Sure foundation is by far the most ad-hoc axiom. But it is also one of the one’s that is easiest to see doesn’t generally matter. For pretty much any natural theorem if a proof uses foundation then there’s a version of the theorem without it. Since not-well founded sets don’t fit most of out intuition for sets as things like boxes that’s not an issue. None of the serious apparent paradoxical properties go away if you remove foundation.
It’s a very cool construction, but it’s a finite one that we can verify by hand or with computer assistance. Of the things that ZF claims exist, some of them have this “verifiability” property and some don’t. At the very least don’t you agree that’s a crucial distinction, and that we ought to be strictly less skeptical of constructible, computable, verifiable things than of things like uncountable ordinals?
Yes, certainly but by how much? If our intuition can go this drastically wrong on small finite objects why should I trust my intuition on objects that are even further removed from my everyday experience? I mean it isn’t like I need 30 or 40 sided dice to pull this off. In fact you can actually make much smaller than 6 sided dice that are non-transitive. Working out the minimum number of sides (assuming that each die in the set doesn’t need to have the same number of sides) is a nice exercise that helps one understand what is going on.
You’re right, I see that it’s the “restriction” of restricted comprehension that actually does the work in avoiding Russell’s paradox, not foundation. Nevertheless, the story is the same: we had an ambitious set-theoretic foundation for mathematics, Russell found a simple and fatal flaw in it, and we should not simply trust that there will be no further problems after patching this one.
If our intuition can go this drastically wrong on small finite objects why should I trust my intuition on objects that are even further removed from my everyday experience?
This is hardly an argument for accepting that infinite sets exist! There may be a counterintuitive contradiction that one can arrive at from ZF, just as Russell’s paradox is a counterintuitive contradiction arrived at from 19th century foundations, and just as all kinds of counterintuitive but non-contradictory behavior is possible in the finite, constructive realm.
I am proposing that we remove the axiom of infinity from foundations, not that we go further and add its negation. (Though I see that there has been work done on the negation of the foundation axiom! And dubious speculation about its role in consciousness.)
Sure foundation is by far the most ad-hoc axiom. But it is also one of the one’s that is easiest to see doesn’t generally matter. For pretty much any natural theorem if a proof uses foundation then there’s a version of the theorem without it. Since not-well founded sets don’t fit most of out intuition for sets as things like boxes that’s not an issue. None of the serious apparent paradoxical properties go away if you remove foundation.
Yes, certainly but by how much? If our intuition can go this drastically wrong on small finite objects why should I trust my intuition on objects that are even further removed from my everyday experience? I mean it isn’t like I need 30 or 40 sided dice to pull this off. In fact you can actually make much smaller than 6 sided dice that are non-transitive. Working out the minimum number of sides (assuming that each die in the set doesn’t need to have the same number of sides) is a nice exercise that helps one understand what is going on.
You’re right, I see that it’s the “restriction” of restricted comprehension that actually does the work in avoiding Russell’s paradox, not foundation. Nevertheless, the story is the same: we had an ambitious set-theoretic foundation for mathematics, Russell found a simple and fatal flaw in it, and we should not simply trust that there will be no further problems after patching this one.
This is hardly an argument for accepting that infinite sets exist! There may be a counterintuitive contradiction that one can arrive at from ZF, just as Russell’s paradox is a counterintuitive contradiction arrived at from 19th century foundations, and just as all kinds of counterintuitive but non-contradictory behavior is possible in the finite, constructive realm.
I am proposing that we remove the axiom of infinity from foundations, not that we go further and add its negation. (Though I see that there has been work done on the negation of the foundation axiom! And dubious speculation about its role in consciousness.)