Axiom:Axiom of Specification/Also known as

Axiom

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.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 2$: The Axiom of Specification
• 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
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 1$ Extensionality and separation

• Weisstein, Eric W. "Axiom of Subsets." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AxiomofSubsets.html