# axiomatize in a sentence

1) Uniform spaces axiomatize ordering the distance between distinct points.

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

3) Grothendieck topologies axiomatize the notion of an open cover.

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

## axiomatize example sentences

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

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

7) Weierstrass began to advocate the arithmetization of analysis, which sought to axiomatize analysis using properties of the natural numbers.

8) The idea still has currency - the history of recent artificial intelligence is replete with stories of systems that seek to "axiomatize" logic inside computers.

9) That is, Howson proposes that we axiomatize our background information k by a set of sentences {A} in such a way that e is logically independent of the other sentences in {A}.

10) 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.

11) We first axiomatize NBG using the . This schema is provably equivalent to 9 of its finite instances, stated in the following section.

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

13) 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.

14) 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.

### example sentences with axiomatize

15) 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.

16) 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).

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

18) These all show that even for rational numbers exponentiation is not nearly as well-behaved as addition and multiplication, which is why one does not in general axiomatize exponentiation.

19) 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).

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