- #1
snits
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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
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Discussion Overview
The discussion revolves around the definition of natural numbers, specifically the inclusion of 0 versus starting from 1. It touches on historical perspectives, modern mathematical conventions, and the implications of these definitions in set theory and analysis.
Discussion Character
- Debate/contested
- Historical
- Conceptual clarification
Main Points Raised
- Some participants note that natural numbers are defined as 0, 1, 2, 3... by some, while others define them as 1, 2, 3... and question the reasons behind these differing definitions.
- One participant mentions that Peano's axioms historically used the definition including 0, while modern treatments are said to often start with 1, 2, 3..., suggesting that consistency is key regardless of the definition chosen.
- Another participant expresses surprise at the claim that modern treatments start with 1 instead of 0, indicating a belief that the opposite is true.
- A participant references Edmund Landau's work, stating that it starts with 1 and defines 0 as an equivalence class, adding to the historical context of the definitions.
- One participant suggests that set theorists tend to include 0 in the natural numbers, while others prefer to exclude it, sometimes referring to the positive integers as Z+ instead.
- Another point raised is that both N (natural numbers) and Z+ (positive integers) are countable sets, implying that the inclusion of 0 may not significantly impact their properties.
Areas of Agreement / Disagreement
Participants express differing views on the definition of natural numbers, with no clear consensus on whether 0 should be included or not. The discussion remains unresolved regarding which definition is more widely accepted in modern mathematics.
Contextual Notes
Participants reference historical definitions and modern conventions, indicating that the discussion may depend on specific contexts or interpretations within mathematics.
- #2
HallsofIvy
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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.
- #3
Hurkyl
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HallsofIvy said:
I don't remember the last time I've seen a modern treatment starting with 1 instead of 0!- #4
CRGreathouse
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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, ...
- #5
HallsofIvy
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Oh, dear,am I living backwards?
- #6
Werg22
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Edmund Landau's famous Foundations of Analysis starts with 1. 0 is then defined as an equivalence class.
- #7
adriank
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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:
- #8
Tac-Tics
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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.
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