Sewing-Machine comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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Comments (653)
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.)