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I heard that ZF- (the ZF which dropped the axiom of foundation) can contain only unique Quine atoms and keep other axioms consistent. Does this mean we can construct Urelement which contains Quine atoms and other well-founded-set in ZF-?

For example: By axiom of pairing, we can construct a Urelement which contains Quine atoms and other well-founded-set. By axiom of union,we can make quine atoms constain itsef and and other well-founded-set.

If it is, is there any applications which using ZF-? Thanks in advance.

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    $\begingroup$ I understand that a "Quine atom" is a set $x$ such that $x=\{x\},$ and an "Urelement" is an object which is not a set but may be an element of a set. From your question, it seems that you are using the terms in some other sense, but it's not clear what it is. What is your definition of "Quine atom" and what is your definition of "Urelement"? $\endgroup$
    – bof
    Commented Dec 27, 2017 at 8:55

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Your question seems to be very confused about terminology. So let's clear some things up.

There is a variant called $\sf ZFA$, or $\sf ZF$ with the existence of "atoms" or "urelements", which are objects that are not sets. In this context we have a predicate for "set" and a predicate for "atom", and we write the axioms of $\sf ZF$ for sets, and atoms are just objects which can be elements, but have no elements of their own (the empty set is a unique exception, it is the "atomic set" in that context). The collection of atoms is often assumed to be a set, but this is not necessary, and there are results using the assumption there is a proper class of atoms.

This system is often used to construct simpler independence results concerning the axiom of choice à la what is known as Fraenkel–Mostowski–Specker permutation models.

One of the ways of producing such models is to have the failure of the axiom of foundation (or regularity), in a way where there is a set of different Quine atoms, where Quine atoms are sets satisfying the property $x=\{x\}$. Then we can use the collection of Quine atoms as our urelements, and create a model of $\sf ZFA$. Thus, all methods applicable to $\sf ZFA$ can be applied to $\sf ZF^-$ when replacing the urelements by Quine atoms, and positing that the universe is "somewhat generated" from the Quine atoms (in some canonical way, e.g. $V(A)$ where $V$ is the von Neumann inner model and $A$ is the class of atoms).

This goes in the other direction as well. Starting from a model with urelements, one can create a model of $\sf ZF^{-}$ where there are Quine atoms. So the two approaches are truly interchangeable.

But Quine atoms are not the only way to violate foundation. Indeed, it is possible that the axiom of foundation is violated in all sort of ways, but there are no Quine atoms. And it is possible there is a single Quine atom, as would follow from Aczel's Anti-Foundation Axiom, something which will not be very useful for FMS permutation models.


To sum up, an "urelement" is an object in the set theoretic universe which is not a set. It does not contain any object, as sets contain objects, and urelements are not sets. Sometimes these are called "atoms", which might cause confusion with Quine atoms, but that's not terribly bad, since we can think about them as Quine atoms in a model of $\sf ZF^-$ instead.

So asking if we can construct an urelement which contains Quine atoms is quite a meaningless question. The answer is negative.

Asking if a Quine atom could contain a well-founded set is also not a very meaningful question, since a Quine atom is $x$ such that $x=\{x\}$, the only member of $x$ is $x$ itself, which is not a well-founded set. But it is true that you can have a set which has both a Quine atom and a well-founded set. For example $\mathcal P(x)=\{\{x\},\varnothing\}=\{x,\varnothing\}$ where $x$ is a Quine atom.

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    $\begingroup$ I do have some confusion about terminology, thanks for your clarification! As you mentioned, can I understand this way: In ZF-, we can have some "classes" like P(x)? If it is, how to apply other axioms of ZF(dropped axiom of foundation) to these "classes"? Do you need extra axioms for these "classes"? Is there any applications which using ZF-? $\endgroup$
    – chansey
    Commented Dec 28, 2017 at 8:27
  • $\begingroup$ I use quotes for "class", since I'm not sure if I can use this term "class" for these objects like P(x) in ZF-. $\endgroup$
    – chansey
    Commented Dec 28, 2017 at 8:29
  • $\begingroup$ I don't understand the question at all. If $x$ is a Quine atom, then it is a set by definition. Since the objects of $\sf ZF^-$ are sets, just not necessarily well-founded sets. One of the axioms of $\sf ZF^-$ states that power sets exists, so $\mathcal P(x)$ which is $\{\varnothing,x\}$ is indeed a set. I'm not sure that I follow your comment to the end, though. Let me set a baseline, here, how much set theory do you know? $\endgroup$
    – Asaf Karagila
    Commented Dec 28, 2017 at 8:42
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    $\begingroup$ More assumptions means that you cannot determine in $\sf ZF^-$ whether or not you have Quine atoms, first of all because you cannot even determine whether or not all sets are well-founded (you only omitted the axiom of foundation, there was no assertion that non-wellfounded sets actually exist). The point is that if $\sf ZF$ is consistent, then $\sf ZF^-$ along with the assumption that there is a set, or even a proper class of Quine atoms; but also with the assumption that there are none, but the axiom of foundation fails. $\endgroup$
    – Asaf Karagila
    Commented Dec 28, 2017 at 10:54
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    $\begingroup$ @chansey "Do we need additional axiom to ensure consistency?" Adding more axioms never makes anything more consistent. (You can't take an inconsistent theory and suddenly make the contradiction vanish by saying more stuff - the exploding cat's already out of the bag.) $\endgroup$ Commented Dec 29, 2017 at 3:28

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