Examples of 'axiom of choice' in a sentence
Meaning of "axiom of choice"
The 'axiom of choice' is a mathematical principle that states that given any collection of sets, it is possible to select one element from each set, even if the collection is infinite. It is an important concept in set theory and has various applications in different branches of mathematics
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- One of the axioms of set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty; any version of said axiom, for example specifying the cardinality of the number of sets from which choices are made.
How to use "axiom of choice" in a sentence
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axiom of choice
The proof needs the axiom of choice.
The axiom of choice is often mentioned when used.
This result depends on the axiom of choice.
Case of the axiom of choice is true.
Their existence relies on the axiom of choice.
If the axiom of choice is not assumed we need to do something different.
Lis always a model of the axiom of choice.
The axiom of choice is true.
A word on notation and the axiom of choice.
The axiom of choice implies the law of excluded middle.
See also use of the axiom of choice below.
The axiom of choice is independent of the other axioms of set theory.
This uses the axiom of choice.
Graduate student from the math department was lecturing on the axiom of choice.
Neither is the axiom of choice.
See also
The statement that all sets are projective is equivalent to the axiom of choice.
The axiom of choice.
Not every situation requires the axiom of choice.
The status of the axiom of choice varies between different varieties of constructive mathematics.
This theorem does not rely on the axiom of choice.
The axiom of choice is not one of the axioms in ZF.
This requires a suitable version of the axiom of choice.
Countable axiom of choice.
The following theorem is equivalent to the axiom of choice.
The Axiom of Choice is the most controversial axiom in the entire history of mathematics.
I is always a model of the axiom of choice.
If the axiom of choice holds, then every successor cardinal is regular.
Was lecturing on the axiom of choice.
If the axiom of choice holds, the law of trichotomy holds for cardinality.
There are many other equivalent statements of the axiom of choice.
In that case, the axiom of choice must be invoked.
Determinacy is however not compatible with the axiom of choice.
If the axiom of choice holds, every cardinal number has an initial ordinal.
Hence this statement is strictly weaker than the axiom of choice.
If the axiom of choice is assumed, then all of these concepts are equivalent.
All these are linked in one way or another to the axiom of choice.
Existing proofs of this require the axiom of choice for the non-separable case.
Perhaps the best known of such assertions is the axiom of choice.
Otherwise, the axiom of choice will fail in LA.
This second definition makes sense without the axiom of choice.
A weaker version of an axiom of choice is the axiom of dependent choice, DC.
This has been used as an argument against the use of the axiom of choice.
Together these results establish that the axiom of choice is logically independent of ZF.
The law of trichotomy for cardinal numbers also implies the axiom of choice.
In the NF axiomatic system, the axiom of choice can be disproved.
The method relies on the axiom of regularity but not on the axiom of choice.
When the axiom of choice is included, the resulting system is ZFC.
We work in set theory ZF without axiom of choice.
If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets.
Every set is a projective object in Set assuming the axiom of choice.
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Examples of using Choice
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And now a choice of viewing on bbc television
She might not have had a choice
The number and choice of courses is up to you
Examples of using Axiom
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By the axiom of extensionality this set is unique
Financial operators are well aware of this axiom
This axiom might lead us to this theorem
