Variables and atomic formulas

Let’s start with the variables. Variables in $\mathcal{L}_1$ function similar to constants. In particular, just as there is a set of constants $\mathfrak{c}_n$ for each natural number $n$, in $\mathcal{L}_1$, there is also a set of variables $\mathbf{x}_n$, for each natural number $n$. In list format, we have:

$$\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}, \mathbf{x}_{5}, \mathbf{x}_{6}, ….$$

Moreover, these variables can take exactly the same places as the constants. For example, if you have a predicate $\mathfrak{P}^{2}_{2}$, then both $\mathfrak{P}^{2}_{2}(\mathfrak{c}_{4}, \mathfrak{c}_{2})$ and $\mathfrak{P}^{2}_{2}(\mathbf{x}_{4}, \mathbf{x}_{9})$ are atomic formulas. Because of this, constants and variables together are usually called terms, to simplify the definition of what it means for something to be an atomic formula. Thus first, we have:

Once we have the notion of a term, which is, again, either a variable or a constant of the language $\mathcal{L}_1$, we can easily define again what it means for something to be an atomic formula – this time for $\mathcal{L}_1$.

As just mentioned, there is nothing more to this than specifying that if you have an $n$-place predicate $\mathfrak{P}^{n}_{k}$, and it is followed by $n$ terms (constants or variables) of $\mathcal{L}_1$ in parentheses, separated by commas, then the resulting expression $\mathfrak{P}^{n}_{k}(t_1, …, t_n)$ is an atomic formula of $\mathcal{L}_1$.

Here are a few examples of atomic formulas of $\mathcal{L}_1$ $$\begin{gathered} \mathfrak{P}^{3}_{3}(\mathfrak{c}_{3}, \mathbf{x}_{5}, \mathbf{x}_{9})\\ \mathfrak{P}^{3}_{3}(\mathbf{x}_{2}, \mathfrak{c}_{6}, \mathfrak{c}_{7})\\ \mathfrak{P}^{2}_{4}(\mathfrak{c}_{3}, \mathfrak{c}_{5})\\ \mathfrak{P}^{2}_{4}(\mathbf{x}_{7}, \mathbf{x}_{8}) \end{gathered}$$ There are a few things you should notice here. First, there are atomic formulas in which both variables and constants occur. Moreover, there are other atomic formulas which are just like the ones before, but in place of variables, there are now constants, and in place of constants, there are now variables. So in general, constants and variables can always take each others’ places. Second, there are atomic formulas that only have constants, and atomic formulas that only have variables, and in each of these cases, for any predicate $\mathfrak{P}^{}_{}$, the situation could have been the converse.

So again, we have: