Download A Guide to NIP Theories by Pierre Simon PDF

Download A Guide to NIP Theories by Pierre Simon PDF

By Pierre Simon

The research of NIP theories has obtained a lot awareness from version theorists within the final decade, fuelled by means of purposes to o-minimal buildings and valued fields. This publication, the 1st to be written on NIP theories, is an creation to the topic that would attract a person attracted to version concept: graduate scholars and researchers within the box, in addition to these in close by parts reminiscent of combinatorics and algebraic geometry. with no living on anyone specific subject, it covers all the uncomplicated notions and offers the reader the instruments had to pursue learn during this quarter. An attempt has been made in every one bankruptcy to offer 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 correct fabric from different fields of arithmetic is made available to the truth seeker.

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Extra info for A Guide to NIP Theories

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Define the following sets: Si = {p ∈ Sφ (Ai ) : p has ≥ extensions to Sφ (A)}, for i < ; S = {p ∈ Sφ (A) : p Ai ∈ Si for all i < }. Note that for every i < , as |Sφ (Ai )| < , we have |Sφ (Ai ) \ Si | < and as each type from that set has less than extensions to a type over A, the cardinality of {p ∈ Sφ (A) : p Ai ∈ / Si } is less that . Summing over i < , we see that |Sφ (A) \ S | < . It follows that every type in Si has at least extensions to a type in S . Let S< = i< Si and S≤ = S< ∪ S . We define a linear order on S≤ in the following way.

28 2. 69. Let φ(x; y) be a formula. Assume that there is some infinite set A with |Sφ (A)| > ded(|A|), then φ(x; y) has IP. Proof. Assume that |Sφ (A)| > ded(|A|) and that A is chosen such that = |A| is minimal. Let = ded(|A|)+ . Enumerate A as {ai : i < } and for i < , set Ai = {aj : j < i}. For each i < , we then have |Sφ (Ai )| ≤ ded(|Ai |) < . Define the following sets: Si = {p ∈ Sφ (Ai ) : p has ≥ extensions to Sφ (A)}, for i < ; S = {p ∈ Sφ (A) : p Ai ∈ Si for all i < }. Note that for every i < , as |Sφ (Ai )| < , we have |Sφ (Ai ) \ Si | < and as each type from that set has less than extensions to a type over A, the cardinality of {p ∈ Sφ (A) : p Ai ∈ / Si } is less that .

Most results there require stronger conditions than NIP (in particular strongly dependent which we will define in Chapter 4). In a completely different direction, Macpherson and Tent have studied pseudofinite NIP groups in [80]. ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ Chapter 3 HONEST DEFINITIONS AND APPLICATIONS A fundamental characteristic property of stable theories is definability of types, namely the property that if A ⊂ U is any subset (big or small), and φ(x; b) ∈ L(U), a formula, then the set φ(A; b) coincides with the trace on A of some A-definable set: there is (x; d ) ∈ L(A) such that φ(A; b) = (A; d ).

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