1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Natural Numbers

  1. Jan 6, 2009 #1
    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... ?
  2. jcsd
  3. Jan 6, 2009 #2


    User Avatar
    Science Advisor

    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.
  4. Jan 6, 2009 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    :eek: I don't remember the last time I've seen a modern treatment starting with 1 instead of 0!
  5. Jan 6, 2009 #4


    User Avatar
    Science Advisor
    Homework Helper

    Ditto. I thought it was the reverse: 1, 2, 3, ... was classic for the Peano axioms, but modern treatments use 0, 1, 2, ....
  6. Jan 6, 2009 #5


    User Avatar
    Science Advisor

    Oh, dear,am I living backwards?
  7. Jan 6, 2009 #6
    Edmund Landau's famous Foundations of Analysis starts with 1. 0 is then defined as an equivalence class.
  8. Jan 7, 2009 #7
    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.
    Last edited: Jan 7, 2009
  9. Jan 7, 2009 #8
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook