Brittish Mathematical Olympiad

Logic

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Eventually, some T E cb is found such that W C T * S, SO that W C S if and only if W k T, and the latter is effectively decidable. But our development in Section 2 offers an apparently more efficient method, which we describe in the proof of our next theorem. , Kleene [ 5 2 ] . We shall describe various operators on formulas and claim that they are primitive recursive. Formally verifying such claims is an arduous but routine task, which we omit (the intuitive idea is that when the application of some operator to a formula P is described as a series of primitive recursive steps, and the number of such steps can be bounded in advance by a primitive recursive function of the number of symbols in F, then the operator itself is primitive recursive.

At the beginning of this section, we remarked that there is 8 close syntac- which w i l l enable US to tical relationship between the theories w and W', deduce from Theorem 19 a similar theorem for W (the Reduction Theorem). The following definitions present some notions needed for this step. DEFINITION 20. fonmcla of of form Vx F there is an 8 E I' is bounded if for each of i t s s u b f o m l a s F v ( V x F ) u {-) Vx F 4, such that Vx(x

4) Now 8 b Mxu(O,-) iff either E 1, or else E = 0 and Xu 2 u or An, # 0 for some x * > x. These are precisely the mame condition8 under which I b Mxu(O,-). 8 Nu,(-) i f f w > xu > 0 or xu w + w*. Assin these are the same conditions under which I b Nx,(-). This proves (21, and the theorem. E. DONER. A. MOSTOWSKI and A. TARSKI COROLLARY 42. Every complete extension of has a prime model. W COROLLARY 43. Every complete extension of W with i n f i n i t e models ha8 non-well-ordered models. In fact, W has complete extensions vithout any wellordered models.

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