Axiom:Axiom of Specification
Contents
Axiom
For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true.
Because we cannot quantify over functions, we need an axiom for every condition we can express.
Therefore, this axiom is sometimes called an axiom schema, as we introduce a lot of similar axioms.
This axiom schema can be formally stated as follows:
Set Theory
For any function of propositional logic $\map P y$, we introduce the axiom:
- $\forall z: \exists x: \forall y: \paren {y \in x \iff \paren {y \in z \land \map P y} }$
where each of $x$, $y$ and $z$ range over arbitrary sets.
Class Theory
The axiom of specification in the context of class theory has a similar form:
Let $\map \phi {A_1, A_2, \ldots, A_n, x}$ be a function of propositional logic such that:
- $A_1, A_2, \ldots, A_n$ are a finite number of free variables whose domain ranges over all classes
- $x$ is a free variable whose domain ranges over all sets.
Then the axiom of specification gives that:
- $\forall A_1, A_2, \ldots, A_n: \exists B: \forall x: \paren {x \in B \iff \paren {x \in B \land \phi {A_1, A_2, \ldots, A_n, x} } }$
where each of $B$ ranges over arbitrary classes.
Also known as
The axiom of specification is also known as:
- The axiom of subsets (although this unnecessarily reduces the scope of this axiom to pure set theory)
- The axiom of comprehension
- The axiom of selection (this axiom allows one to select the elements of a subset)
- The axiom of separation or separation principle (although this can be confused with the Tychonoff separation axioms, which arise in topology, so this name is rarely used)
- The axiom of segregation, under its German name Aussonderungsaxiom
- The limited abstraction principle, in apposition to the unlimited abstraction principle, also known as the comprehension principle.
- In the context of class theory, the term axiom of class formation is often seen.
Also see
- Definition:Comprehension Principle -- do not confuse that with this
The axiom of specification can be deduced from the Axiom of Replacement.
Historical Note
The axiom of specification was created by Ernst Zermelo as a replacement for the comprehension principle of Frege set theory.
The latter had been demonstrated, via Russell's Paradox, to lead to the conclusion that Frege Set Theory is Logically Inconsistent.
Thus, rather than allowing a set to be constructed of any elements at all which satisfy a given property $P$, the elements in question are restricted to being elements of some pre-existing set.
This in turn leads to the further question of how to create such a pre-existing set in the first place.
Hence the need to develop further axioms in order to allow the creation of such sets.
As a result of this, Ernst Zermelo found it necessary to create:
- the axiom of existence, allowing for the existence of $\O := \set {}$
- the axiom of pairing, allowing for $\set {a, b}$ given the existence of $a$ and $b$
- the axiom of unions, allowing for $\bigcup a$ given the existence of a set $a$ of sets
- the axiom of powers, allowing for the power set $\powerset a$ to be generated for any set $a$
- the axiom of infinity, allowing for the creation of the set of natural numbers $\N$.
Internationalization
Axiom of specification is translated:
In German: | Aussonderungsaxiom | (literally: axiom of segregation) |
Sources
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.1$: Sets