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Every modern presentation of ZF includes the axiom of regularity (or foundation): every nonempty set $x$ has an element $y$ with $x\cap y=\emptyset$. Of course, ZF is a system which developed over time, and going back far enough we can find various different versions and predecessors of the modern system.

That said, I was quite surprised to find out recently that in Mendelson's text on logic states that the axiom of regularity is not necessarily a standard one:

In recent years, ZF is often assumed to contain [the axiom of regularity]. The reader should always check whether [regularity] is included in ZF. $\quad$(pg. 288)

Moreover, a review by Van Dalen doesn't mention this, suggesting that Mendelson's commentary on the axiom isn't out of place. This is all especially surprising to me since I was under the impression that it was adopted in the 30s.

My question is:

When did regularity become universally accepted as an axiom of ZF?

Of course I'm making an assumption here - that regularity is universally accepted as an axiom of ZF in the modern era. While it's impossible to prove that there isn't some recently written text somewhere which doesn't include it, in lieu of an example of such a text (say, from the last 40 years) I think I'm more than justified in making this claim; e.g. every major textbook from the last 40 years of which I'm aware (Bar Hillel/Fraenkel/Levy/Van Dalen, Kunen, Jech, Enderton, Ciesielski,...).

Noah Schweber
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    "First edition 1964", i.e., not from the last 40 years. So why wasn't that passage updated? "In recent years, ZF is often assumed to contain [the axiom of regularity]." That's still true, isn't it? "The reader should always check whether [regularity] is included in ZF." Seems like a good idea. After all, the reader may be reading an old classic. – bof Sep 16 '17 at 21:57
  • @bof I'm not quite sure what you're getting at? I don't know why the passage wasn't updated in later editions, but I don't know how much updating was done at all so I don't know if that's weird. And yes, the original is more than 40 years old - that's why I said that all the texts that are from within the last 40 years include regularity. And re: the advice to double-check whether regularity is assumed, this actually confuses things more than it helps, since e.g. papers in set theory don't begin by writing out the axioms of ZF; given that regularity is now part of ZF, it's good to know that. – Noah Schweber Sep 17 '17 at 02:19
  • That is: while it might be good advice for reading old classics (although even then I don't know of a text other than Mendelson which doesn't include regularity), it's terrible advice for actually working in set theory. – Noah Schweber Sep 17 '17 at 02:20
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    You can see Adam Rieger, Paradox ZF and the Axiom of Foundation (2007), page 10: "have been considered by Mirimanoff ([1917a] and [1917b]), who distinguishes “ordinary” sets which do not have infinite descending membership chains from “extraordinary” ones which do. He does not assert, however, that there is anything wrong with the extraordinary sets. Von Neumann [1925] describes non-wellfounded sets as “superfluous” and gives an axiom which excludes some, but not all, of them. Three years later [1928] he formulates the axiom of foundation ... 1/2 – Mauro ALLEGRANZA Sep 17 '17 at 12:16
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    However, it is not until the paper Zermelo [1930] that the axiom of foundation is explicitly adopted as a postulate. With this paper all the axioms of standard modern set theory are in place." 2/2 – Mauro ALLEGRANZA Sep 17 '17 at 12:17
  • @MauroALLEGRANZA Yes, I'm aware of that - which is why I'm surprised to see Mendelson's comments in the early 60s. – Noah Schweber Sep 17 '17 at 12:21
  • Suppes' textbook (1960) as well as Bernays' one (1958) have the axiom. – Mauro ALLEGRANZA Sep 17 '17 at 16:26
  • @MauroALLEGRANZA So was there any basis at all for Mendelson's comment? – Noah Schweber Sep 17 '17 at 16:39

1 Answers1

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Jean Louis Krivine's book Introduction to Axiomatic Set Theory (1971) defines the theory ZF (chapter 1) as not containing the Axiom of Foundation (introduced in chapter 3). This edition is the English translation of the French version of this book, published in 1969.

The same author published a new book Theorie des ensembles (2007, only in French) keeping such a definition of ZF.

In my opinion, the fact of non including the Axiom of Foundation in the definition of ZF is quite popular at least in France. I don't know if this is due to Krivine's first book, or if Krivine was following an already consolidated French tradition. Bourbaki's Théorie des ensembles (1970, only in French) does not mention the Axiom of Foundation. Anyway, this definition of ZF is still valid today in France. For instance, Cori's and Lascar's undergraduate textbook Mathematical Logic: A Course With Exercises, Part II (2001, largely used in courses of mathematical logic in French universities) follows this definition in chapter 7.

Taroccoesbrocco
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  • @NoahSchweber - I edited my answer. Sorry, I have misinterpreted your question. – Taroccoesbrocco Sep 18 '17 at 14:21
  • Not at all, I've changed my -1 to a +1. I think this is really interesting - I didn't know at all that there was any disagreement on this point anymore! – Noah Schweber Sep 18 '17 at 14:36
  • The sentence "Anyway, this is still valid today in France" sounds funny when taken out of context. ........+1 – DanielWainfleet Sep 18 '17 at 23:17
  • @NoamSchweber - Also Bourbaki's Théorie des Ensembles (1970, only in French) does not consider the Axiom of Foundation as a part of ZF. Actually it does not even mention it, but it mentions all the other axioms. – Taroccoesbrocco Sep 20 '17 at 14:21
  • @Taroccoesbrocco I don't have Bourbaki on hand - do they actually name the set theory ZF? I thought Bourbaki used their own set theory. Regardless, I've accepted - this definitely answers my question, and is fascinating! Interestingly, all set theory research papers I've ever seen do include regularity in ZF (e.g. by using induction on rank to prove something about all sets); I'd be interested to see a counterexample here, to. (BTW my name is "Noah," not "Noam.") – Noah Schweber Sep 20 '17 at 14:40