2.6.6: Construction of a Model
Right now we are not concerned about \(\eq[][]\) , i.e., we only want to show that a consistent set \(\Gamma\) of sentences not containing \(\eq[][]\) is satisfiable. We first extend \(\Gamma\) to a consistent, complete, and saturated set \(\Gamma^*\) . In this case, the definition of a model \(\Struct{M(\Gamma^*)}\) is simple: We take the set of closed terms of \(\Lang{L'}\) as the domain. We assign every constant symbol to itself, and make sure that more generally, for every closed term \(t\) , \(\Value{t}{M(\Gamma^*)} = t\) . The predicate symbols are assigned extensions in such a way that an atomic sentence is true in \(\Struct{M(\Gamma^*)}\) iff it is in \(\Gamma^*\) . This will obviously make all the atomic sentences in \(\Gamma^*\) true in \(\Struct{M(\Gamma^*)}\) . The rest are true provided the \(\Gamma^*\) we start with is consistent, complete, and saturated.
Let \(\Gamma^*\) be a complete and consistent, saturated set of sentences in a language \(\Lang L\) . The term model \(\Struct M(\Gamma^*)\) of \(\Gamma^*\) is the structure defined as follows:
- The domain \(\Domain{M(\Gamma^*)}\) is the set of all closed terms of \(\Lang L\) .
- The interpretation of a constant symbol \(c\) is \(c\) itself: \(\Assign{c}{M(\Gamma^*)} = c\) .
- The function symbol \(f\) is assigned the function which, given as arguments the closed terms \(t_1\) , …, \(t_n\) , has as value the closed term \(f(t_1, \dots, t_n)\) : \[\Assign{f}{M(\Gamma^*)}(t_1, \dots, t_n) = f(t_1,\dots, t_n)\nonumber\]
- If \(R\) is an \(n\) -place predicate symbol, then \[\tuple{t_1, \dots, t_n} \in \Assign{R}{M(\Gamma^*)} \text{ iff } \Atom{R}{t_1, \dots, t_n} \in \Gamma^*.\nonumber\]
A structure \(\Struct{M}\) may make an existentially quantified sentence \(\lexists{x}{A(x)}\) true without there being an instance \(A(t)\) that it makes true. A structure \(\Struct{M}\) may make all instances \(A(t)\) of a universally quantified sentence \(\lforall{x}{A(x)}\) true, without making \(\lforall{x}{A(x)}\) true. This is because in general not every element of \(\Domain{M}\) is the value of a closed term ( \(\Struct{M}\) may not be covered). This is the reason the satisfaction relation is defined via variable assignments. However, for our term model \(\Struct{M(\Gamma^*)}\) this wouldn’t be necessary—because it is covered. This is the content of the next result.
Let \(\Struct M(\Gamma^*)\) be the term model of Definition \(\PageIndex{1}\) .
- \(\Sat{M(\Gamma^*)}{\lexists{x}{A(x)}}\) iff \(\Sat{M}{A(t)}\) for at least one term \(t\) .
- \(\Sat{M(\Gamma^*)}{\lforall{x}{A(x)}}\) iff \(\Sat{M}{A(t)}\) for all terms \(t\) .
Proof.
- By Proposition 5.12.4 , \(\Sat{M(\Gamma^*)}{\lexists{x}{A(x)}}\) iff for at least one variable assignment \(s\) , \(\Sat[,s]{M(\Gamma^*)}{A(x)}\) . As \(\Domain{M(\Gamma^*)}\) consists of the closed terms of \(\Lang{L}\) , this is the case iff there is at least one closed term \(t\) such that \(s(x) = t\) and \(\Sat[,s]{M(\Gamma^*)}{A(x)}\) . By Proposition 5.13.3 , \(\Sat[,s]{M(\Gamma^*)}{A(x)}\) iff \(\Sat[,s]{M(\Gamma^*)}{A(t)}\) , where \(s(x) = t\) . By Proposition 5.12.3 , \(\Sat[,s]{M(\Gamma^*)}{A(t)}\) iff \(\Sat{M(\Gamma^*)}{A(t)}\) , since \(A(t)\) is a sentence.
- Exercise.
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Complete the proof of Proposition \(\PageIndex{1}\) .
Suppose \(A\) does not contain \(\eq[][]\) . Then \(\Sat{M(\Gamma^*)}{A}\) iff \(A \in \Gamma^*\) .
Proof. We prove both directions simultaneously, and by induction on \(A\) .
- \(\indcase{A}{\lfalse}\) \({\SatN{M(\Gamma^*)}{\lfalse}}\) by definition of satisfaction. On the other hand, \(\lfalse \notin \Gamma^*\) since \(\Gamma^*\) is consistent.
- \(\indcase{A}{R(t_1, \dots, t_n)}\) \(\Sat{M(\Gamma^*)}{\Atom{R}{t_1, \dots, t_n}}\) iff \(\tuple{t_1, \dots, t_n} \in \Assign{R}{M(\Gamma^*)}\) (by the definition of satisfaction) iff \(R(t_1, \dots, t_n) \in \Gamma^*\) (by the construction of \(\Struct M(\Gamma^*)\) ).
- \(\indcase{A}{\lnot B}\) \(\Sat{M(\Gamma^*)}{\indfrm}\) iff \({\SatN{M(\Gamma^*)}{B}}\) (by definition of satisfaction). By induction hypothesis, \({\SatN{M(\Gamma^*)}{B}}\) iff \(B \notin \Gamma^*\) . Since \(\Gamma^*\) is consistent and complete, \(B \notin \Gamma^*\) iff \(\lnot B \in \Gamma^*\) .
- \(\indcase{A}{B \land C}\) exercise.
- \(\indcase{A}{B \lor C}\) \(\Sat{M(\Gamma^*)}{\indfrm}\) iff \(\Sat{M(\Gamma^*)}{B}\) or \(\Sat{M(\Gamma^*)}{C}\) (by definition of satisfaction) iff \(B \in \Gamma^*\) or \(C \in \Gamma^*\) (by induction hypothesis). This is the case iff \((B \lor C) \in \Gamma^*\) (by Proposition 10.3.1 (3) ).
- \(\indcase{A}{B \lif C}\) exercise.
- \(\indcase{A}{\lforall{x}{B(x)}}\) exercise.
- \(\indcase{A}{\lexists{x}{B(x)}}\) \(\Sat{M(\Gamma^*)}{\indfrm}\) iff \(\Sat{M(\Gamma^*)}{B(t)}\) for at least one term \(t\) (Proposition \(\PageIndex{1}\) ). By induction hypothesis, this is the case iff \(B(t) \in \Gamma^*\) for at least one term \(t\) . By Proposition 10.4.2 , this in turn is the case iff \(\lexists{x}{B(x)} \in \Gamma^*\) .
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Complete the proof of Lemma \(\PageIndex{1}\) .