where it looked like I could make a quick “gotcha” point, which is a bad way to converse.
Reading the original comment more carefully, I’m seeing how I disagree with it. It says (emphasis mine)
in practice the problems of infinite ethics are more likely to be solved at the level of maths, as opposed on the level of ethics and thinking about what this means for actual decisions.
I highly doubt this problem will be solved purely on the level of math, and expect it will involve more work on the level of ethics than on the level of foundations of mathematics. However, I think taking an overly realist view on the conventions mathematicians have chosen for dealing with infinities is an impediment to thinking about these issues, and studying alternative foundations is helpful to ward against that. The problems of infinite ethics, especially for uncountable infinities, seem to especially rely on such realism. I do expect a solution to such issues, to the extent it is mathematical at all, could be formalized in ZFC. The central thing I liked about the comment is the call to rethink the relationship of math and mathematical infinity to reality, and that doesn’t necessary require changing our foundations, just changing our attitude towards them.
If the only alternative you can conceive of for ZFC is removing the axiom of choice then you are proving Jan_Kulveit’s point.
I don’t think ZF(C) is the problem. If whatever alternative you come up with doesn’t have equivalent results, then I think it isn’t expressive enough (or you’ve possibly even assumed away infinites, which would be empirically questionable). And whatever solution you might come up with can probably be expressed in ZF, and with less work than trying to build new foundations for math. I think it’s better to work within ZF, but with additional structure on sets or using different ethical axioms.
On further thought I want to walk back a bit:
I confess my comment was motivated by seeing something where it looked like I could make a quick “gotcha” point, which is a bad way to converse.
Reading the original comment more carefully, I’m seeing how I disagree with it. It says (emphasis mine)
I highly doubt this problem will be solved purely on the level of math, and expect it will involve more work on the level of ethics than on the level of foundations of mathematics. However, I think taking an overly realist view on the conventions mathematicians have chosen for dealing with infinities is an impediment to thinking about these issues, and studying alternative foundations is helpful to ward against that. The problems of infinite ethics, especially for uncountable infinities, seem to especially rely on such realism. I do expect a solution to such issues, to the extent it is mathematical at all, could be formalized in ZFC. The central thing I liked about the comment is the call to rethink the relationship of math and mathematical infinity to reality, and that doesn’t necessary require changing our foundations, just changing our attitude towards them.