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I'm curious about the history of the word "regularity" in the axiom of foundation/regularity. Both of those terms are used interchangeably (Wikipedia and most texts seem to use "regularity", while MathWorld uses "foundation", for one). But in actually describing the meaning of the axiom, the term "well-founded" is common, while I've never seen sets described as "regular".

It the use of the term "regularity" based on the assumption that it is intuitive a "regular" set means one which conforms to basic principles like being well-founded and extentional? Or does this term have some other meaning of which I'm unaware?

For reference, I'm writing a book which touches on set theoretical concepts and I'm undecided on which term to use. Advice is appreciated.

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