Why the language of set theory involves only $\in$?

Question

I’m reading A Friendly Introduction to Mathematical Logic and in Example 1.2.3, they write:

The language of set theory is not very complicated at all. We will include one binary relation symbol, $\in$, and that is all:$$ L_{ST} ~is~ \{\in\}$$

But I think, when we use set theory we usually write things like: $$ x \in X \\ 1,2,3 \cdots n \in \mathbb{N} $$

That is, the language involves the letters and numbers also. So, why the book has not included that in its $L_{ST}$?

Answer 1

There are many set theories, but the most common one, is ZFC. I will talk only about ZFC, because I do not know every set theory that exists and I do not want to make generalizations that do not hold. ZFC is a theory in first order logic. That means that it has not only the symbol "$\in$", but also all symbols that the language of first order predicate calculus has, like the symbols of variables, the parenthesis "$($" and "$)$", the logical connectives "$\neg$", "$\land$", "$\vee$", "$\rightarrow$", "$\leftrightarrow$", or some of them, the quantifiers "$\forall$" and "$\exists$", or at least one of them and perhaps, the equality symbol "$=$". All the symbols that belong to the language of logic, are called "logical" symbols.

Of course, one cannot express everything using logical symbols alone. In order to apply logic in other fields of mathematics, one must expand its language with all the symbols that are necessary for that field. Those are called "non logical" symbols and they are listed in the "signature" of the language. Because the logical symbols are taken for granted, it is convenient to describe a language only by its non logical symbols. When somebody says "the language of set theory has only the symbol "$\in$" ", he means that it has only "$\in$" in its signature. That is: beyond the logical symbols, the only symbol is "$\in$".

Now, why does ZFC only needs "$\in$" as non logical symbol? Because it is enough to formulate all the axioms of that theory, and because everything else results from the axioms. So, every statement of set theory can be expressed by the logical symbols and "$\in$", alone. Then, what about symbols like "$\varnothing$", "$\cup$", "$\subseteq$" etc? Well, those symbols are used for convenience, but they are not necessary. Everything that can be written with them, it can also be written without them. For example, "$a\subseteq b$" is equivalent to: "$\forall x\left(x\in a\rightarrow x\in b\right)$". The first is just an abbreviation of the second. Or, lets take $a\cup b\neq\varnothing$. This is equivalent to $\exists c\left[\forall x\left(x\in a\vee x\in b\leftrightarrow x\in c\right)\wedge\exists y\left(y\in c\right)\right]$. It is not forbidden to include non necessary symbols to a signature - in fact, we often do - but it is not necessary either. If we do not include such symbols, the formal language does not have them, and every formula that is made of them, does not belong - strictly speaking - to the formal language. In practice, we often use them, but it is easy to translate our formulas in the strictly formal form, whenever is needed.

In your question, you also mention natural numbers. There are is a definition of natural numbers, from the polymath John von Neumann, based on ZFC. He defines 0 as the empty set, and the successor of a number $n$ is $n\cup\{n\}$. This gives: $0=\varnothing$, $1=\{0\}=\{\varnothing\}$, $2=\{0,1\}=\{\varnothing,\{\varnothing\}\}$, $3=\{0,1,2\}=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}$... Then, the set $\mathbb{N}$ of all natural numbers, is the smallest set that contains $\varnothing$, which is closed under the function $s(n)=n\cup\{n\}$. In this way, every proposition that contains numerals, can be translated to a proposition that contains only logical symbols and "$\in$".

However, if you do not use von Neuman definition, and your numerals have a different meaning, which is not definable inside ZFC, then you may have to include some of them in the signature, together with other necessary symbols, from which you will define the rest. In that case, you have created an extension of your language, which is not the language of set theory any more.