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These day, I am studying set theory (with Jech's book) and I can find many notations of formula (or statement) such as $\phi(x,y,p)$ or $\psi(x,y,z,w_1,w_2,\cdots , w_n)$. In Jech's book, there is an explanation: Concerning formulas with free variables, we adopt the notational convention that all free variables of a formula $\phi(u_1,u_2,\cdots ,u_n)$ are among $u_1,u_2,\cdots, u_n$.

And in wikipedia context about ZFC, it introduce the detail of Axiom schema of specification as: Let $\phi$ be any formula in the language of ZFC with all free variables among $x,z,w_1,w_2,\cdots ,w_n$. Then, $\forall z \forall w_1 \forall w_2 \cdots \forall w_n \exists y \forall x [x\in y \iff (x\in z \land \phi)]$.

My question is why we need to care about free variables when we compute any formula (or statement). What important roles free variables do? According to Pinter's set theory, the definition of free variable is written: $x$ is free in $\phi(x)$ if $x$ is not governed by a quantifier $\exists$ or $\forall$; thus, in $\exists y (x<y)$, $x$ is free whereas $y$ is not.

Yet, I cannot relate the definition and its importance. Can anyone elaborate??

Andrés E. Caicedo
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user301120
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    Well, how would you make sense of a quantified formula if you couldn't tell which variables the quantifiers were binding or if they were binding the variables? How would the quantifier rules of logic work without knowing anything about free variables? In the ZFC example, the free variables of $\phi$ tell you how to check if a formula is actually an instance of the axiom schema. – Malice Vidrine Apr 23 '17 at 03:15
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    https://math.stackexchange.com/questions/2173317/axiom-schema-of-separation-parameters/ – Asaf Karagila Apr 23 '17 at 04:32
  • Thank you for all of you. And I have another simple question. Let $u$ a set. Clearly, $x\in u$ is a formula (or predicate). Can we say $x$ and $u$ are free since there is no quantifier which governs $x$ or $u$??? – user301120 Apr 23 '17 at 08:05
  • Yes; in $x \in u$, both $x$ and $u$ are free. In $\forall x \ (x \in u)$, $u$ is free while $x$ is bound. – Mauro ALLEGRANZA Apr 23 '17 at 08:29
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    There is nothing specific in $\mathsf {ZFC}$ regarding free variables... It is a theory in the language of first-order logic and thus it uses all the resources of that language; predicates, variables, quantifiers and (boolean) connectives. – Mauro ALLEGRANZA Apr 23 '17 at 08:32
  • That's interesting. Thanks – user301120 Apr 23 '17 at 08:37
  • @Herace It might be useful to distinguish between free variables and parameters. If I have an $L$-structure $M$ in mind, I can consider an expanded language $L'$ containing a new constant for every element of $M$; there's an obvious way to expand $M$ to an $L'$-structure, call it "$M'$." Now in $L'$, I have formulas like "$x\in c_u$", where $u\in M$ and $c_u$ is the new constant symbol in $L'$ corresponding to $u$. In this formula, $x$ is a free variable but $c_u$ is a constant symbol. We often abbreviate this situation by writing "$x\in u$" when we have some fixed model in mind. (contd) – Noah Schweber Jun 17 '17 at 16:11
  • In this context we'd say $u$ is a parameter. But parameters only make sense with a specific model in mind and writing these parametrized formulas as "$x\in u$" rather than "$x\in c_u$" is sloppy (but convenient and common) notation. This isn't directly relevant to your question (hence I'm posting this as a comment rather than an answer) but it might be helpful since you may run into the word "parameter" being thrown around rather informally in this context. In particular, the idea of parameters and passing to $L'$ is necessary for inductively evaluating truth of a sentence in a structure. – Noah Schweber Jun 17 '17 at 16:15

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