Axiom:Axiom of Subsets
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 scheme, as we introduce a lot of similar axioms.
This axiom scheme is formally stated as follows:
For any function of propositional logic $P \left({y}\right)$, we introduce the axiom:
- $\forall z: \exists x: \forall y: \left({y \in x \iff \left({y \in z \land P \left({y}\right)}\right)}\right)$
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.
Alternative specification
Alternatively, this axiom can be specified as follows:
- If $\phi$ is a property (with parameter $p$), then for any $X$ and $p$ there exists a set:
- $Y = \left({u \in X: \phi \left({u, p}\right)}\right)$
- that contains all those $u \in X$ that have the property $\phi$.
- $\forall X: \forall p: \exists Y: \forall u: \left({u \in Y \iff \left({u \in X \land \phi \left({u, p}\right)}\right)}\right)$
Also known as
Otherwise known as:
- The Axiom of Specification (this axiom allows one to specify the elements of a subset)
- The Axiom of Comprehension
- The Axiom of Selection (this axiom allows one to select the elements of a subset)
- The Axiom of Separation (although this can be confused with the Tychonoff separation axioms, which arise in topology, so this name is rarely used).
This can be deduced from the Axiom of Replacement.
Also see
- Definition:Comprehension Principle -- do not confuse that with this
Internationalization
Axiom of Subsets is translated:
In German: | Aussorderungsaxiom | (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.
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.1$: Sets
- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html