axiom schema of specification

The axiom schema of specification must be used to reduce this to a set with exactly these two elements.

Accepting only the axiom schema of specification was the beginning of axiomatic set theory.

In a typed language where we can quantify over predicates, the axiom schema of specification becomes a simple axiom.

Because the axiom schema of specification is not independent, it is sometimes omitted from contemporary statements of the Zermelo-Fraenkel axioms.

They also independently proposed replacing the axiom schema of specification with the axiom schema of replacement.

For existence, we will use the Axiom of Infinity combined with the Axiom schema of specification.

The role of Spec in S is analogous to that of the axiom schema of specification of Z.

Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension)

Burgess's theory ST is GST with Null Set replacing the axiom schema of specification.

To show that the natural numbers themselves constitute a set, the axiom schema of specification can be applied to remove unwanted elements, leaving the set N of all natural numbers.

The axiom schema of specification, the other axiom schema in ZFC, is implied by the axiom schema of replacement and the axiom of empty set.

Then we use the Axiom Schema of Specification to define our set - i.e. is the set of all elements of which happen also to be elements of every other inductive set.

It implies the axiom schema of specification, axiom schema of replacement, axiom of global choice, and even, as noticed later by Azriel Levy, axiom of union at one stroke.

The axiom schema of specification is characteristic of systems of axiomatic set theory related to the usual set theory ZFC, but does not usually appear in radically different systems of alternative set theory.

The existence of a set with at least two elements is assured by either the axiom of infinity, or by the axiom schema of specification and the axiom of the power set applied twice to any set.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is a schema of axioms in Zermelo-Fraenkel set theory.