# Terms (free variables or classes) with inputs bound to a well formed formula

I know that a term is a free variable or a class. I am having trouble understanding a definition where a term is written as a well formed formula. As in, the term is followed by a set of bound variables in a set of parentheses.

$$\Gamma$$ is a term and $$\phi$$ is a well formed formula. $$\{\Gamma(x_1,···, x_n)|\phi(x_1,...,x_n)\}=$$
$$\{y|(\exists x_1,...,x_n)[y=\Gamma(x_1,...,x_n)\land \phi (x_1,···,x_n)]\}$$

The way I understand it is that the term has some bounds that are bounded by the well formed formula to create a class. This is equivalent to there being a bound that contains those previous bounds to create the same class.

It's being described as notational convenience. This leads me to believe that the class described by the wff and bound to a term must be bound to a single variable to be considered a class. Is this because a term is a free variable or a class, both of which are not allowed to be the bound of a class? Why must a class be bound by a single variable?

"a term is a free variable or a class."

Not exactly; in predicate logic a term is either a variable or a constant or a "complex" term built from function symbols. Examples: $$x, 0, x+0, (x+1)\times 0$$.

In predicate logic a term is a "name" for an object of the domain of interpretation.

If we are working in set theory, a term will be a name for a set or class.

"a definition where a term is written as a well formed formula."

Wrong: a formula is not a term alone, but one or more terms "glued" by predicates and connectives, like e.g. $$x=0, 0 < 1, (x=0) \land (0 < 1)$$ and quantifiers: $$\forall x (x \ge 0)$$.

The symbol $$Γ(x_1,\ldots,x_n)$$ is a term with free variables, like $$x+0$$ in my example above, and $$y=Γ(x_1,\ldots,x_n)$$ is a formula.

The set definition you quoted is the definition of a set $$\{ y \mid \phi(y) \}$$ that reads: "the set of all and only those 'objects' $$y$$ that satisfy the condition expressed by formula $$\phi(y)$$ on the left of the "$$|$$" sign".

The set "specifying condition" in your example is the formula:

$$(\exists x_1,\ldots, x_n) \ [y= \Gamma (x_1, \ldots, x_n) \land \phi(x_1,\ldots, x_n)]$$,

where only variable $$y$$ is free.

A more simple example will be $$\{ y \mid \exists x (y=x+1) \}$$.

In this case, if we assume that free variables range over natural numbers, the set will contains all numbers that are successors of some number, i.e. all naturals except for $$0$$.