I am trying to prove that all recursive functions are representable in the theory $R$ whose language is $L$ and whose theorems are the consequences in $L$ of the following infinitely many sentences:
- $\mathbf{i}\ne\mathbf{j}$ for all $i,j$ such that $i\ne j$
- $\mathbf{i}+\mathbf{j}=\mathbf{k}$ for all $i,j,k$ such that $i+j=k$
- $\mathbf{i}\cdot\mathbf{j}=\mathbf{k}$ for all $i,j,k$ such that $i\cdot j=k$
- $\forall x(x<\mathbf{i}\to x=\mathbf{0}\lor\dots\lor x=\mathbf{i-1})$ for all $i$
- $\forall x(x<\mathbf{i}\lor x=\mathbf{i}\lor\mathbf{i}
for all $i$
I am having trouble proving minimization, composition, successor, projection, addition and multiplication.
I attempted to prove the theorems using Boolos' model of proof from $Q$ but I quickly got stuck, since the proof of representability using $Q$ in Boolos uses axioms $R$ doesn't have. The notes I am using can be found here starting on page 94.