The truth about incommensurable truths: iterated truth predicates in a nonstandard world
If is a transitive model of , then either admits a strongly amenable iterated truth predicate of length or else there is a unique so that admits a strongly amenable iterated truth predicate of length but not one of length . (We can go beyond , but this starts getting into subtleties I want to avoid in a paragraph introducing another topic.) However, if has ill-founded , this is no longer the case. We can find different full satisfaction classes for which can be iterated different lengths. Indeed, any reasonable length is possible.
In my previous post, I gave an application of iterated truth predicates. The only structures dealt with there were transitive. In a transitive world, things are simple. It is easy to check that the truth predicate for a structure is unique. Since transitive structures have the correct , they are correct about what formulae are. Thus, the only class which can be added to a transitive structure which looks like its truth predicate is its actual truth predicate.
For nonstandard structures, things are not so nice. If we want to be able to use a class for much, then we want the expanded structure to satisfy the axiom schemata of our theory—Separation and Collection for set theory, Induction for arithmetic—with a predicate symbol added for the class. When this happens, we say that is, in the case of set theory, strongly amenable or, in the case of arithmetic, inductive.
Hereon, I’ll focus on models of arithmetic, but similar things can be said about models of set theory.
It’s easy to see that the actual truth predicate, which consists only of standard-length formulae, for a nonstandard model can never be inductive. Otherwise, apply the instance of induction for the property “there is a -formula in the truth predicate”, which is true for and closed under successor, to get that the model must be the standard model, a contradiction. As such, if we want to do much we need something that looks like a truth predicate but includes formulae of all length, even nonstandard length.
The notion used by model theorists of arithmetic is that of a full satisfaction class. A satisfaction class for a model is a class which satisfies the recursive Tarskian definition of truth. For instance, is in the satisfaction class only if there is so that is in the satisfaction class. It is full when it decides every formula, i.e. either is in the satisfaction class or else is in, even for nonstandard .
Full satisfaction classes are well studied; see, for example, this survey article by Kotlarski. One important fact is that if a model admits a full satisfaction class (even a non-inductive one) then it must be recursively saturated. If the model is countable, then the converse also holds.
Another important fact is that full satisfaction classes are far from satisfying the uniqueness property from the standard world; Krajewski constructed countable models of arithmetic with continuum many different full satisfaction classes. Even if we restrict to inductive full satisfaction classes, things are no better. We cannot even guarantee that different full satisfaction classes give the same theory in the expanded language.
Proposition: There is and inductive full satisfaction classes so that .
(I do not know who first observed this fact. I learned of it from Joel David Hamkins. In a joint paper with Ruizhi Yang he attributes a stronger form of this proposition to Jim Schmerl—see page 30. It is more-or-less his argument I now sketch.)
Proof sketch: Consider the theory consisting of plus the assertion, formalized as an axiom schema, that is an inductive full satisfaction class. (To be clear, the language for this theory is the language of arithmetic augmented with an extra predicate symbol for .) While the reduct of this theory to the language of arithmetic is complete, is not itself complete. This is just a version of Tarski’s theorem on the undefinability of truth. Now let and be two consistent but incompatible extensions of . By a standard resplendency argument, find a model and so that and . ∎
It must be that and agree about standard formulae. Specifically, restricted to standard formulae they are both . Their disagreement can only occur for nonstandard formulae— may think that nonstandard is true while know better and think that is false.
The above proposition shows, however, that if we are interested not in truth but rather in truth about truth then disagreement can happen at a standard level. Any full satisfaction class for the structure must have as an initial segment. As such, if we want inductive iterated full satisfaction classes starting with and —say we only want to iterate one step to get and respectively—then there is no longer any guarantee that they line up if we restrict to looking only at standard truths.
We have no hope that the theory we get from iterating truth is independent of our choice of satisfaction class, but perhaps we can find other invariants. In my previous post, I showed how to construct transitive models of set theory which admit a strongly amenable iterated truth predicate of length but one of length . As we’ve seen, for nonstandard models we have a choice of satisfaction class. But we might hope that how far out we can iterate a satisfaction class (where we have to make a choice at each level how to proceed) is an invariant.
Put differently, we can think of the inductive iterated satisfaction classes as forming a tree. The root is simply the empty set. Given a node in the tree, i.e. an iterated inductive full satisfaction class of length , its children are all the -iterated inductive full satisfaction classes which extend . Is it the case that all branches through this tree have the same length?
The answer is no.
Before seeing why, let’s fix some notation. Let be the theory consisting of the axioms of plus the assertions that is an inductive full satisfaction class. (This theory is in the language of arithmetic augmented with a symbol for .) Note that this theory is computably enumerable; it is a single sentence to say that is a full satisfaction class and a computable schema to say that is inductive. ( stands for “compositional truth”; cf. this SEP article.)
Next, let be all theorems of which are in the language of arithmetic. That is, consists of all theorems of which don’t make reference to truth. It is not hard to see that is a proper extension of . For instance, is a theorem of but, by a famous theorem of Gödel, is not a theorem of .
As a warm-up, let’s prove the following.
Proposition: Let be countable and recursively saturated. Then, there is so that . Moreover, can be chosen so that is also recursively saturated.
In informal language, the only thing that could possibly stop a countable, recursively saturated model from admitting an inductive full satisfaction class is its theory.
Proof: This is a standard argument by resplendency. For the sake of the reader unfamiliar with this kind of argument (and because I’d feel a little silly giving a one sentence argument :/) I’ll provide a more detailed argument.
The first step is to note that contains nonstandard which codes the theory of . This is a fun exercise in writing down the correct recursive type. Next, let be an effective enumeration of the theorems of . Finally, let be the formula expressing , where is .
Observe that for all standard . By overspill, we can pick nonstandard so that . Working inside , apply the arithmetized completeness theorem to . This gives a definable class of which codes a model of . (Of course, is an ersatz theory which consists of real, standard formulae as well as nonstandard formulae, so the preceding sentence is a bit of nonsense. What is really meant is that has a definable class which codes what it thinks is the satisfaction predicate of and that this satisfaction predicate includes .) Restricting to the standard formulae, we get that .
Further, we get that end-extends . This is because thinks it is the standard model and that it embeds into any model of arithmetic. This is sufficiently absolute that if thinks it then it really is true. As a consequence, we get that and have the same standard system. But then, and are isomorphic. (This is a consequence of the more general fact that countable recursively saturated are isomorphic if they have the same theory and the same standard system.) Letting be the image of under the isomorphism we get that , as desired.
The “moreover” part of the proposition is because for countable models, recursive saturation implies resplendency implies chronic resplendency. ∎
Similar to the definitions of and , we can define and for truth about truth instead of just truth. That is, consists of plus the assertions that is a -iterated inductive full satisfaction class while consists of the theorems of in the language of arithmetic.
Proposition: Suppose is countable and recursively saturated. Then, there are inductive full satisfaction classes so that can be iterated one step further to get a -iterated inductive full satisfaction class while cannot.
Proof: Constructing is very similar to what we have already done. The easiest way to go about it is to first find a -iterated inductive full satisfaction class for by essentially the same argument as above. Then, restrict to the first stage to get a -iterated inductive full satisfaction class , i.e. an inductive full satisfaction class.
To construct yet again apply the same argument from above. To ensure that it cannot be iterated further to produce an inductive -iterated full satisfaction class, it suffices to work with a theory which is incompatible with . This can be done because, by a version of Tarski’s theorem on the undefinability of truth, is independent over . This gives us which cannot be extended to a model of . ∎
This gives us that the length a full satisfaction class be iterated to produce inductive iterated full satisfaction classes is not invariant under choice of satisfaction class. We can push this a little bit further and get that the length can be anything we desire.
WordPress is telling me that I’m nearly at two thousand words, so let me skimp on details. Similar to how we can write down a theory for having a -iterated inductive full satisfaction class, we can do the same for any and get . Indeed, if we work over a fixed we can talk about for any . We also get , the theorems of in the language of arithmetic. By considering a theory extending but incompatible with we can get , an -iterated inductive full satisfaction class which cannot be iterated any further. Therefore, any possible length can be achieved.
Proposition: Let be countable and recursively saturated and fix . Suppose that . Then, for any in , there is so that but cannot be iterated one step further to an -iterated inductive full satisfaction class. ∎
This completely determines what can happen for iterated inductive full satisfaction classes of -finite length. But, at least in some circumstances, we can go further. For instance, proves the existence of iterated truth predicates of transfinite length. (Naturally, any in the second-order part of a model of must be inductive over the first-order part.) Thus, starting with a model of we can find, say, which is an -iterated inductive full satisfaction class for . What can be said in general about iterated inductive full satisfaction classes of transfinite length?