The number of countable models
نویسندگان
چکیده
Definition 1.1 T is small if for all n < ω, |Sn(∅)| ≤ ω. Remark 1.2 If T is small, then there is a countable L0 ⊆ L such that for every φ(x) ∈ L there is some φ′(x) ∈ L0 such that in T , φ(x) ≡ φ′(x). Hence, T is a definitional extension of the countable theory T0 = T L0. Proof: See Remark 14.25 in [4]. With respect to the second assertion, consider some n-ary relation symbol R ∈ LrL0. There is some formula φ(x1, . . . , xn) ∈ L0 equivalent to Rx1 . . . xn in T . If we add all the definitions ∀x1 . . . xn(Rx1 . . . xn ↔ φ(x1, . . . , xn)) (and similar definitions for constants and function symbols) to T0 we obtain T . 2
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