A remark on Africk's paper on Scott's interpolation theorem for L_ω1ω

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In order to prove that the Scott's interpolation theorem fails in L_ω1ω, H. Africk proved the following lemma in [1]. (See [1] for notations.)

Africk's Lemma. Suppose that and . Then every -sentence in L_ω1ω is equivalent to a sentence of the form and a sentence of the form , where A_i,j, B_i,j, are F_j-sentences and there are only countable distinct A_i,j or B_i,j together.

But this lemma is false as is shown in the following: Suppose that Z = {0,1}, F_j = {P_k,j}_k⊂ω and . Let A be the sentence ⋀_k⊂ω(∀xP_k,0(x) ∨ ∀xP_k,1(x)). If Africk's Lemma is true, then there are countably many F_j-sentences , such that A is equivalent to Since is countable, there are only countably many distinct pairs (A_i,0, A_i,1). So, we get a countable set {(B_i,0, B_i,1)}_i⊂ω, such that A is equivalent to ⋁_ik⊂ω(B_k,0 ∧ B_k,1). Since B_k,0 ∧ B_k,1 → ∀xP_k,0(x) ∨ ∀xP_k,1 is l-valid(i.e. valid in all first-order structures of cardinality 1) for every k ϵ ω, we have that either B_k,0 → ∀xP_k,0(x) is 1-valid or B_k,0 → ∀xP_k,1(x) is 1-valid. Let I be the set of all the k ϵ ω such that B_k,0 → ∀xP_k,0(x) is 1-valid and the first-order structure defined by , , if k ϵ I and , if k ∉ I.