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JoshuaZ comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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The axiom of foundation seems pretty ad hoc to me. It's there to patch Russell's paradox. I see no reason not to expect further paradoxes.
We arrived at the axiom of infinity from a finite amount of experience, which seems troubling to me.
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?
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?