In Robert Roger's 'Mathematical logic and formalized theories', there's the following paragraph:
We need next the concept of an arbitrary infinite sequence of individuals within D . The variables of F1 are arranged in an infinite sequence. Thus, each infinite sequence of individuais within D correlates with each of the variables of F1 some individual within D . Thus, in particular, given any formula A and infinite sequence of individuais S = (b1, b2, ...), each of the free variables within A has some individual within D correlated with it by S.
I'd like to understand what he means by an infinite sequence of individuals correlating other things. How can sequences of individuals (b1, b2...) correlate anything whatsoever? Isn't correlating something that only relations can do? I don't understand.
Thank you.
See:
We may consider a function $s : \text {Var} \to D$, where $\text {Var}$ is the set of individual variables $\{ x_i \}$, as a way to correlate to each variable $x_i$ an element $b_i$ of the domain $D$, i.e. $s(x_i)=b_i$.
In this way, we may "give meaning" to a formula $A$ with free variables.
Consider for example the language for arithmetic and the set $\mathbb N$ of natural numbers.
Consider the function $s$ that correlates $x_1$ with the number $0$, $x_2$ with $1$ and so on: $s(x_{n+1})=n \in \mathbb N$.
Consider now as $A$ the formula $(x_2=1)$, with $x_2$ free.
We have that, with the interpretation above and with the "variable assignment" function $s$, the formula $A$ is satisfied; in symbols:
exactly because $s$ assigns to $x_2$ the "meaning" $1$ and $1=1$ is true in $\mathbb N$.
Things are different with a different function $s'$ such that e.g. $s'(x_2)=0$. In this case: $\mathbb N, s' \nvDash (x_2=1)$.