Example 2

The following symbols represent assemblies (and very long ones at that) :

“3 and 4”

“the real line”

“the $\Gamma$ function”

“Every finite division ring is a field”

“The zeros of $\zeta(s)$ other than $-2$, $-4$, $-6$, ... lie on the line $R(s)=1/2$.”

In general, the symbol used to represent an assembly contains all the letters which appear in the assembly. Nevertheless, this principle can sometimes be infringed without risk of confusion. *For example, “the completion of X” represents an assembly which contains the letter X, but which also contains the letter which represents the set of entourages of the uniform structure of X. On the other hand,

represents an assembly in which the letter $x$ (and the letter $d$) do not appear; and the assemblies represented by $\mathbf{N}$, $\mathbf{Z}$, “the $\Gamma$ function” do not contain any letters.*

A mathematical theory (or simply a theory) contains rules which allow us to assert that certain assemblies of signs are terms or relations of the theory, and other rules which allow us to assert that certain assemblies are theorems of the theory.

The description of these rules, which will appear in this chapter, does not belong to formal mathematics; the rules involve assemblies which are more or less undetermined, for example undetermined letters. To simplify the exposition it is convenient to denote such assemblies by less cumbersome symbols. We shall use, especially, combinations of signs (of a mathematical theory), bold-face italic letters (with or without indices or accents), and particular symbols, of which some examples will be given. Since our object is only to avoid circumlocutions (cf. note (*), § 3, no. 1, p. 28) we shall not enunciate strict general rules for the use of these symbols; the reader will be able to reconstruct without trouble the assembly in question, in each particular case. By abuse of language we shall often say that the symbols are assemblies, rather than that they denote assemblies : expressions such as “the assembly $A$” or “the letter $x$”, in the statements of the following rules, should therefore be replaced by “the assembly denoted by $A$” or “the letter denoted by $x$”.

Let $A$ and $B$ be assemblies. We shall denote by $AB$ the assembly obtained by writing the assembly $B$ on the right of the assembly $A$. We shall denote by $\vee A\neg B$ the assembly obtained by writing, from left to right, the sign $\vee$, the assembly $A$, the sign $\neg$, the assembly $B$. And so on.

Let $A$ be an assembly and let $x$ be a letter. We shall denote by $\tau_x(A)$ the assembly constructed as follows : form the assembly $\tau A$, link each occurrence of $x$ in $A$ to the $\tau$ written on the left of $A$, and then replace $x$ everywhere it occurs by the sign $\square$. The assembly denoted by $\tau_x(A)$ therefore does not contain $x$.