axiomatize
axiomatise BrE

There are typically multiple ways to axiomatize a given mathematical domain.

We prefer, however, to axiomatize not "set" but "function".

Cauchy spaces axiomatize the ability to test whether a net is Cauchy.

Grothendieck topologies axiomatize the notion of an open cover.

The Peano Axioms (described below) thus only partially axiomatize this theory.

To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms).

Hilbert's sixth problem is to axiomatize those branches of physics in which mathematics is prevalent.

These axioms axiomatize Euclidean solid geometry.

The following axiom schema and three inference rules axiomatize the Boolean algebra of n-ary terms.

Such types are called principal types, and the formulas that axiomatize them are called complete formulas.

This result shows that it is possible to axiomatize ZFC with a single infinite axiom schema.

Attempts to formalize the principles of the empirical sciences, use an interpretation to model reality, in the same way logicians axiomatize the principles of logic.

Alexandre Grothendieck solved this problem by introducing Grothendieck topologies, which axiomatize the notion of covering.

We axiomatize predicate calculus without equality, i.e. there are no special axioms expressing the properties of equality as a special relation symbol.

In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct.

There have been several attempts to axiomatize the underlying rings of von Neumann algebras, including Baer *-rings and AW* algebras.

However, it is possible to axiomatize the above procedure and give conditions under which the above spectral sequence holds for a general (co)homology theory, see Smith's original work or the introduction in ).

In the early 20th century, David Hilbert led a program to axiomatize all of mathematics with precise axioms and precise logical rules of deduction which could be performed by a machine.

Attempts to axiomatize the empirical sciences, Carnap said, use a descriptive interpretation to model reality.: the aim of these attempts is to construct a formal system for which reality is the only interpretation.

In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only.

Frege did explicitly axiomatize a theory, in which the formalized version of naive set theory can be interpreted, and it is this formal theory which Bertrand Russell actually addressed when he presented his paradox.

Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability (Solovay 1976) and set-theoretic forcing (Hamkins and Löwe 2007).

Montague (1961) included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms.

In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent finitely axiomatizable system-such as necessary to axiomatize the elementary theory of arithmetic-a statement that can be shown to be true, but that does not follow from the rules of the system.

On the converse, however, if the fusion asserted by M8 is assumed unique, so that M8' replaces M8, then - as Tarski (1929) had shown - M3 and M8' suffice to axiomatize GEM, a remarkably economical result.