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mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. An axiom schema is a formula in the metalanguage...

axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom), subset axiom, axiom of class construction, or axiom...

In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under...

proposed replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, as well as the axiom of regularity (first...

\ldots ,x_{n})].} Then the axiom schema of replacement is replaced by a single axiom that uses a class. Finally, ZFC's axiom of extensionality is modified...

range of f, which can be seen to be a set from the axiom schema of replacement. Applying the axiom of regularity to S, let B be an element of S which...

Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of...

axiom schema containing a separate axiom for each possible predicate. The article Peano axioms contains further discussion of this issue. The axiom of...

(logic) Axiom schema, in formal logic Image schema, a recurring pattern of spatial sensory experience Database schema XML schema Body schema, a neural...

holds. This is more properly an axiom schema (that is, each condition S ( x ) {\displaystyle S(x)} gives rise to an axiom). "Condition" here means a "sentence"...

theorem. If separation is not postulated as an axiom schema, but derived as a theorem schema from the schema of replacement (as is sometimes done), the situation...

operation symbols and replacing the second-order induction axiom with a first-order axiom schema. The term Peano arithmetic is sometimes used for specifically...

intuitively true, by definition). Axiom schema 1 is P → (Q → P). Axiom schema 2 is (P → (Q → R)) → ((P → Q) → (P → R)). Axiom schema 3 (Peirce's law) is ((P →...

\lnot B))} are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modus ponens, one can...

the axiom of union. Together with the axiom of empty set and the axiom of union, the axiom of pairing can be generalised to the following schema: ∀ A...

use the Axiom of Infinity combined with the Axiom schema of specification. Let I {\displaystyle I} be an inductive set guaranteed by the Axiom of Infinity...

of the following countably infinitely many axioms added (these can be easily formalized as an axiom schema): ∃ x 1 : ∃ x 2 : ¬ ( x 1 = x 2 ) {\displaystyle...

the postcondition. Formally, this result is obtained by applying the axiom schema with P being ( y = 43 {\displaystyle y=43} and x + 1 = 43 {\displaystyle...

Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Fragments of ZFC include: Zermelo set theory, which replaces the axiom schema of replacement with that...

Schwabhäuser, Szmielew, and Tarski (1983), which set out the 10 axioms and one axiom schema shown below, the associated metamathematics, and a fair bit of...