Arithmetic With Satisfaction

A language in which we can express arithmetic and which contains its own satisfaction predicate (in the style of Kripke’s theory of truth) can be formulated using just two nonlogical primitives: ′ (the successor function) and Sat (a satisfaction predicate). Let L be a language with vocabulary: , ( ) ∃ ¬ ∨ = ′ Sat plus the variables x0, x1, x2, . . .. A term is a variable followed by zero or more occurrences of ′. An atomic formula is any formula of the form t0 = t1, Sat(t0), or Sat(t0, . . . , ti) (for any finite string of terms t0, . . . , ti). Nonatomic formulas are defined in the normal way. (Note that, though for simplicity we let Sat take any number of terms, this is not necessary for our purposes. We could consider just a 5-place predicate, Sat(x0, . . . , x4). More will be said about this later.) We will be concerned with partial interpretations of L in which the variables range over the natural numbers, ′ is interpreted as the successor function, and a disjoint pair of sets (S1, S2) of finite sequences of natural numbers is assigned to Sat. Let L(S1, S2) represent such an interpretation of L . Let s be an infinite sequence of natural numbers, and let s∗ be the corresponding assignment of natural numbers to terms (thus s∗(xi) = s(i) and s∗(t′) = the successor of s∗(t)). Then we say: L(S1, S2) |= Sat(t0, . . . , ti)[s] (i.e., L(S1, S2) satisfies Sat(t0, . . . , ti) with s) iff 〈s∗(t0), . . . , s∗(ti)〉 ∈ S1 . On the other hand, we say: L(S1, S2) =| Sat(t0, . . . , ti)[s] (i.e., L(S1, S2) falsifies Sat(t0, . . . , ti) with s) iff 〈s∗(t0), . . . , s∗(ti)〉 ∈ S2. And finally, L(S1, S2) leaves Sat(t0, . . . ti) undefined with respect to s if L(S1, S2) neither Received October 20, 1993; revised November 14, 1994