Truth and Nonsense

Continues this post.

I think now that it is relatively straightforward to establish a correspondence between the Tarski hierarchy of truth and my hierarchy of nonsense.

Basically, the two hierarchies diverge thanks to two different notions of the correct way to add a "truth" predicate to a base language. The Tarski hierarchy adds a metalanguage that only talks about truth in the base language. The nonsense hierarchy instead prefers Kripke's method, in which we construct a metalanguage that contains its own truth predicate. Both approaches can be thought of as constructing a metalanguage on top of a base language, but the Tarski hierarchy keeps doing so, resulting in a hierarchy of truth, whereas the Kripke fixed-point construction cannot be iterated- doing so adds nothing more. To continue upwords in the Kripke construction, we proceed in a different direction, adding nonsense predicates.

When we use the Kripke truth construction, we can clearly interpret the first Tarski iteration: all the truth-sentences that talk only about truth of base-language statements will be there, provided we have enough machinery to interpret the restricted quantifiers. (Details here will depend on the exact construction.) The semantics assigns them the same truth values; none of these sentences will come up undefined. (I'm talking about semantic interpretations, not proof-theoretic ones... again, details need to be worked out.) The second iteration of Tarskian truth will similarly be inside the Kripke construction; since the first iteration gets definite truth values, the second does. So it goes for as long as the Kripke construction can interpret the restricted quantifiers; that is, for as long as the characteristics of a particular level of the Tarski hierarchy are definable given the tools that the Kripke construction has at its disposal. For example, if these tools can only define computable structures, I'd suppose that the Kripke construction would interpret the portions of the Tarski hierarchy corresponding to the computable ordinals. (That's just a guess. More research required!)

In any case, given a particular amount of expressiveness in the base langauge, the Kripke construction will add a definite amount of expressiveness, corresponding to climbing a particular number of Tarski-hierarchy steps. (Probably this is known; I imagine someone has researched the semantic expressiveness of the Kripke least-fixed-point...) So what happens when we add in more nonsense predicates? Well, adding in nonsense predicates basically allows us to climb that same number of levels again; each nonsense predicate plays the role of allowing us to talk fully about the semantic structure of the construction-so-far (the role that the truth predicate plays in the Tarski hierarchy). This can be thought of as adding that amount of structure to the base language. Then, the Kripke truth construction can do its work on that increased amount of structure. So, we jump up the same number of steps on the Tarski hierarchy for every nonsense predicate added.

Eventually, since the amount of structure added by the truth predicate is always fixed, the scene will be dominated by the hierarchical structure added by the nonsense predicates. Still, it seems clear that each level will correspond in a definite way to a level on the Tarski hierarchy. The nonsense hierarchy merely forces one to make larger jumps at a time.