By Pierre Simon

ISBN-10: 1107057752

ISBN-13: 9781107057753

The learn of NIP theories has obtained a lot consciousness from version theorists within the final decade, fuelled via purposes to o-minimal buildings and valued fields. This e-book, the 1st to be written on NIP theories, is an creation to the topic that may entice somebody attracted to version thought: graduate scholars and researchers within the box, in addition to these in within reach components equivalent to combinatorics and algebraic geometry. with no living on anybody specific subject, it covers all the simple notions and offers the reader the instruments had to pursue learn during this quarter. An attempt has been made in each one bankruptcy to provide a concise and chic route to the most effects and to emphasize the main invaluable rules. specific emphasis is wear sincere definitions, dealing with of indiscernible sequences and measures. The suitable fabric from different fields of arithmetic is made obtainable to the philosopher.

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**Example text**

54. We say that a formula φ(x; y) has the binary tree property if there are |y|-tuples (bf : f ∈ < 2) such that for every f ∈ 2, the partial type {φ(x; bf i )f(i) : i < } is consistent. 55. Let φ(x; y) be a formula. The following are equivalent: (i) for any infinite set |A|, |Sφ (A)| ≤ |A|; (ii) for some infinite cardinal , for every set A of size , |Sφ (A)| ≤ |A|; (iii) φ(x; y) does not have the order property; (iv) φ(x; y) does not have the binary tree property; (v) Any φ-type over a set A is definable.

Xn ; z) depends only on φ(x1 , . . , xn ; y) and not on the parameters b. By A being “big or small”, we mean that there is no cardinality restriction on A: it could be a small subset, a definable set, a type-definable set etc. 4. Let M = (R; ≤), seen as a model of DLO. Then M is (uniformly) stably embedded in U. More generally, a small set A is weakly stably embedded in U if and only if all types over A are definable. The theory T is stable if and only if any set A is weakly stably embedded if and only if any set A is stably embedded.

Induced structure. Let A ⊆ U be a set big or small (as above, the relevant cases are: A is a small set, A is a definable set or A is a type-definable set). We would like to consider A as a structure in its own right. 6. Let A ⊆ U be any subset and let B ⊂ U be a small set of parameters. We define Aind(B) to be the structure in the language LB = {Rφ(x) ¯ : φ(x) ¯ ∈ L(B)} whose universe is A and where each Rφ (x) ¯ ¯ (x) is interpreted the obvious way: for every a¯ ∈ A, Aind(B) |= Rφ (a) ¯ ⇐⇒ U |= ¯ φ(a).

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