Proof of the division algorithm

matqkks

New member
Joined
Jun 25, 2019
Messages
3
In many books on number theory they define the well ordering principle (WOP) as:
Every non- empty subset of positive integers has a least element.
Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it possible to apply the WOP to a subset of non-negative integers? Am I being too pedantic?
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
4,395
Since the principle as you state it requires positive integers, it can't be applied directly.

But given a set \(\displaystyle A\) of non-negative integers, you could form the set \(\displaystyle B = \{a+1 : a\in A\}\), and apply the principle to that.
 

matqkks

New member
Joined
Jun 25, 2019
Messages
3
Thanks.
 
Top