Can Mathematical Axioms and Postulates Prove That 1+1 Equals 2?

  • Context: Undergrad 
  • Thread starter Thread starter richardnub
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Discussion Overview

The discussion revolves around the foundational aspects of mathematical axioms and postulates in relation to proving that 1+1 equals 2. It explores the definitions of addition and the natural numbers, as well as the implications of various mathematical frameworks.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions whether there exists an axiom or postulate that defines addition.
  • Another participant suggests that defining natural numbers as an ordered set allows for the concept of "succession," which can lead to a definition of addition through repeated succession, although they note that this approach requires further formalization.
  • A third participant provides a link to the Peano axioms, indicating a potential framework for understanding the axioms related to natural numbers.
  • One participant argues that set theory is not necessary unless one aims to prove that integer addition adheres to set axioms, and they propose that one could simply define 2 as equal to 1+1.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of set theory and axioms in defining addition and proving that 1+1 equals 2. No consensus is reached on the best approach or the foundational requirements for such a proof.

Contextual Notes

There are limitations regarding the assumptions made about the definitions of addition and natural numbers, as well as the implications of using set theory versus axiomatic definitions.

richardnub
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Is there some axiom or postulate that defines addition?

I've always wondered this.
 
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Well, when you get to that level it becomes rather difficult. I believe that if you define the natural numbers as an ordered set, you can define an operation on them that essentially says "go to the 'next' number in the set", which we can call "succession" and denote as s(n), where n is a natural number. In normal terms, s(n)=n+1, but since we haven't defined "addition" yet, we can't really call it that yet. Then, you can define addition as repeated succession in some way, and from there define 1+1 and find it to be equal to 2.

At least, I think the above method can work. Obviously someone better than I would have to come along and codify it.
 
You do not really need to delve into set theory, unless you want to prove that the rules we set up for integer addition follow from the rules we have from set axioms (i.e, mathematical reductionism).

You can, of course, DEFINE 2 to be equal to 1+1.
 

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