## Does well-ordering principle apply natural numbers?

**contains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element**. Depending on the framework in which the natural numbers are introduced, this (second order) property of the set of natural numbers is either an axiom or a provable theorem.

## Why is the well-ordering principle important?

Quote from video on Youtube:*So here's a statement of the well ordering principle. Every non-empty set of non-negative integers.*

## How do you prove the natural numbers are well-ordered?

The Well-Ordering Principle can be used to prove all sort of theorems about natural numbers, usually by assuming some set $T$ is nonempty, finding a least element $n$ of $T$, and “inducting backwards” to find an element of $T$ less than $n$–thus yielding a contradiction and proving that $T$ is empty.

## What do you mean by well-ordering principle?

Definition: The Well Ordering Principle. **A least element exist in any non empty set of positive integers**. This principle can be taken as an axiom on integers and it will be the key to proving many theorems.

## How do you prove using the well-ordering principle?

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 meant by well-ordering list few examples?

The Well Ordering Principle says that **the set of nonnegative integers is well ordered, but so are lots of other sets**. For example, the set r\mathbb{N} of numbers of the form rn, where r is a positive real number and n \in \mathbb{N}.

## Is well-ordering principle an axiom?

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).

## Can C be well-ordered?

The Well-Ordering Principle states that every set can be well-ordered. This result is equivalent to the Axiom of Choice. It is therefore true that **C can be “well- ordered,”** but this should not be confused with the idea of ordering C in a way that generalizes the ordering of R.