What exactly is a formula in set theory?

Question

I've taken a look at this: Set theory formula

But I'm still a little confused. Is the formula supposed to take x, y, z as input and produce "TRUE" as an output? Are all set theory formula's TRUE?

Answer 1

A formula is an expression of the language of set theory built up acoording to the rules of the syntax.

Examples : $∃y \ ∀x \ ¬(x ∈ y), ∀x \ ¬(x ∈ \emptyset)$.

A formula can be a sentence, i.e. without free variables (like the two previous examples) ore an open one, like e.g. : $(x∈y)$.

A sentence has a definite truth value : $∀x \ ¬(x ∈ \emptyset)$ means "the empty set has no elements" and it is true in set theory.

An open formula, like $(x∈y)$ has not a definite truth value; its truth value depends on the "reference" assigned to the variables.

Consider some simple arithmetical examples : $\forall n (n \ge 0)$ is true in $\mathbb N$.

Consider $(n > 0)$ instead : it is false if $n$ denotes $0$ and is true otherwise.

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For a formal definition, see the post : In Mathematical Logic, What is a Language?

Answer 2

Put simply in the language of set theory we start with atomic formulas $$(x\in y)$$ or $$(x=y)$$ where $x$ and $y$ are variables.

Then we expand the definition of formulas to so that it is closed under $$\neg(\cdot)$$ and $$(\cdot)\wedge(\cdot)$$ and $$(\exists x)(\cdot)$$ where $x$ is a variable.

Then we introduce the following notations.

$$(\varphi\vee\psi)\equiv\neg(\neg\varphi\wedge\neg\psi)$$ $$(\varphi\Rightarrow\psi)\equiv\neg\varphi\vee\psi$$ $$(\varphi\Leftrightarrow\psi)\equiv(\varphi\Rightarrow\psi)\wedge(\psi\Rightarrow\varphi)$$ $$(\forall x)\varphi\equiv\neg(\exists x)\neg\varphi$$

We take variables, punctuations, $=$, $\in$, $\neg$, $\wedge$, $\exists$ as primitive notions. That is to say, they are undefined symbols which are informally trying to capture the notion of variables, punctuations, equality, membership, negation, conjunction, existential quantifier. Establishing a collection of axioms is how we try to capture desired notions.

Let me speak in the language of set theory:

$$(\exists x)(\neg(x=x))$$

What I just said is false because of established axioms of set theory; most popular being $\mathsf{ZFC}$.

Let $\varphi$ be an arbitrary formula. Then certain variables within $\varphi$ are "free." We typically denote those variables by writing $\varphi(x_1,\ldots,x_n)$ instead of simply $\varphi$.