Ref.to Raymond Smullyan, First-Order Logic (1968 – Dover reprint).
Some background :
[pag.44] - individual variables (to be used bound) and individual parameters (to be used free)
[pag.47] - first-order valuation (basically, the standard semantics for f-o logic) : based on a non-empty set $U$ (universe of individuals or domain) with the quantifier clause : $\forall xA$ is true iff for every $k \in U$, $A(k/x)$ is true
[pag.50] – “ for a set $S$ of *sentences with parameters [we] consider a mapping $\phi$ form the set of parameters of $S$ into a universe $U$; for any $A \in S$, by $A^{ \phi}$ we mean the result of substituting for each parameter $a_i$ of $A$ its image $\phi(a_i)$ under $\phi$. Now we say that $S$ is (simultaneously) satisfiable in $U$ if there exists an interpretation $I$ of the predicates of $S$ into elements of $U$ and there exists a ”substitution” $\phi$ mapping the parameters of $S$ into elements of $U$ such that for any $A \in S$, $A^{ \phi}$ is true under $I$.”
[pag.51] – “consider the universe $V$ whose elements are the parameters themselves”
[pag.53-60] – the basic construction in the proof of Completeness Th for a f-o formula $A$ amount to generating a complete tableaux with an open branch $\theta$ for a satisfiable formula $A$. In this way we obtain for $A$ an interpretation with universe $V$.
[pag.64] – Skolem-Lowenheim theorem : let $S$ a denumerable set of formulas. ”We shall restrict ourselves for a while to pure sets – i.e. sets of closed formula with no parameters.”
Question - Why the restriction ? May we say that this restriction is due in order to avoid to “run out” of parameters in building the interpretation with universe $V$ ?
Shall we say that Henkin’s construction avoid this restriction assuming a denumerable set of new individual constants ?
