Would " S = { x | x belongs to S} " count as a syntactically correct sentence in set theory?

Question

Are there in ( standard) set theory syntactic rules preventing from writing this :

S = { x | x belongs to S}?

Sure, this sentence does not provide any information; it is semantically vacuous. But is it also syntactically incorrect?

Answer 1

You have to consider a formal presentation of set theory.

We can consider e.g. $\mathsf {ZF}$ :

it is an axiom system formulated in first-order logic with equality and with only one binary relation symbol $\in$ for membership.

This means that there are only two types of atomic formulas : $A=B$ and $A \in B$.

It is typical to introduce abbreviations (i.e. defined symbols) based on the "braces".

We can start defining : $\{ A,B \} = w \leftrightarrow \forall z (z \in w \leftrightarrow (z=A \lor z=B))$, denoting the pair, whose existence is licensed by the Pair axiom.

From it, we derive $\{ A \}$ for the singleton.

More generally, we define a definition schema for the set-builder operator (a syntactical object mapping a formula to a term) :

$\{ x \mid \varphi(x) \}=y \leftrightarrow \forall x (x \in y \leftrightarrow \varphi(x))$ and $y$ is a set.

The previous example (the pair symbol) is an instance of this schema.

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As said above, $x \in S$ is an atomic formula; thus we can use it as the formula $\varphi(x)$ in the definition schema above.

Obviously, if $S$ is a set :

$S = \{ x \mid x \in S \}$.

Answer 2

Syntactically is correct. Is like saying that 2=2 but outside of the context describing what “2” means. There is no purpose at describing a set S as the collection of its own elements because that is the general definition of a set and not a description of a particular set S.

If I were to investigate the properties of the set S or of an element x, I would start with:

$x: x\epsilon S$

This sentence describes x as an element of a set S by relating the symbol x to the sentence that x belongs to S.