well ordering principle

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Aug 18th 2009, 11:50 AM
flower3

well ordering principle

Use the WOP of N to prove that there is no $n \in N$ such that 0<n<1
Aug 18th 2009, 12:59 PM
putnam120

Well if there were then since $n\in\mathbb{N}$ and $n>0$ , it follows that $(n-1)\in\mathbb{N}$ . Now you have that $(n-1)\neq{0}$ , so continue in this way to get a contradiction.
Aug 18th 2009, 01:37 PM
Plato

Quote:

Originally Posted by flower3

Use the WOP of N to prove that there is no $n \in N$ such that 0<n<1

I am a bit puzzled by the wording of this question.
I my experience the following is a standard theorem: $\left( {\forall x \in \mathbb{R}} \right)\left( {\exists !n \in \mathbb{Z}} \right)\left[ {n \leqslant x < n + 1} \right]$ .
The unique integer is of course the floor of $x$ .
I see what putnam120 is trying to do. But I an not at all sure where W.O. be used.

The theorem I quoted above with $x=0$ give the result at once.

PS. Of course, this is all mute if your course includes a rigorous construction of the natural numbers.
Aug 18th 2009, 02:18 PM
putnam120

W.O. states that every set of non-negative integers has a least element. Thus since $\mathbb{N}$ only contains non-negative integers it to must have a minimal element. So using the construction I started you show that the set doesn't have a minimal element.

I usually see $\mathbb{N}$ defined as follows:

1) $0\in\mathbb{N}$

2) $(n\neq{0})\in\mathbb{N}\Longrightarrow (n\pm{1})\in\mathbb{N}$

3) $(n=0)\in\mathbb{N}\Longrightarrow (n+1)\in\mathbb{N}$