By Martin Zeman

This quantity is an advent to internal version conception, a space of set thought that is occupied with fantastic structural internal types reflecting huge cardinal houses of the set theoretic universe. The monograph incorporates a unique presentation of basic superb constitution conception in addition to a latest method of the development of small center types, particularly these versions containing at so much one powerful cardinal, including a few of their functions. the ultimate a part of the publication is dedicated to a brand new strategy encompassing huge internal versions which admit many Woodin cardinals. The exposition is self-contained and doesn't suppose any distinct prerequisities, which may still make the textual content understandable not just to experts but additionally to complex scholars in Mathematical good judgment and Set idea.

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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.

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