I began reading Paul Halmos' "Naive Set Theory", and encountered the "Axiom of Specification".
To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly those elements $x$ of $A$ for which $S(x)$ holds.
Earlier in the same section, I learned that statements in set theory should be "sentences". A sentence was defined by
There are two basic types of sentences, namely, assertions of belonging,
$x \in A$,
and assertions of equality,
$A = B$;
all other sentences are obtained from such atomic sentences by repeated applications of the usual logical operators…
A more complete definition of a sentence follows, which can be read on Google Books here: http://goo.gl/XvK2B
I tried to translate the axioms and theorems in the book into sentences, but it seems like the Axiom of Specification is not a sentence. It refers to "every condition", but I have no way to build a sentence that refers to "every condition" because the atomic sentences only refer to sets.
Is the Axiom of Specification a sentence? If not, does that mean that statements about set theory do not need to be sentences? What other sorts of statements are allowed? (I'm using "statement" colloquially since I don't know the technical term.)
Best Answer
The axiom of specification is not a sentence. It's an "axiom scheme", which is to say that it is a family of sentences. This is one of the things that becomes more clear when you move to axiomatic set theory, instead of naive set theory.
For each sentence $S(x)$ that does not mention $B$, the axiom of specification includes the axiom $$ \forall A \exists B \forall x ( x \in B \Leftrightarrow x \in A \land S(x)). $$
What that axiom says, informally, is that given a set $A$ and a definition $S$ of a subset of $A$, that subset actually exists. The scheme is slightly more general than my previous formula, because the scheme allows sentences with "parameters".
The restriction that $S$ does not mention $B$ is to avoid paradoxes. Otherwise we would have as an axiom (letting $A = \{0\}$ and letting $S$ be "$x \not \in B$") $$ \exists B \forall x ( x \in B \Leftrightarrow x \in \{0\} \land x \not \in B). $$ That set is paradoxical - it contains 0 if and only if it doesn't contain $0$.
The reason that we cannot quantify over sentences is that set theory is formalized using the logical system of "first order logic". That system is not able to quantify over sentences. This isn't an arbitrary choice; the inability to quantify over sentences is a necessary result of certain logical properties of first-order logic that are desirable. There are other logics in which one can quantify over sentences, but these logics do not have nice properties (and some have argued these logics themselves include set theory).
All of this is explained, in great detail, in books on axiomatic set theory. One reasonable book is Levy's Basic set theory. The standard graduate textbook is Kunen's Set theory: an introduction to independence proofs, and it can be used to learn axiomatic set theory, but it is somewhat terse at the beginning and is better as a second book on axiomatic set theory in my opinion.