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Last time we looked at Tarski’s inductive definition of truth formalized inside ZF set theory. Recall the setup: a first-order language L and a structure S for L. The domain of S is a set (not a proper class), which means that the relations and functions on the domain are too. The fruit of the effort was a formula in L(ZF), True(L,S,φ), which expresses “S is a set-based structure for L and φ is a sentence of L and S ⊧ φ”. (With minor changes, we get a parametrized version, True(L,S,φ,u). Here u is a finite list of elements of the domain of S to be assigned to the free variables in φ.)
In the first post in this series, we looked at {Trued}, an infinite sequence of formulas in L(PA); Trued expresses truth for formulas with parse-depth at most d. In this post, we’ll look at SatΣn, satisfaction for Σn formulas.
SatΣn is the “professional-grade” version of Trued; it’s what you’ll find in the textbooks in the bibliography, like Kaye. (I suppose “satisfaction” sounds more professional than “truth”.) Like Trued, it’s an infinite sequence of formulas. Like Trued, it limits its ambitions by complexity: parse-tree depth for Trued, and the number of quantifier alternations for SatΣn. So (for example) the axiom ∀x∀y∀z((z=0 ∨ ¬x<y) ∨ x·z<y·z) has depth 6, but belongs to Π1. (SatΠn amounts to the negation of SatΣn, since a Πn sentence is true iff its negation, a Σn sentence, is false.)
The main effort in defining SatΣn takes place at the ground level, defining SatΔ0. Surprising! And this definition shares DNA with True(L,S,φ) from ZF. Here are some analogies to conjure with:
PA : infinite sets :: ZF : proper classes
PA : finite sets :: ZF : sets.
In ZF, quantification over a set is a “bounded search”, in some sense. In PA, the same holds for quantification over a finite range.
The construction of True(L,S,φ) rested on the concept of an instantiation tree. We can do nearly the same thing for Δ0 sentences. Here’s an example, for the sentence ∃x<3 ∀y<x ψ(x,y):
I’ve made a different graphic choice from the diagram for the ZF case. Instead of regarding the instantiations “∀y<0 ψ(0,y)”, “∀y<1 ψ(1,y)”, and “∀y<2 ψ(2,y)” as all “living in the same node”, I’ve split them into separate nodes. This reflects a more fundamental difference: in the ZF case, all the quantifications are “bounded” by the same set, the domain of S. Here the bound varies from node to node.
The instantiation tree must be grown from the root down. Our Δ0 formula is a sentence, i.e., no free variables. Of course subformulas need not be closed. Any outermost quantifier must have a fixed bound—that is, a closed term for the bound, which can be evaluated. The associated quantifier node will have a bounded list of children, in which the quantified variable has acquired a fixed value. The example above is particularly simple, but something like
∀x<10 (¬ξ(x) ∨ ∃y<3x ∀w<x+y (ζ(x,y,w)∧η(x,y,w)) )
goes much the same way. We’ll have 10 nodes under root, each a disjunction. The one with x=5 will have a right hand child ∃y<15… . One of those children has y=7, and is labeled ∀w<12 (i.e., x+y=5+7). Its 12 children are all conjunctions.
Truth evaluations proceed bottom up, of course, using the obvious rules. Now, can we code all this into PA? The diagrams look pretty when the bounds are all elements of ℕ, but over a nonstandard model the bounds can be nonstandard. That’s OK, provided we can code all this into a formula of PA. (And by ‘formula’, I mean an ordinary vanilla formula, not one of “nonstandard length” or anything like that.)
The short answer is, sure, no problem! PA is “basically the same as” ZF¬∞. The instantiation tree and its truth evaluation “look like” finite combinatorial objects, so of course PA can encode them. The “local behavior” aspect of the ZF case—a thousand quantifiers in the formula don’t bleed through to give a thousand quantifiers in True(…)—applies here too.
For the long answer, consult Kaye, Chapter 9. He begins by saying
This is the chapter that no one wanted to have to write: the material here is technical and difficult to describe clearly, yet it is very necessary for later work. I have put it off as long as possible, but cannot do so any longer.
Nineteen pages later, he gives the definition of SatΔ0. A few pages more establish its chief properties. Then in a quarter of a page, he defines the sequences SatΣn and SatΠn, building on top of SatΔ0. It’s much like Trued: a syntactic ∃x becomes semantic. Life is just a touch more complicated, because SatΣn must handle a block of existential quantifiers prepended to a Πn−1, formula; likewise for SatΠn. But PA has mastered lists, so this is barely a hiccup.
Just a few more remarks. If you do decide to read Kaye’s Chapter 9, be aware he does not explicitly use parse-trees. Instead, he ‘parses’ the Gödel numbers that represent strings of symbols. You will also find data structures that you can construe as instantiation trees, although he does not make this explicit. I confess I have not had the patience to go through his treatment line by line. My sympathy for the author’s pains!
As a reward for the detailed development, Kaye can show that SatΔ0 is a Δ1(PA) formula, and SatΣn and SatΠn are Σn(PA) and Πn(PA) formulas (respectively) for n>0. (The quantifiers “bleed through”.) A diagonal argument now shows that the Σn/Πn hierarchy is a true hierarchy: for any model N of PA, there is a relation on N that is Σn(N) but not Πn(N), and vice versa.
As with Trued, PA proves the expected equivalence for any sentence φ with the right complexity. That is, if φ belongs to Σn(PA), then PA⊢φ↔SatΣn(⌜φ⌝). Likewise for the parametrized version. Note that this is a collection of proofs, one for each formula φ.
That’s all for now!
Diagonal Argument 
By the way, I like “satisfaction” much better than “truth” in this context. We’re talking about whether a sentence is satisfied in some model. “Truth” in its full cosmic sense has very little to do with it.
Similarly I think that “Tarski’s theorem on the undefinability of truth” is a somewhat overblown name.
To me mathematical logic (as opposed to philosophical logic) has a lot to say about provability and satisfaction, but very little to say about truth, except perhaps indirectly. The axioms of ZFC could be false (whatever that means), but that wouldn’t affect mathematical logic—unless they were also inconsistent, which would affect what you can prove with them, and their models.
Ah, but “Truth” has such a satisfying ring to it!
A little more seriously, I don’t believe “Truth” has a single “full cosmic sense”. Just a lot of local senses. Artistic truth, scientific truth, emotional truth, mathematical truth, true colors, true friend…
I don’t think “Truth” has a single “full cosmic sense” either. I just meant I want to stay away from the vagueness and grandiosity of this concept when making mathematical claims. I don’t mind saying a theorem is true, meaning the usual thing: some sort of fuzzy mix of “I’ve proved it”, “I can prove it”, “I bet I can prove it”, and “I feel sure in my bones that someone could prove it”. But I don’t want my theorem statements making claims about “truth”.