I am working on the following homework problem for a logic class on Godel's incompleteness theorems and the following question is asked.
Is the converse of Theorem $13.1$ true? Explain.
Theorem 13.1 states, "If a function is $\Sigma_{1}$ it is also $\Pi_{1}$." So, we are asked to prove or disprove that, "If a function is $\Pi_{1}$ it is also $\Sigma_{1}$." I think that the converse is true and I have attempted the question by giving the following "proof",
Assume that $f$ is a $\Pi_{1}$ function. That is, $f$ can be expressed by a strictly $\Pi_{1}$ wff. Let $\Phi(x,y):=\forall \eta_{1} \cdot \cdot \cdot \forall \eta_{k} \varphi(x,y)$ be such a $\Pi_{1}$ wff, where $\varphi(x,y)$ is $\Delta_{0}$. Well, by DeMorgan's law for quantifiers, we have that $\forall \eta_{1} \cdot \cdot \cdot \forall \eta_{k} \varphi(x,y) \equiv \exists \eta_{1} \cdot \cdot \cdot \exists \eta_{k} \neg \varphi(x,y)$, where $\neg \varphi(x,y)$ is again $\Delta_{0}$. So, $\Phi(x,y) = \exists \eta_{1} \cdot \cdot \cdot \exists \eta_{k} \neg \varphi(x,y)$ still expresses $f$ and so since $\Phi(x,y)$ is not only $\Pi_{1}$, but $\Sigma_{1}$, we conclude that $f$ is also $\Sigma_{1}$.
This seems to be a sufficient argument for my tastes (disregard my idiotic proof), but the proof the book gave for Theorem 13.1 seems to be of quite a different style, I will post it here:
Suppose the one-place function $f$ can be expressed by the strictly $\Sigma_{1}$ wff $\varphi(x,y)$. Since $f$ is a function, and maps numbers of unique values, we have $f(m) =n$ if and only if $\forall z ( f(m) = z \rightarrow z = n)$. Hence $f(m)=n$ if and only if $\forall z(\varphi(\bar{m},z) \rightarrow z= \bar{n})$ is true. In other words, $f$ is equally well expressed by $\forall z(\varphi(x,z) \rightarrow z=y)$. But it is a trivial exercise of moving quantifiers around to show that if $\varphi(x,y)$ is strictly $\Sigma_{1}$, then $\forall z(\varphi(x,z) \rightarrow z=y)$ is $\Pi_{1}$.
(Note that, $\bar{x}$ means the value of $x$, meaning $\underbrace{S \cdot \cdot \cdot S}_\text{x times}0$).
It seems that the book is actually using an explicit wff to express $f$, in my case I just keep $\Phi$ as some ethereal wff which we assume expresses $f$; is there any problem with this? I essentially want to know if my proof is adequate, and if not where I went wrong or what I misunderstand. It seems to me utterly trivial to prove both Theorem 13.1 and its converse if the method I provided is correct, which is why I suspect that my method is incorrect. Any suggestions and nudges in the right direction would be greatly appreciated, thank you!