Models of ZFC with one or zero extensions to a model of GBC
This is more or less notes from a talk I gave in the CUNY Graduate Center’s set theory seminar on March 13, 2015.
I wish to consider the question of how many different class structures can be put on models of to get a model of second-order set theory. I will focus here on as my second-order set theory.
includes for the first-order part with the following extra axioms/axiom schema for classes:
- Replacement for class functions—if is a class function then for any set , is a set;
- -comprehension schema—if is a formula with only set quantifiers (but possibly class parameters), then for all , is a class; and
- Global choice—there is a class which is a global well-order. Equivalently, there is a class function .
In general, models of look like where and are the elementhood relations. However, the extensionality axiom for classes implies that every model of is isomorphic to a model , where . Of course, doesn’t have to be transitive, so may fail to be the true elementhood relation. Nevertheless, for the sake of readability, my convention will be to use the symbol for the elementhood relation for models of set theory, even if isn’t the true elementhood relation. As such, models of can be shortly written as .
The following theorem implies that is conservative over for first-order formulae; if proves some sentence with only set variables, then proves it.
Theorem 1 (Solovay) If is countable, then there is countable so that . Indeed, , all classes definable from some global well-order of .
Solovay’s theorem is proved by finding so that , i.e. the theory where a predicate symbol for is allowed in the formulae for the axiom schemata. Any global well-order added by forcing will satisfy this. One way to force this is to force with the poset consisting of set well-orders in , ordered by extension.
To fix some terminalogy, if and , we call a –realization of . If a -realization of exists, we say is -realizable. Under this terminology, Solovay’s theorem says that every countable model of is -realizable.
If has a definable global well-order, then we don’t need to do any work to find a -realization of ; in this case. This holds even if is uncountable.
Remark 1 If is countable, then has infinitely many -realizations.
Proof: Let where . The key fact used is that Solovay’s argument goes through for , giving us a global well-order which is -generic. Obviously, . I claim that as well . Otherwise, defines by some formula which doesn’t mention . Thus, . This happens iff it is forced by some condition , i.e. . But then for all , iff . But is a formla which doesn’t mention , so this is true in , getting that iff . That is, is definable in , contradicting our assumption that was definable in .
By iterating this argument, we get infinitely many -realizations for .
It’s not terribly difficult to tweak this argument to get that has continuum many -realizations. This gives us models of with as many -realizations as possible. The remainder of this blog post will be devoted to constructing the opposite, models of with few -realizations. In particular, we will see that there are models with a unique -realization and models with no -realization.
Necessarily, such models must be uncountable. A natural first place to look as at -like models, in some sense the smallest uncountable models. To spoil any suprise, this turns out to be the right place to look.
Definition 2 is –like if and for all , is at most countable. Equivalently, we could require that this holds just for s or that it holds just for ordinals.
We can build -like models of set theory by growing them up from countable models of set theory. If, when we extend, we only add new sets “on top”, then after many extensions, we will have an -like model.
Definition 3 . is a top extension of , written if for all and , . We will be concerned just with elementary top extensions, written .
Lemma 4 If is an elementary chain of top extensions of countable models, then is -like.
Proof: Obviously . We have only to check that each element in only at most countably many predecessors. Fix . Then, for some . As , , which is countable.
One method of constructing elementary top extensions goes through Skolem ultrapowers. Starting with a model of , we want to have the class functions on the ordinals of the model represent points in the extension, where we mod out by an ultrafilter on the classes of ordinals.
Let’s fix some notation: . Set and . If is an ultrafilter on the boolean algebra and , say that if and if . Let .
Theorem 5 (Łoś) iff .
Proof: This is proved by induction. The only interesting case is the existential step, i.e. when we are trying to prove that iff . The forward direction of the implication is trivial. For the backward direction, fix a global well-order . For -many , let be -least such that . Then, if , . Thus by inductive hypothesis, .
This gives that , where embeds via the map . We needed the second-order structure on so that we could use a global well-order to pick witnesses, but the ultrapower doesn’t carry this second-order structure up to . If has a definable global well-order, then we could take and carry out the construction.
In general, Skolem ultrapowers don’t have to create top extensions, but if we choose the ultrafilter correctly, we can ensure this happens.
Lemma 6 Suppose for all that if there is so that is bounded in , then there is so that is constant on . Then, .
Proof: Consider and suppose and . Then, , so is constant on some set in . Thus, , for some .
Rather classless models
We now have the groundwork laid in place to build models of set theory with or -realizations.
Definition 7 . is amenable (to ) if every initial segment of is in : for all , there is so that iff and . Write for the collection of amenable subsets of .
Examples of amenable sets are easy to find. If , then any is amenable. If is definable in , then is necessarily amenable. However, there are models of set theory with many non-amenable subsets. If is -nonstandard, then is not amenable as is the true , which cannot be a set in . Even in transitive models, we can see this. If exists, then is not amenable to as .
Definition 8 is rather classless if .
We’ve already seen that for any . We can think of being rather classless as saying that has as few amenable subsets as possible. We can trivially observe that no countable is rather classless: any cofinal -sequence is amenable, but never definable. However, there are -like rather classless models.
But before actually constructing any rather classless models, let’s see that they answer our original question.
Let be rather classless. If has a definable global well-order, then has a unique -realization. If does not have a definable global well-order, then has no -realization.
Proof: Observe that if , then ; this is just separation. Hence, the only possible -realization for rather classless is . iff has a definable global well-order.
Remark 2 Note that having a definable global well-order is expressible as a single first-order sentence. This is as having a global well-order definable from parameter is equivalent to . Hence, the sentence asserts the existence of a definable global well-order.
We are now ready to construct -like rather classless models of set theory. The key step is contained in the following lemma:
Lemma 9 (Key Lemma) countable, . Then there is so that is not coded in , i.e. for all , .
Proof: We will find an ultrafilter on so that is as desired. Well order as . We will define a descending sequence of unbounded classes in which will generate . Start with .
Suppose has already been defined and look at . If is bounded in , then use the pigeonhole principle to find unbounded on which is constant.
Otherwise, is unbounded in . We want to ensure that does not represent a set in which codes . To see how to do this, let’s introduce an auxiliary tree. Set , where iff and . is not quite a tree in the usual set theoretic sense; if is ill-founded, then chains in don’t have to be well-founded. However, if we slightly generalize the notion of tree to allow chains to be ill-founded but linear, then is a tree in this less restrictive sense. Any induces a cofinal branch through : .
For , set . Then, is the collection of nodes for which it is possible to have extend while still being able to continue the construction of the ultrafilter. The sets for are the possibilities for . Clearly, and for every . That is, picking one of these to be will keep us within .
Further note that is unbounded. This is easy to see: for any , there are only -set many possibilities for . Hence, there must be an unbounded class on which they are all the same.
Observe that if on -many , extends a node in , then does not code . Hence, if , we can find so that will not code . to see this, suppose otherwise that . As is unbounded, this would imply that . But this first subset of is in , as , and . Hence, we were wrong to suppose that . We can thus pick unbounded which ensures does not code .
Let be the filter generated by . is an ulrtafilter as every characteristic function of a is bounded and hence constant on a class in . , by the earlier lemma. By construction, no codes .
Theorem 10 (Keisler) Assume and let be countable. Then there is an elementary top extension of which is -like and rather classless.
Proof: Fix a diamond sequence . We will construct an chain of elementary top extensions of countable models . At limit stages, we just take unions. At successors of successors, we let . The only interesting case is in defining successors of limits. Without loss of generality, ‘s universe is . Hence, . If , then let be any top extension of . We can use the Key Lemma to get this top extension. Otherwise, if is not definable in , then we can find countable so that and : if then take where is -generic; if then take any . We can now apply the Key Lemma and find which kills . Finally, let be the union of this elementary chain. is -like.
To see that is rather classless, consider . I claim there is a club of so that . Closedness is easy. To see unboundedness, fix . By taking Skolem closures, we can find
so that . Then, if , .
then gives us a so that and . Pick . Then, . This element of codes , so by construction, must have been definable in . By elementarity, , as desired.
We have constructed rather classless models of set theory, but only from the assumption of . However, this assumption can be eliminated.
Theorem 11 (Shelah) The assumption of can be eliminated from Keiser’s theorem.
Working through an argument for this will have to wait for the future.
Corollary 12 Let be countable. If , then there is an elementary top extension of which has a unique -realization. If then there is an elementary top extension of which has no -realization.
Proof: In either case, take to be -like and rather classless.
It is natural to ask if a model of set theory can have few, but more than one, -realization.
For which are there models of set theory with exactly -realizations?
1. Some references
The original construction of rather classless models is due to Keisler. He proved a more general result than the one presented here, showing that, for example, every countably model of has an elementary end extension to an -like rather classless model. Keisler drew the corollary that there are models of set theory with unique -realizations, but said nothing about models with no -realizations. The elimination of is due to Shelah.
- H. Jerome Keisler, “Models with tree structures,” Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1974, pp. 331–348.
- Saharon Shelah, “Models with second-order properties. II. Trees with no undefined branches,” Ann. Math. Logic 14 (1978), no. 1, 73–87.