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Some time ago I enrolled in a class called "Introduction to Real Analysis" in which we learned a lot about Set theory and it's axioms, I learned about the specification axiom, about Russell's paradox and (my favorite part) the nonexistence of the set of all sets. Later on discussing with friends they told me that this was nonsensical since the axiom of regularity (which I didn't knew) solves the Russell's paradox.

Reading in Wikipedia and in other materials about the axiom of regularity I couldn't understand why it is needed. It isn't clear to me how it is true and why we should 'believe' in it, Isn't the axiom of specification more than enough to define a set? Aren't the two getting in the way of each other?

And is my favorite proposition false? The universe exists?

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Russell's paradox, as a disproof of the existence of the universe, goes through just fine, and in much weaker theories than ZFC. If you look at the proof, you'll see that the only axioms that are actually used are those from the Specification scheme. Since it's a theorem of Zermelo set theory (even less), you can't make it not a theorem by adding more axioms like Regularity.

Regularity and Specification do not "get in the way of each other" (why would they?). In fact Regularity and Specification do entirely different sorts of things. Regularity isn't an axiom that lets you form certain kinds of sets, but an assertion that sets have a certain kind of structure (that of being well-founded); it doesn't provide you with new sets, so much as rule out some kinds of sets that could potentially exist without it, like Quine atoms and their relatives. Having such sets around would muck up $\in$-induction, which is a very nice thing to have for some applications; and since Regularity doesn't disprove the existence of any sets we would otherwise depend on for most ordinary mathematical purposes, we may as well use it.

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