I am reading the handbook of philosophical logic, elementary predicate logic by Wilfred Hodges and have a question in section 17 covering the Consequences of the construction of Models.
In the section previous (16) the book describes how every Hintikka set has a model and then shows how we construct a Hintikka set from a given consistent theory which must also be a model of the theory. It shows three arguments on how to construct this set, namely direct, tree and maximising arguments.
In the consequences section it changes the Hintikka set construction slightly. Firstly instead of using the individual constants of the language as elements we can number the constants and then use the numbers themselves as elements. Since numbers can be thought of as pure sets the structure we end up with will be a pure set structure.
It then states:
Theorem 10 Suppose T is a theory and $\psi$ a sentence of L such that the calculus doesn't prove $\psi$ from T. Then there is a pure set structure which is a model of T and not of $\psi$.
My Proof Since $T\not\vdash\psi$ then $T\cup\{\neg\psi\}$ is consistent and so has a model. We can construct the model ( as described above ) so that it is a pure set structure. This structure will be a model of $T\cup\{\neg\psi\}$ and therefore a model of $T$ and $\neg\psi$ so that it is a model of $T$ and not of $\psi$
The section then goes on to say that we can go one step further and encode all symbols and formulas of L as natural numbers. So a theory in L will be a set of numbers. Suppose the theory T is in fact the set of all numbers that satisfy the first order formula $\phi$ in the langage of arithmetic. Then by analysing the proof of theorem 10 we can find another first order formula $\chi$ in the language of arithmetic which defines a structure with natural numbers as it's elements so that
Theorem 11 In first order Peano Arithmetic we can prove that if some standard proof calculus doesn't prove T is inconsistent then the structure defined by $\chi$ is a model of T.
My Question
I don't understand where $\chi$ comes from analysing the proof of theorem 10. How do we find the first order formula $\chi$ ?
How do we then prove theorem 11 ?
Cheers