Difference between transitive and non-transitive models

Question

In set theory, a transitive model has the property that every set in the model has, as elements, only other sets which are also in the model. I'm trying to understand what difference this makes in terms of properties such models can have.

One way I tried to build a nontransitive model is to start with a transitive one and introduce a new element $x$. $x$ isn't added to the model directly, but we then create a nontransitive model by adding in new sets which contain $x$, but without adding in the element $x$ itself.

If I am on the right page here, such "nontransitive sets" will seem to have very strange properties. For instance, compare the sets $A = \{1, 2\}$ and $B = \{1, 2, x\}$. The model will think that these two sets are equal, because "for all" elements $s$ in the model, we have $s \in A \iff s \in B$. The model's notion of "for all" doesn't have $x$ in it, and since our model satisfies extensionality it will think $A = B$.

If this is the correct view, then this is a very bizarre situation because equality within the model would seem to be different from equality in the ambient theory. For instance: we can then look at the set $\{ A, B \}$. This set really has two elements in it, and it is part of the model. Does the model think that $\{ A, B \} = \{ A \} = \{ B \}$? Even though these are three different sets, does the model think that all of them have (unique?) bijections to what it is calling $1$?

Basically, my questions are:

1. Is this the right idea about what nontransitive models are?
2. In the example above, would the model think that $\{ A \} = \{ B \} = \{ A, B \}$? And that both have one element?
3. If we are basically adding extra sets that the model thinks are equal to other sets, how could this be used to do anything interesting, e.g. change cardinalities of sets in the model?

Answer 1

I think it's useful to bear in mind that a model of set theory is really just some set $M$, equipped with some relation $E$, satisfying all of the axioms you want to be satisfied. A model being transitive is quite a special property - it asserts that the elements of $M$ have some special intrinsic properties, and that $E$ is a very specific relation!

I think the following two definitions are kind of analogous to the concept of "transitive model of set theory", in terms of the character of what they assert (I'm not claiming they're necessarily useful or meaningful).

1. Say a model $N$ of first-order Peano Arithmetic is tronsitive if $N$ is a set of ordinals, and the additive identity of $N$ is the ordinal $0$, and the successor operation on $N$ is given by the ordinal successor operation.
2. Say a group $G$ (ie, a model of the theory of groups) is trunsitive if the elements of $G$ are permutations of some set $S$, the composition operation on $G$ is composition of permutations, and the identity of $G$ is the identity permutation on $S$.

Now there are some ways in which each might fail. Firstly, because these definitions assert something about the fundamental nature of the actual elements of the model, and the nature of the operations/relations we have, they don't respect isomorphism of models. So there are some silly ways to make a non-trounsitive model, by transporting the structure along silly bijections ("relabelling the elements of the model", which I was alluding to in my comment).

1. Consider the map $\omega \to \omega$ given by $n \mapsto 2n$. By transporting structure along this map, we get a model $\{0, 2, 4, 6, \dotsc\}$ of PA, whose elements are ordinals. But it's not tronsitive, because the successor operation is given by adding $2$. (Although it does contain $0$!)
2. Considering $n \mapsto n + 1$, we get a model $\{1, 2, 3, \dotsc\}$ of PA, whose elements are ordinals. But it's not tronsitive, because its additive identity is $1$. (Although the successor operation is the actual successor operation!)
3. Consider the map that fixes all of $\omega$, except it swaps $2$ and $3$. This makes $\omega$ into a non-tronsitive model of PA, because the successor of $3$ is $2$ in this model.
4. Consider the map $\omega \to \Bbb R$ sending $n \mapsto n \sqrt 2$. This makes $\{0, \sqrt 2, 2\sqrt 2, \dotsc\} \subseteq \Bbb R$ into a model of PA, which is definitely non-tronsitive because its elements aren't even ordinals (unless you constructed $\Bbb R$ in a really weird way).
5. Similarly, the group $(\Bbb Z, 0, +)$ is almost certainly not trunsitive, because its elements aren't permutations. And in any trunsitive group, you can apply a silly relabelling to get a group whose elements are permutations, but whose composition isn't actually composition of permutations.

This approach works just as well to manufacture a non-transitive model of set theory, if you start with a transitive one!

1. You can just swap eg $\emptyset$ and $\{\emptyset\}$, to get a model which is a transitive set, but whose membership relation is wrong.
2. You can do something crazy like considering the map $M \to M \times \{M\}$, given by $X \mapsto (X, M)$. This gives a model which isn't a transitive set.
3. You fix any set $X_0$ (which could be in $M$) and define an operation $F$ on $M$ by $\in$-recursion: $F(Y) = \{F(Z) \mid Z \in Y\} \cup \{X_0\}$. Then the image of $M$ under $F$ is a model of set theory, which is a non-transitive set, but since $A \in B$ iff $F(A) \in F(B)$, the membership relation on it is the usual one! I think this is possibly closest to what you were suggesting about adding a new set - but exactly what you had written can't be a model of ZFC, because equality is usually interpreted as genuine equality in first-order logic (certainly when you're doing set theory with it). See here and here.

However, all of these examples are a bit silly, in my opinion, because they give a model that is clearly isomorphic to a traounsitive model. Often if you're concerned with the theory of XYZs, then really the interesting properties of models are those which respect isomorphism. So I think perhaps it may be fruitful to think about "essentially traounsitive models", by which I mean the models $M$ such that there is some traounsitive model $M'$, which is isomorphic to $M$. A nice thing about such models is that if you're given an essentially traounsitive model, you can usually say "WLOG the model is traounsitive".

1. An essentially tronsitive model of $PA$ is one that's isomorphic to $\Bbb N$ with the usual operations (sometimes called "standard models"). There are models which are not of this form - for example, any uncountable model, or any countable model containing an element not of the form $0$ or $S0$ or $SS0$ etc (which exist by the compactness theorem, or by taking an ultrapower). In fact, a model of PA is nonstandard iff it's not well-ordered (foreshadowing...)
2. An essentially trunsitive group is in fact just any group, by Cayley's Theorem.
3. An "essentially transitive" model $M$ of ZFC is in fact exactly the same thing as a well-founded model of ZFC, by considering the Mostowski collapse of $M$. Again, you can build a non-wellfounded model of ZFC from a well-founded one by using the compactness theorem, or taking an ultrapower. There is much to read about these things and why their existence doesn't contradict for instance the Axiom of Foundation on this site.