Also one other remark: Foundation isn’t there to repair any Russel issues. You can get as a theorem that Russell’s set doesn’t exist using the other axioms because you obtain a contradiction. Foundation is more that some people have an intuition that sets shouldn’t be able to contain themselves and that together with not wanting sets that smell like Russell’s set caused it to be thrown in.
I’m really tempted to be obnoxious and present an axiomatic system with a primitive called a “paradox” and then just point out what happens one adds the axiom that there are no paradoxes. This is likely a sign that I should go to bed so I can TA in the morning.
Also one other remark: Foundation isn’t there to repair any Russel issues. You can get as a theorem that Russell’s set doesn’t exist using the other axioms because you obtain a contradiction. Foundation is more that some people have an intuition that sets shouldn’t be able to contain themselves and that together with not wanting sets that smell like Russell’s set caused it to be thrown in.
And of course more generally, for those not familiar, you can never get rid of paradoxes by adding axioms!
I’m really tempted to be obnoxious and present an axiomatic system with a primitive called a “paradox” and then just point out what happens one adds the axiom that there are no paradoxes. This is likely a sign that I should go to bed so I can TA in the morning.
How about by legislating? Has that been tried?