axiom of choice in a sentence
1) The proof needs the axiom of choice.
2) Not every situation requires the axiom of choice.
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 sentences5) 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 axiom15) 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 sentence28) 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 proofs.
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 proof of the well-ordering theorem.
35) This follows from Zorn's lemma, an equivalent formulation of the axiom of Choice.
36) The proof 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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