Axiomatized

All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting the origin"), rather than being axiomatized independently.

Earlier, Menger and Birkhoff had axiomatized complex projective geometry in terms of the properties of its lattice of linear subspaces.

Effective axiomatization A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set (Franzén 2004, p. 112).

Fuzzy set theory main In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not.

His attempt to support axiomatized mathematics with definitive principles, which could banish theoretical uncertainties, ended in failure.

It is not even possible for an infinite list of axioms to be complete, consistent, and effectively axiomatized.

Moreover, no effectively axiomatized, consistent extension of Peano arithmetic can be complete.

The Löwenheim–Skolem theorem implies that infinite structures cannot be categorically axiomatized in first-order logic.