a function to specify arity-- what do these sentences mean?? I have encountered a function $\pi$ mapping the set of function and predicate symbols to the natural numbers so that for each $k\ge1$, each of the sets {$i \in N | \pi (F_i)=k$}, {$i \in N|\pi(P_i)=k$} is infinite. Then it says that the purpose of the function $\pi$ is to specify the number of arguments or $arity$ of each function and predicate symbol. I guess my question is: How do they succeed in specifying arity? Thanks for your time!

Each function and predicate symbol has an arity, which is the number of arguments to the function or predicate. We can pack all the arities in a function $\pi$. If we have infinitely many function and predicate symbols of arbitrary positive arity, then we are in the situation which you describe. (We could also have infinitely many function symbols of zero arity, i.e. constants, and infinitely many predicate symbols of zero arity, i.e. constant logical values.)

Here is one way to arrange this situation: for each $i,j\geq 1$, define $F_{ij}$ to be a function symbol of arity $i$ and $P_{ij}$ to be a predicate symbol of arity $i$.