snits
- 14
- 0
Why do some people define the natural numbers as the integers 0,1,2,3... while others define them as the integers 1,2,3... ?
The Debate Over Natural Numbers: 0 vs 1
- Context: Undergrad
- Thread starter snits
- Start date
Join the discussion
Registration is free. Ask a follow-up in this thread, or start your own.
7 replies · 4K views
Discussion
- 42,895
- 983
The former, 0, 1, 2, ..., (now usually referred to as the "whole numbers"), historically, was used by Peano when he set up "Peano's axioms" for the natural numbers. Modern treatments usually start with 1, 2, 3, ... It really doesn't matter which you use as long as you are consistent.
- 14,922
- 28
HallsofIvy said:
I don't remember the last time I've seen a modern treatment starting with 1 instead of 0!Hurkyl said:
Ditto. I thought it was the reverse: 1, 2, 3, ... was classic for the Peano axioms, but modern treatments use 0, 1, 2, ...
- 42,895
- 983
Oh, dear,am I living backwards?
Werg22
- 1,431
- 1
Edmund Landau's famous Foundations of Analysis starts with 1. 0 is then defined as an equivalence class.
adriank
- 534
- 1
The way I see it, set theorists like to include 0 in the natural numbers, and everyone else doesn't. :)
Either that, or they omit any mention of "natural numbers" completely and call it, say, Z+, the set of positive integers.
Either that, or they omit any mention of "natural numbers" completely and call it, say, Z+, the set of positive integers.
Last edited:
Tac-Tics
- 816
- 7
The important property of both N and Z+ is that they are countable. Whether 0 is cool enought to join the party usually doesn't matter.