# Name of meta-properties

How are properties like "definability" called (in which formulas are involved):

A set $X$ is definable when there is a formula $\phi(x)$ such that $X = \lbrace x : \phi(x)\rbrace$.

It is not a first-order property (not even equivalent to a first-order property!), and it is not a second-order property (but maybe equivalent to a second-order property?)

Is it called a "semantic property" or a "meta(-theoretical) property"?

What is the usual name?

A companion definition of the one above is

A formula $\phi$ is a set-builder when there is a set $X$ such that $X = \lbrace x : \phi(x)\rbrace$.

How are such properties (of formulas) called?

Caveat: I am aware of the fact that treating formulas as sets will lead to problems when definability, truth, or satisfaction comes into play. So I explicitly want to treat them as linguistic entities, different from the objects they are talking about, so no contradictions can arise.

• See this question at MO: mathoverflow.net/questions/146307/… – Hans-Peter Stricker Oct 29 '13 at 22:44
• In Computability Theory one usually talks about representability of a given set/predicate/function in a given theory (in that sense, for instance, computable functions can be represented in a formal theory containing enough arithmetic). Is that what you're interested on? – J Marcos Oct 30 '13 at 0:50