Axiom schema information

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...

axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension...

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...

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

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...

and replacing the second-order induction axiom with a first-order axiom schema. When Peano formulated his axioms, the language of mathematical logic was...

\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...

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...

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...

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...

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...

several of the standard ZF axioms for constructing new sets with a principle known as Ackermann's schema. Intuitively, the schema allows a new set to be constructed...

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...

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...

the axiom schema of specification replaced by the axiom of empty set. Note that Specification is an axiom schema. The theory given by these axioms is not...

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 →...

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

theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory. This axiom was introduced by Ernst Zermelo. Informally, the axiom states that...

formulas permitted in one's adopted Separation schema, by Diaconescu's theorem. Similar results hold for the Axiom of Regularity existence claim, as shown below...

paradox in 1902. If the axiom schema of unrestricted comprehension is weakened to the axiom schema of specification or axiom schema of separation, If P is...

usually denoted Q. Q is almost[clarification needed] PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language...

the axiom schema of predicative separation, or of restricted, or Δ0 separation, is a schema of axioms which is a restriction of the usual axiom schema of...

well-ordered, so the axiom of choice is not needed to well-order them. The following construction of the Vitali set shows one way that the axiom of choice can...

paradox by permitting only stratifiable formulas to be defined using the axiom schema of comprehension. For instance, x ∈ y is a stratifiable formula, but...