Can you prove that 1+1=2 or is it an axiom?

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Discussion Overview

The discussion revolves around whether the statement "1 + 1 = 2" can be proven or if it is merely an axiom. Participants explore definitions, axioms, and the nature of mathematical proof within the context of foundational mathematics.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest using Peano axioms as a basis for discussing the proof of "1 + 1 = 2".
  • One participant proposes that "2" is defined as the successor of "1", leading to the expression "1 + 1 = S(1)".
  • Another participant argues that the statement is more of a definition than a proof, asserting that "2" is simply the number next to "1".
  • A participant elaborates on the definition of "2" as S(S(0)) and provides a reasoning process to show that "1 + 1 = 2" follows from this definition through the properties of addition.

Areas of Agreement / Disagreement

Participants express differing views on whether "1 + 1 = 2" can be considered a proof or if it is fundamentally a definition. No consensus is reached regarding the nature of the statement.

Contextual Notes

Some participants reference specific mathematical frameworks, such as Peano axioms, but the discussion does not resolve the implications of these frameworks on the proof status of "1 + 1 = 2".

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Can you prove that 1+1=2 or is it an axiom?
 
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Well, use peano axioms. Look it up yourself! :smile:
 


Unredeemed said:
Can you prove that 1+1=2 or is it an axiom?
Most commonly, that (or something very similar) is taken as the definition of the symbol '2'.
 


Unknot said:
Well, use peano axioms. Look it up yourself! :smile:

Could you say that:

x+1=S(x)

and, therefore
1+1=S(1)
S(1)=2

Would that work as a proof?
 


I wouldn't call that a proof, unless you were proving it a la Russell/Whitehead.

As Hurkyl said, it probably is a definition more than anything else. 2 is the number next to 1.
 


Okay, thanks :)
 


I consider S(S(0)) to be the definition of "2". As such, "1 + 1 = 2" unpacks as "S(0) + S(0) = S(S(0))". From the definition of addition, S(0) + S(0) is S(S(0) + 0). Also from the definition of addition, S(0) + 0 is S(0). Thus S(0) + S(0) = S(S(0)) and so 1 + 1 = 2.
 

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