The translation of Godel's original Completeness Theorem I am reading is contained in the book "From Frege to Godel." I know people may not have read through the original proof (and they will need to have done so to get the context of my question) but I thought I would ask it anyways. I have left a lot of the terms undefined, I can define all the relevant terms if people don't have access to the original proof.
So I have understood every step of the proof, up until the last part. I understand he is using Zorn's lemma, but I can't mentally visualize the construction he is making. Here is the part I don't understand (the confusing sections are in bold):
"Assume that the functional variables $ F_1, F_2, ... , F_k $ and the propositional variables $X_1,X_2,...,X_l$ occur in $A$. Then $A_n$ consists of elementary components of the form:
$F_1, (x_{p_1},...x_{q_1}), F_2 (x_{p_2},...,x_{q_2}),..., X_1,X_2,...,X_l$
compounded solely by means of the operations $\lor $ and $ \neg $. With each $A_n$ we associate a formula $B_n$ of the propositional calculus by replacing the elementary components of $A_n$ by propositional variables, making certain that different components (even if they differ only in the notation of the individual variables) are replaced by different propositional variables. Furthermore, we understand by "satisfying system of level $n$ of $P(A)$" a system of functions $f^n_1, f^n_2,...,f^n_k $ defined in the domain of integers $z (0 \leq z \leq ns)$ as well as truth values $w^n_1, w^n_2, ..., w^n_l $ for the propositional variables $X_1, X_2,...,X_l$ such that a true proposition results if in $A_n$ the $F_i$ are replaced by the $f^n_i $, the $x_i$ by the numbers $i$, and the $X_i$ by the corresponding truth values $w^n_i$. Satisfying systems of level $n$ obviously exist if and only if $B_n$ is satisfiable.
Each $B_n$, being a formula of the propositional calculus, is either satisfiable or refutable (Lemma 7). Thus only two cases are conceivable.
- At least one $B_n$ is refutable. Then, as we can easily convince ourselves (Rules of inference 2 and 3, Lemma 1(c)), the corresponding $(P_n)A_n$ is refutable also, and consequently, because of the provability of $(P)A \rightarrow (P_n)A_n$, so is $(P)A$.
- No $B_n$ is refutable; hence all are satisfiable. Then there exist satisfying systems of every level. But, since for each level there is only a finite number of satisfying systems (because the associated domains of individuals are finite) and since furthermore every satisfying system of level $n+1$ contains one of level $n$ as a part (as is clear from the fact that the $A_n$ are formed by successive conjunctions), it follows by similar arguments that in this case there exists a sequence of satisfying systems $S_1, S_2,..., S_k,... $ ($S_k$ being of level $k$) such that each contains the preceding one as a part. We now define in the domain of $all$ integers $0 \geq$ a system $S= \{\phi_1,\phi_2,...,\phi_k ; \alpha_1,\alpha_2,...,\alpha_l\}$ by means of the following stipulations:
1) $\phi_p(a_1,...,a_i) (1\leq p \leq k)$ holds if and only if for at least one $S_m$ of the sequence above (and then for all subsequent ones also) $f^m_p (a_1,...,a_i)$ holds.
2) $\alpha_i=w^m_i (1\leq i \leq l)$ for at least one $S_m$ (and then for all those that follow).
Then it is evident at once that $S$ makes the formula $(P)A$ true. In this case, therefore, $(P)A$ is satisfiable, which concludes the proof of the completeness of the system of axioms given above."
I am confused because the functions $\phi_p$ are defined by an if and only if statement instead of an explicit definition. Is $\phi_p$ simply one of the $f^m_p$ functions but extended to the domain of integers? I don't that is it.