8/12/2019 What is Model Theory (Jhonatan Kirby)

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8/12/2019 What is Model Theory (Jhonatan Kirby)

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cardinality - then we can deduce a lot about the structure S itself, and

about all of the other models ofT. IfThas a few models in each cardinalitythen we can say something else about the models and if it has lots we can saysomething else again. In general, the fewer models a theory has, the more wecan say about them. Theories with few models are called stable, and thosewith many models are called unstable.

Definability

The other main idea in model theory is that of definability. Given a structureM, some of its subsets will be definable in the language, and others wont. Forexample, in the structure of the natural numbers with the language , 2,the set of all even numbers is definable because in this language we canexpress the idea that a number, n, is even. The sentence (m)[n= 2m]says that there is some number m such that n is equal to 2 m, which isprecisely what it means for nto be an even number. However, the set of allnumbers divisible by 3 is not definable in this structure. I havent explainedenough about constructing sentences of the language from the building blocksto prove this, so youll have to take my word for it or read about it elsewhere.

For some structures, all the definable sets can be described very easily. Ifthis holds for a structure S then it also holds for every other model of thetheory ofS. This is typically true of structures with stable theories, but can

also be true of those with unstable theories. In this case we might have astructure which is very complicated (which we know because its theory is un-stable) but where the definable sets are very simple, so all the complicationsare invisible to the language. A typical example of this is the field of all realnumbers, often thought of as all decimal expansions like 95.235233177 . . ., inthe language with building blocks representing addition and multiplication.

The idea of definability is very important to applications of model theory,since there are many structures in mathematics which are very complicatedbut which people want to study. In particular, measurements in the realworld use real numbers, and its quite common to want to add and multiply

them, so the field of real numbers is a structure which is used all the time.Many questions about such structures turn out to be expressible in the lan-guage of the structure, and so relate to definable sets. In this case, modeltheory allows us to reduce the questions about the complicated structure toquestions about the simple definable sets, which are much easier to answer.

This is actually a formula, not a sentence, but this distinction isnt important here.