Axiom in a sentence

Axiom counters a lifting Liger Bomb with a truly wonderful Canadian Destroyer, that should have been a finish but Lee kicks out at 2. Lee, who cares not for selling, kicks Axiom in the head then heads up top, but he takes too long and Axiom kicks him.

Hence, the axiom of regularity is equivalent, given the axiom of dependent choice, to the alternative axiom that there are no downward infinite membership chains.

Apollo Crews is watching the Carmelo Hayes/Trick Williams video from earlier tonight when Axiom comes in. Axiom thanks Crews for having his back last week but Crews goes on a rant about Hayes and Williams.

Back up and Melo whips Axiom into the hostile corner, but Axiom dropkicks Trick!

Last week, Scrypts agreed that he and Axiom would face Lucien Price and Bronco Nima this week, but Axiom said they aren’t a regular team.

Axiom Space astronaut(s) intend to wear and operate these technologies remotely throughout their training, launch, and during Axiom Mission 4 (Ax-4), which is scheduled to launch next spring.

Into the ropes and Page nails Axiom; back into the ropes and Axiom with a kick to Page!

A formulation of set theory that does not include the axiom of replacement will likely include some form of the axiom of separation, to ensure that its models contain a sufficiently rich collection of sets.

Although the axiom schema of replacement is a standard axiom in set theory today, it is often omitted from systems of type theory and foundation systems in topos theory.

Axiomatizing arithmetic induction in first-order logic requires an axiom schema containing a separate axiom for each possible predicate.

Axiom of infinity Ramified types and the axiom of reducibility In simple type theory objects are elements of various disjoint "types".

Because at least one such infinite schema is required (ZFC is not finitely axiomatizable), this shows that the axiom schema of replacement can stand as the only infinite axiom schema in ZFC if desired.

Because there are models of Zermelo–Fraenkel set theory of interest to set theorists that satisfy the axiom of dependent choice but not the full axiom of choice, the knowledge that a particular proof only requires dependent choice can be useful.

By the closure of W under scalar multiplication (specifically by 0 and −1), the vector space's definitional axiom identity element of addition and axiom inverse element of addition are satisfied.

Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice.

Examples of these axioms include the combination of Martin's axiom and the Open colouring axiom which, for example, prove that (N*) 2 ≠ N*, while the continuum hypothesis implies the opposite.

However, the axiom of empty set can be shown redundant in either of two ways: *There is already an axiom implying the existence of at least one set.

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

It is possible to prove many theorems using neither the axiom of choice nor its negation; such statements will be true in any model of Zermelo–Fraenkel set theory (ZF), regardless of the truth or falsity of the axiom of choice in that particular model.

Most of the axioms of equality still follow from the definition; the remaining one is : and it becomes this axiom that is referred to as the axiom of extensionality in this context.