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I am taking a course in logic which is based off the text $\textit{Set Theory}$ by Thomas Jech. The course is really jumping around rapidly, and not really paying attention to details, which get skipped over with no regard. As a result, everything in the course and text regarding models of Set Theory seem pretty hand-wavy, especially in proofs.

As far as I know about model theory, we are given a language $L = (R,F,C, \alpha)$, where $R$ is a set of relation symbols, $F$ is a set of function symbols, $C$ is a set of constant symbols, and $\alpha: R\cup F \longrightarrow \mathbb{Z}_{\geq0}$ is a function detailing the intended arity of the respective relations and functions. We then define (this is at least one way of doing it) an $L$-structure as a tuple $\mathcal{U} = (A,I_R,I_F,I_C)$ where $A$ is a set, and the latter three are the respective interpretation functions, and we may adjoin all of them into one extended general interpretation function $I$. Now we can define a $\mathcal{U}$-interpretation as $\beta : V_L \longrightarrow A$, where $V_L$ is the set of variables of the language $L$. We then define $\mathcal{U} \models_\beta \phi$ for a given formula $\phi$ and interpretation $\beta$ by recursion on formula complexity, and then define $\mathcal{U} \models \phi$ if $\mathcal{U} \models \phi$ for every variable interpretation $\beta$, and this could be extended to arbitrary sets of sentences. Further, for a set $\Gamma$ of $L$-sentences, $\Gamma \models \phi$ if for every $L$-structure $\mathcal{U}$ with $\mathcal{U} \models \Gamma$, $\mathcal{U} \models \phi$.

In Jech's book, we have the language of set theory as $L_\in = (\in)$, and a model of set theory would be of the form $(M, \in)$ where $M$ is now allowed to be a class (often transitive) instead of merely a set. Also, we would then say the $\phi$ holds in $\textbf{ZFC}$ (where $\textbf{ZFC}$ denotes the collection of $L_\in$-formulas containing the standard axioms) if $\mathbf{ZFC}\models \phi$ in the above sense.

Now, Jech's proofs rarely even mention such underlying machinery in his exposition, and every proof seems kind of hand-waved to me as a result. One issue I have is the frequent use of the term $\textit{parameters}$ for a formula $\phi$, and when it is an element of a particular set or not. Jech gives the standard definition for the revitalization of a formula $\phi$ denoted $\phi^M$, and states that when using $\phi^M(x_1,\dotsc, x_n)$, it is assumed that the variables range over $M$. What exactly does that mean in terms of the above constructions? What is a rigorous definition for the process of substitution into a formula? He then goes to say things like $$\phi^M(a_1,\dotsc,a_n) \longleftrightarrow (M,\epsilon)\models [\phi][a_, \dotsc,a_n]$$ where he makes a distinction between metalogical formulas $\phi$ and the rigorous $L_\epsilon$-formulas $[\phi]$ does this essentially mean to say that $\textbf{ZFC}\models \phi^M$ if and only if $(M,\in) \models \phi$? He then goes on to define $\Delta_0$ formulas and proves that for $M$ a transitive class and $\phi$ a $\Delta_0$ formula, $$\phi^M(x_1, \dotsc,x_n) \longleftrightarrow \phi(x_1, \dotsc x_n).$$ Does this mean to say that $\textbf{ZFC}\models \phi^M(x_1, \dotsc, x_n)$ if and only if $\textbf{ZFC} \models \phi(x_1, \dotsc, x_n)$? One thing that confuses me is not only his inconsistent use of parameters and when making them explicit, but these model-theoretic theorems are not proven in terms of taking arbitrary structures and interpretations, etc. Rather, the proofs of these theorems, especially in the proof that the cumulative hierarchy minus the Axiom of Regularity model the relativized axioms, seem to be proven like how one would prove a typical theorem with the formula's meaning taken to be literal rather than a semantic or syntactical argument. I am having trouble understanding Jech's book in terms of this. Any helps is greatly appreciated.

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  • $\begingroup$ We define what it means for a variable-symbol $x$ to occur free or to occur bound (by a quantifier $\exists$ or $\forall$) in a formula . We define what it means to substitute a value $a\in M$ for a variable $x$ in a formula. I can say much more but first I would like to know if you are familiar with this. $\endgroup$ – DanielWainfleet Feb 21 '18 at 4:54
  • $\begingroup$ parameters are variables. See page 7: Separation Schema with formula $\varphi(u, p)$: the parameter $p$ is quantified; thus, it must be a variable. $\endgroup$ – Mauro ALLEGRANZA Feb 21 '18 at 7:55
  • $\begingroup$ For $\vDash$, see page 155. When a formula $\varphi(x_1,\ldots,x_n)$ is satisfied by the $n$-uple $\langle a_1,\ldots,a_n \rangle$ of elements of the domain $A$ of the structure $\mathfrak A$ (i.e. $a_i \in A$) we write: $\mathfrak A \vDash \varphi[a_1,\ldots,a_n]$, meaning that we have assigned to the free var $x_i$ of the formula the "object" $a_i$ of the domain of the structure as its reference. $\endgroup$ – Mauro ALLEGRANZA Feb 21 '18 at 8:01
  • $\begingroup$ Regarding relativization, if you review (12.15) page 161, you can see that in the relevant case when $E$ is $\in$, notingh change regarding formulas except for quantified one, that become: $(\exists x \in M)\varphi^M$. I.e., the domain of the varibale is restricted to the class $M$. $\endgroup$ – Mauro ALLEGRANZA Feb 21 '18 at 8:05
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    $\begingroup$ Thus, the gist of (12.16) is that the relativized formula holds, i.e. $\varphi^M(a_1,\ldots,a_n)$ iff the "original formula" (denoted $\ulcorner \varphi \urcorner$) is satisfied in $M$ by the tuple $\langle a_1,\ldots, a_n \rangle$, i.e. $M \vDash \ulcorner \varphi \urcorner [ a_1,\ldots, a_n]$. $\endgroup$ – Mauro ALLEGRANZA Feb 21 '18 at 8:10

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