# axiom of choice in a sentence

1) The pro

**of**needs the

**axiom**

**of**choice.

2) Not every situation requires the

**axiom**

**of**choice.

axiom collocations

3) This requires a suitable version

**of**the

**axiom**

**of**choice.

4) There are many other equivalent statements

**of**the

**axiom**

**of**choice.

## axiom of choice example sentences

5) However, specifying an isomorphism requires the**axiom**

**of**choice.

6) Lis always a model

**of**the

**axiom**

**of**choice.

7) If one accepts the

**axiom**

**of**choice, the answer is yes.

8) Assuming the

**axiom**

**of**choice, addition

**of**infinite cardinal numbers is easy.

9) Thus the

**axiom**

**of**

**choice**is not generally available in constructive set theory.

10) Within ZF, it is strictly weaker than the

**axiom**

**of**choice.

11) The status

**of**the

**axiom**

**of**

**choice**varies between different varieties

**of**constructive mathematics.

12) Assuming the

**axiom**

**of**choice, multiplication

**of**infinite cardinal numbers is also easy.

13) The existence

**of**the long line depends on the

**axiom**

**of**Choice.

14) If the

**axiom**

**of**

**choice**is not assumed we need to do something different.

### example sentences with axiom

15) Indeed, their existence is a non-trivial consequence**of**the

**axiom**

**of**choice.

16) Two famous statements in set theory are the

**axiom**

**of**

**choice**and the continuum hypothesis.

17) Together these results establish that the

**axiom**

**of**

**choice**is logically independent

**of**ZF.

18) Otherwise, the

**axiom**

**of**

**choice**will fail in L(A).

19) There are models

**of**Zermelo-Fraenkel set theory in which the

**axiom**

**of**

**choice**is false.

20) If the

**axiom**

**of**

**choice**is assumed, then all

**of**these concepts are equivalent.

21) The statement that every vector space has a basis is equivalent to the

**axiom**

**of**choice.

22) In the absence

**of**the

**axiom**

**of**choice, the reverse implications are all unprovable.

23) The

**axiom**

**of**

**choice**is not the only significant statement which is independent

**of**ZF.

24) NAV's best contemporary world music album:

**axiom**

**of**

**choice**(band).

25) There are several results in category theory which invoke the

**axiom**

**of**

**choice**for their proof.

26) A weaker version

**of**an

**axiom**

**of**

**choice**is the

**axiom**

**of**dependent choice, DC.

27) Under the assumption

**of**the

**axiom**

**of**choice, this transfinite sequence includes every cardinal number.

#### How to use axiom in a sentence

28) Skepticism about the**axiom**

**of**

**choice**was reinforced by recently discovered paradoxes in naive set theory.

29) This paper led to the general acceptance

**of**the

**axiom**

**of**

**choice**in the mathematics community.

30) The pieces in this decomposition, constructed using the

**axiom**

**of**choice, are non-measurable sets.

31) Using the

**axiom**

**of**choice, they select and memorize a representative sequence from each equivalence class.

32) The former result opened the door for mathematicians to assume the

**axiom**

**of**

**choice**in their pro

**ofs**.

33) Without the

**axiom**

**of**choice, these theorems may not hold for mathematical objects

**of**large cardinality.

34) In 1904, Ernst Zermelo promotes

**axiom**

**of**

**choice**and his pro

**of**

**of**the well-ordering theorem.

35) This follows from Zorn's lemma, an equivalent formulation

**of**the

**axiom**

**of**Choice.

36) The pro

**of**is based on Hausdorff's paradoxical ball decomposition, whose production requires the

**axiom**

**of**choice.

37) Assuming the

**axiom**

**of**Choice, one can show that __FORMULA__ for any infinite cardinal __FORMULA__.

38) That last one, called Tychonoff's theorem, is yet another equivalence to the

**axiom**

**of**choice.

39) It is well known that the

**axiom**

**of**

**choice**is equivalent to the ability to well-order every set.

40) There are several weaker statements that are not equivalent to the

**axiom**

**of**choice, but are closely related.

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