Mike, my first concern is that one can still prove everything important that used to be provable. Fwiw, I've found assertions in Wikipedia (which for those who don't know it is an amazingly good source of math info: www.wikipdia.com) saying that without Regularity, virtually everything that was true for ordinary sets still is. I don't know what to say about the examples you bring up. If there were a concrete example of something that ought to be true, but without Regularity seems impossible to prove, then I'd say there is a problem. --Dan ----------------------------------------------------------------- Mike Speciner writes: << Intuitively, you prove things by induction by showing it true in one place (typically the null set), and then showing it true for the "next", and so on. Equivalently, you assume the "least" place where something is false, and then find a smaller example. But with sets that can chase their own tail, how do you do either? If you start somewhere and go "upwards", you never create a set which chases its own tail. If you start with such a set, you can't necessarily find a smaller thing. Note that for set theory, we do transfinite induction. Integers are merely countable. Daniel Asimov wrote:

Mike, can you give an example of where a problem might arise?

Ordinary induction is one of the Peano Axioms for the integers. Would this be torpedoed by tossing out Regularity?

Mike Speciner asks: << Well, Dan, how would you do induction? Or can you do without it?

Dan Asimov wrote: << . . . . . . What do others think of possibly tossing out Regularity as a set theory axiom?

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