Hi,


Is set membership *defined* in the signature of ZFC, or is it \*specified" in the signature of ZFC? The wikipedia article <http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory> says that the signature *has* set membership, but what does it mean by *has*?


If I understand correctly, the axioms specify the properties of set membership. Given set membership, how come ZFC can be foramlised in FOL when some axioms, e.g., axiom of infinity, require quantification of sets? Aren't sets unary predicates since S(a) = True iff a is a member of S? Quantification over unary predicates is a feature of second-order though.


I must be missing something...


Thanks



ZFC, set membership and FOL
A *specification* is a prescription of how things are supposed to be. A *model* (or *implementation* in computer-sciency talk) is an instance of something that satisfies a specification.


In first-order logic and model theory a specification is called a *theory*, and it consists of two parts:


* a *signature* which prescribes the non-logical part of the language (constants, function symbols, and relation symbols)
* *axioms*, which are first-order statements written in the language that includes the parts provided by the signature


The division of a theory into two parts is convenient, but is not always possible outside first-order logic. For example, if we wanted to include Russell's definite operator $\iota x . \phi(x)$ ("the $x$ such that $\phi(x)$") then our syntax would get intertwined with logic because in order to even form the term $\iota x . \phi(x)$ one first has to prove something.


Anyhow, the *theory* ZFC *specifies* that there is one relation symbol $\in$, which we usually read as "element of". A model of ZFC *interprets* the symbol $\in$ as a binary relation, so in this sense it *defines* its meaning.


In a context where theory and its intended semantics are bunched together, such as the Wikipedia page seems to be, the distinction between the specification and implementation will be obsucred. In fact, most mathematicians seem to be a bit unclear about this distinction, as they never really have to distinguish the language they use from its meaning.


