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# 45.14 - Henkin's Model and Metatheorem 45.14 Branden...

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Henkin’s Model and Metatheorem 45.14 Branden Fitelson 04/10/07 Henkin’s Model . Let T be a consistent, negation-complete, and closed first order theory. Henkin’s model M is a a denumerable interpretation for T such that for each WFF A of T , A is true on M iff T A . The existence of such a model M undergirds metatheorem 45.14. Characterizing M will involve doing five things: (1) specifying M ’s (denumerable) domain D , (2) saying for each constant symbol c of T which object d in the domain M assigns to c , (3) saying for each n -place function symbol f which n -ary function f is assigned to f by M , (4) saying for each n -place predicate symbol F which n -ary property F ( i.e. , which set of ordered n -tuples of closed terms of T , since we identify properties with their extensions ) is assigned by M to F , and (5) saying for each propositional symbol p of T , what truth-value is assigned to p by M . Here is Henkin’s M , followed by a proof of 45.14 (arguably the most important metatheorem of the entire course). 1. The domain D of M is the set of closed terms of T . This set contains all the constant symbols a 0 , a 00 , a 000 , . . . , b 0 , b 00 , b 000 , . . . , c 0 , c 00 , c 000 , . . . of T (the b ’s and c ’s are effectively enumerable sets of new constant symbols that may be added to Q for Q + purposes and/or for the purpose of ensuring T is closed). D also contains all the closed terms with function symbols: f *0 a 0 , f **0 b 0 a 00 , . . . of T . Important Digression on Symbols, Abstract Objects, Types, and Tokens . It is important to note that the symbols of T are abstract objects , and they are types not tokens . You should not confuse a token of a symbol with the symbol itself. For instance, when I write a token inscription “ a 0 ” (the physical inscription between the quotation marks preceding this parenthetical remark), I have not written down the symbol itself. It is not tokens of symbols of T that get assigned to objects by M , but rather the symbols themselves. For instance, when I say that the numeral “1” gets interpreted as the number one (which is

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45.14 - Henkin's Model and Metatheorem 45.14 Branden...

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