Redirected from Axiom schema of comprehension
Suppose P is any predicate in one variable that doesn't use the symbol B. Then in the formal language of the ZermeloFraenkel axioms, the axiom schema reads:
To understand this axiom schema, note that the set B must be a subset of A. Thus, what the axiom schema is really saying is that, given a set A and a predicate P, we can find a subset B of A whose members are precisely the members of A that satisfy P. We can use the axiom of extension to show that this set B is unique. We denote the set B using setbuilder notation as {A : P}, or {C ∈ A : P(C)} to be more clear. Thus the essence of the axiom is:
The axiom schema of specification is generally considered uncontroversial as far as it goes, and it or an equivalent appears in just about any alternative axiomatization of set theory. Indeed, many alternative formulations of set theory try to find a way to use an even more generous axiom schema, while stopping short of the axiom schema of (unrestricted) comprehension mentioned below.

Relation to the axiom schema of replacement
The axiom schema of specification can almost be derived from the axiom schema of replacement.
First, recall this axiom schema:
The only problem is if no such E exists. But in this case, the set B required for the axiom of specification is the empty set, so the axiom schema follows in general using also the axiom of empty set.
For this reason, the axiom schema of specification is often left out of modern lists of the ZermeloFraenkel axioms. However, it's still important for historical considerations, and for comparison with alternative axiomatisations of set theory, as can be seen for example in the following sections.
The axiom schema of unrestricted comprehension reads:
This axiom schema was tacitly used in the early days of naive set theory, before a strict axiomatisation was adopted. Unfortunately, it leads directly to Russell's paradox by taking P(C) to be (C is not in C). Therefore, no useful axiomatisation of set theory can use unrestricted comprehension, at least not with classical logic. Accepting only the axiom schema of specification was the beginning of axiomatic set theory. Most of the other ZermeloFraenkel axioms (but not the axiom of extension or the axiom of regularity) then became necessary to serve as an additional replacement for the axiom schema of comprehension; each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy.
In the von NeumannBernaysGödel axioms[?] of set theory, a distinction is made between sets and classes. A class C is a set iff it belongs to some class E. In this theory, there is a theorem schema that reads:
In second order predicate logic[?], we can quantify over predicates, and the axiom schema of specification becomes a simple axiom. This is much the same trick as was used in the NBG axioms of the previous section, where the predicate was replaced by a class that was then quantified over.
In the New Foundations[?] approach to set theory pioneered by W.V.O. Quine, the axiom of comprehension for a given predicate takes the unrestricted form, but the predicates that may be used in the schema are themselves restricted. The predicate (C is not in C) is forbidden, because the same symbol C appears on both sides of the membership symbol; thus, Russell's paradox is avoided. However, by taking P(C) to be (C = C), which is allowed, we can form a set of all sets.
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