## Is axiom of choice equivalent to well-ordering theorem?

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In mathematics, the well-ordering theorem, also known as Zermelo’s theorem, states that every set can be well-ordered. The well-ordering theorem together with Zorn’s lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents).

### Is the well-ordering principle true?

As pointed out in the introduction, not every ordered set is well-ordered, but it is in fact true that every set has an ordering under which it is well-ordered, if one assumes the axiom of choice.

#### Does the well-ordering principle have a proof?

The proof is by well ordering. Let C be the set of all integers greater than one that cannot be factored as a product of primes. We assume C is not empty and derive a contradiction. If C is not empty, there is a least element, n 2 C, by well ordering.

**What is the meaning of axiom of choice?**

The axiom of choice is an axiom in set theory with wide-reaching and sometimes counterintuitive consequences. It states that for any collection of sets, one can construct a new set containing an element from each set in the original collection. In other words, one can choose an element from each set in the collection.

**Why is Zorn’s lemma equivalent to axiom of choice?**

Zorn’s lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that within ZF (Zermelo–Fraenkel set theory without the axiom of choice) any one of the three is sufficient to prove the other two.

## Why is Z not well-ordered?

Then x−1∈Z. But x−1contradicts the supposition that x∈Z is a smallest element. Hence there can be no such smallest element. So by Proof by Contradiction, Z is not well-ordered by ≤.

### What does well-ordered mean in math?

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set.

#### How do you use axiom of choice?

The axiom of choice allows us to pick elements from ‘indexed sets’. When dealing with ‘finite things’, this seems kinda obvious. For instance, if A={1,2,3}, B={3,4,5}, and C={5,6}, then it is easy to pick an element from each. Just pick, say, 1 from A, 3 from B, and 6 from C.

**What is meant by well-ordering?**

Definition of well-ordered 1 : having an orderly procedure or arrangement a well-ordered household. 2 : partially ordered with every subset containing a first element and exactly one of the relationships “greater than,” “less than,” or “equal to” holding for any given pair of elements.

**Why do we need axiom of choice?**

The Axiom of Choice tells us that there is a set containing an element from each of the sets in the bag. Basically, this allows us to meaningfully extract elements from infinitely large collections of sets. In fact, it allows us to do this even if each set contains an infinite number of elements themselves!