Axiom:Axiom of Specification/Set Theory
Axiom
The axiom of specification is an axiom schema which can be formally stated as follows:
For any well-formed formula $\map P y$, 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.
This means that if you have a set, you can create a set that contains some of the elements of that set, where those elements are specified by stipulating that they satisfy some (arbitrary) condition.
Also presented as
The Axiom of Specification can also be specified as follows:
- If $\phi$ is a property (with parameter $p$), then for any $X$ and $p$ there exists a set:
- $Y = \paren {u \in X: \map \phi {u, p} }$
- that contains all those $u \in X$ that have the property $\phi$.
- $\forall X: \forall p: \exists Y: \forall u: \paren {u \in Y \iff \paren {u \in X \land \map \phi {u, p} } }$
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 deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$)
- 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
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 the Empty Set, 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
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 2$: The Axiom of Specification
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.11$ which proves it as a theorem.
- 1982: Alan G. Hamilton: Numbers, Sets and Axioms ... (previous) ... (next): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF6}$
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.1$: Sets
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 9$ Zermelo set theory
- Weisstein, Eric W. "Axiom of Subsets." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AxiomofSubsets.html