kenny_handout - Notes on Isomorphism, Elementary...

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Unformatted text preview: Notes on Isomorphism, Elementary Equivalence, the Generalized Lwenheim-Skolem-Tarski Theorem, Categoricity, Non-Standard Models, etc. Kenny Easwaran 05/10/05 1 Preliminaries and Setup I will use the notation , rather than F 1 , for the two-place predicate whose intended (normal) interpretation is the identity relation. And, I will use the notation x y , rather than p x y q , to express the claim that x and y are non-identical. A theory is just a set of formulas in a particular language. A model for a theory is a interpretation on which every formula in the set is true (and thus every conse- quence of the set is true on such an interpretation as well). We will only consider normal models here, i.e. , models where the symbol is interpreted as the identity relation. The size of a model is the cardinality of the domain of the interpretation. Thus, the standard model of any theory of the natural numbers is countable, while any model of the theory consisting just of x y p x y q has size 1. A consistent theory is one from which no contradictions can be derived. A max- imal consistent theory is a theory T in a language L such that for every sentence S P L , if S R T then T Yt S u is inconsistent. Clearly, any maximal consistent theory is negation-complete , meaning that for any sentence S P L , either S P T or S P T . T is said to be Henkinized if, for every formula A of one free variable, some sentence H called its Henkin sentence given as p Ac { v v A q is in T , where v is the one free variable in A and c is some constant. A Henkinized theory is clearly one where if T $ QS = Ad { v for all d in the language, then T $ QS = v A , because some d will be identical to the c in the Henkin sentence. 2 Isomorphism and elementary equivalence Let M 1 and M 2 be two interpretations for a language L . That is, each M i will specify a domain D i , together with an object in that domain for each constant symbol in the language, a function from the n-tuples of the domain to the domain for each n-place function symbol, and a set of n-tuples from the domain for each n-place relation symbol. In particular, since were only considering normal models, both must specify that the symbol gets interpreted as the actual identity relation on the domain. We will say that M 1 is isomorphic to M 2 if there is a way to pair up the elements of the two domains that preserves all the structure mentioned in the language. That is, there should be a function f from D 1 to D 2 such that no two elements of D 1 get sent to the same element of D 2 and every element of D 2 is the image of some element of D 1 under f . In addition, if x is the object assigned to constant c by M 1 , then f p x q must be the object assigned to c by M 2 . Similarly, x x 1 ,..., x n y is in the interpretation of some n-place relation R under M 1 iff x f p x 1 q ,..., f p x n qy is in the interpretation of R under...
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kenny_handout - Notes on Isomorphism, Elementary...

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