Download A Set Theory Workbook by Iain T. Adamson PDF
By Iain T. Adamson
Offers a proper description of set idea according to the Von Neumann-Bernays-Godel axiomatic method utilizing the idea that of sessions. Covers the basis of the idea, kin, ordinals, cardinals, and the axiom of selection. Paper. DLC: Set conception.
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A type p(x) ∈ S(B) which does not fork over A ⊆ B has also a global nonforking extension p(x) ∈ S(C) which does not fork over A. Therefore, in a simple theory any type has a global nonforking extension. Proof. The same argument as for a nonforking extension over a small set. 9. Let (I, <) be a linearly ordered set. The sequence (ai : i ∈ I ) is A-independent (or independent over A) if for every i ∈ I , ai | a
12. 6. For each set ∆ = {ϕ1 (x, y1 ), . . , ϕn (x, yn )} of formulas ϕi (x, yi ) ∈ L, there is a formula ∆ (x, z) ∈ L such that for all (x), for all k, D( , ∆, k) = D( , ∆ , k). Proof. 4. By induction on α we see that for each and k, D( , ∆, k) ≥ α if and only if D( , ∆ , k) ≥ α. This is clear for α = 0 and follows from the induction hypothesis for limit α. The case α + 1 is easy and only requires noticing that ∆ is finite and therefore any infinite sequence of ∆-formulas contains an infinite subsequence of instances of a single formula.
Proof. The direction from right to left is obvious from the definitions of D-rank and dividing. For the other direction, assume D( (x), ∆, k) ≥ α + 1. Let > 2|T |+|A| . 3 and compactness, we see that there are ϕ(x, y) ∈ ∆ and (ai : i < ) such that for each i < , D( (x) ∪ {ϕ(x, ai )}, ∆, k) ≥ α and {ϕ(x, ai ) : i < } is k-inconsistent. By choice of , there is an infinite subset I ⊆ such that ai ≡A aj for all i, j ∈ I . Then it suffices to take a = ai for some i ∈ I . 11. For any partial type (x) over A, any ϕ = ϕ(x, y) ∈ L, any k < , and any ordinal α ≤ the following are equivalent: 1.