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Please give the most illuminating example of a model for a formal system, and a simple example of its use. I also wish an example of an interpretation, and what its useful for.

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The simplest example would be a single axiom system, $\exists x (x=x)$, with model a set with a single element. Are you really sure you want to ask for "the simplest example"? – Arturo Magidin Oct 15 '11 at 21:50
I'm not really sure what you mean by "useful". – Asaf Karagila Oct 15 '11 at 22:29
@Asaf Please undelete your answer. – turian Oct 15 '11 at 22:39
@turian: Sure, but I'm not sure that it has answered the "useful" part. – Asaf Karagila Oct 15 '11 at 22:41
up vote 5 down vote accepted

A formal system is a syntactical concept. It includes a language in which there are relation symbols, functions and constant symbols. We can write sentences and we can prove things from assertions written in the language. If we have a collection of such assertions, from which we cannot derive contradiction then we say that this collection is a theory.

A model is a semantical concept. It interprets the language, and through this interpretation we can say whether or not a statement is true in that model .

When we say "Of course $0\neq 1$", we know that this is a true statement, but formally we can define a theory and a model of this theory which interprets $0$ and $1$ the same way, and so in that model $0=1$.

So we first need to understand that a model interprets the language, and is bounded by the assertions of the theory, which we will call axioms.

For example, take the language in which there is one constant $c$ and two relation symbols $=$ and $<$, both are binary relations. We can now write the following axioms:

  1. $\forall x(\lnot(x<x))\land\forall x\forall y\forall z(z<x\land x<y\rightarrow z<y)\land\forall x\forall y(x=y\lor x<y\lor y<x)$, The relation $<$ is a strict total order relation;
  2. $\forall x(c<x)$ the constant $c$ is the $<$-minimum element;
  3. $\forall x\exists y(x<y)$ the order $<$ has no maximal element.

We can interpret this system in the model $N$ which we define as:

  • $|N|=\mathbb N$, that is the element of the model $N$ are the natural numbers.
  • $<^N$, the interpretation of $<$ is going to be the "usual" interpretation, what does that mean? It means that $0<1<2<3<\ldots$, as we are used to think informally.
  • $c^N=0$, we interpret the constant $c$ as our minimal element, that is $0$.
  • $=^N$ is interpreted, as anywhere else in mathematics as equality. $x=y$ if and only if they are the same number.

Note that we could not have interpreted $c$ as any other number without altering $<$ too.

It is quite simple to see that this is indeed a model of the formal system. However we could have chosen a different model, we could have defined it to be $Q$ which will be composed of the rational numbers which are not negative, or any other way.

Note that the axioms of this theory cannot be satisfied by a finite model, this is because we require there will be no $<$-maximal element, and a finite ordering always has a maximal element.

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