I'm working on this homework problem:
We know that all primitive recursive formulas can be expressed in $L_A$. We could have just define new function symbols for them then, to save space. Show that although all PR functions can be expressed in $L_A$, they can't all be expressed as terms.
My first guess was that the solution would involve showing that the set of PR functions is not countable (I posted another question asking about that earlier today), but it turns out that this is not the case. I don't see how this question could possibly have a solution, then.
Can someone give me a suggestion?
In the first-order language of arithmetic, terms do not provide a lot to work with. You have the variable that represents the input to your function, and then $0$, $S$, $+$, $\times$ and naught else.
You should be able to prove (by induction on the structure of the term) that every function you can express in that way is non-decreasing.
If you can then show just one primitive recursive function with, say, $f(0)=1$ and $f(1)=0$, you will have completed your proof.