# require axiom in a sentence

1) Not every situation

**requires**the

**axiom**of choice.

2) However, specifying an isomorphism

**requires**the

**axiom**of choice.

axiom collocations

3) Many of these theorems are independent of ZFC, requiring stronger

**axioms**for their proof.

4) Note that solid geometry

**requires**no new

**axioms**, unlike the case with Hilbert's

**axioms**.

## require axiom example sentences

5) The proof is based on Hausdorff's paradoxical ball decomposition, whose production**requires**the

**axiom**of choice.

6) Reverse mathematics is a program in mathematical logic that seeks to determine which

**axioms**are

**required**to prove theorems of mathematics.

7) However, in 1949 A.P. Morse showed that the statement about Euclidean polygons can be proved in ZF set theory and thus does not

**require**the

**axiom**of choice.

8) The construction of these pieces

**requires**the

**axiom**of choice; the pieces are not simple regions of the ball, but complicated subsets.

9) But in general, for non-commutative operations (such as the next example of the quaternions), they are not equivalent, and therefore

**require**separate

**axioms**.

10) Hence relations can be defined by set theory, thus the theory of relations does not

**require**any

**axioms**or primitive notions distinct from those of set theory.

11) The de Broglieâ€“Bohm theory, by contrast,

**requires**no such measurement

**axioms**(and measurement as such is not a dynamically distinct or special sub-category of physical processes in the theory).

12) Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets

**requires**additional

**axioms**related to determinacy and large cardinals.

13) Indeed, the

**axioms**

**required**of a monoid operation are exactly those

**required**of morphism composition when restricted to the set of all morphisms whose source and target is a given object.

14) In a FOL system additional

**axioms**are

**required**to infer facts such as a block does not change position unless it is moved.

### example sentences with axiom

15) Here are some statements that**require**the

**axiom**of choice in the sense that they are not provable from ZF but are provable from ZFC (ZF plus AC).

16) A proof requiring the

**axiom**of choice may establish the existence of an object without explicitly defining the object in the language of set theory.

17) Since "X" is not measurable for any rotation-invariant countably additive finite measure on "S", finding an algorithm to select a point in each orbit

**requires**the

**axiom**of choice.

18) One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem,

**require**the

**axiom**of choice for their proofs.

19) This

**requires**a so-called frame axiom, a formula like:

20) It is easy to check that this system obeys the

**axioms**

**required**of a projective plane:

21) The abstractly

**required**field

**axioms**reduce to standard properties of rational numbers, such as the law of distributivity

22) " Von Neumann avoided

**axiom**schemas by formalizing the concept of "definite propositional function" with his functions, whose construction

**requires**only finitely many

**axioms**.

23) (These last two assertions

**require**the

**axiom**of choice in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as L"p" spaces.)

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