Fundamental Property of Integers

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Discussion Overview

The discussion centers around the properties of integers, specifically focusing on the proof that the sum of any two even numbers is an even number. Participants explore the foundational aspects of integers and addition, questioning whether these properties are axiomatic or derived from proofs.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents a proof that the sum of two even numbers is even, questioning the assumption that the sum of two integers (k + l) is also an integer.
  • Another participant mentions the closure axiom as relevant to the discussion.
  • A participant emphasizes the need for definitions of integers and addition, suggesting that the lack of formal definitions may lead to vagueness in the argument.
  • Concerns are raised about whether different definitions of integers and addition could lead to different number theories, similar to how different axioms can lead to different geometries.
  • Another participant argues that while starting from axioms may be overkill for this discussion, some assumptions, like the sum of two integers being an integer, are intuitive and generally accepted.
  • There is a suggestion that all theories of integers can be considered "isomorphic," indicating that while axioms may vary, they do not fundamentally change the nature of integers.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of formal definitions for integers and addition, with some suggesting that intuitive understanding suffices while others argue for the importance of clarity in definitions. The discussion remains unresolved regarding the implications of different definitions on number theory.

Contextual Notes

Participants acknowledge the intuitive nature of integers and addition but highlight the potential for ambiguity without formal definitions. The discussion touches on foundational mathematical concepts without reaching a consensus on the necessity of axiomatic definitions.

ChuckleFox
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So there is a proof that the sum of any two even numbers is an even number.

2k + 2l = 2(k +l)

We have written the sum as 2 times an integer. Therefore the sum of any two even numbers is an even number.

An essential part of this proof is that k + l is an integer. How do we know this? Is it an assumed property of integers, an axiom, or is there a proof out there that this is true?
 
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Closure axiom
 
ChuckleFox said:
So there is a proof that the sum of any two even numbers is an even number.

2k + 2l = 2(k +l)

We have written the sum as 2 times an integer. Therefore the sum of any two even numbers is an even number.

An essential part of this proof is that k + l is an integer. How do we know this? Is it an assumed property of integers, an axiom, or is there a proof out there that this is true?

How did you define integers, and how did you define +? The answer to your problem relies on this.
 
micromass said:
How did you define integers, and how did you define +? The answer to your problem relies on this.

We didn't define anything. We just used our intuitive understanding of integers and addition. I assume this makes what we were doing uselessly vague?

Your answer makes me think that integers and addition can be defined in different ways. Do different theories exist depending on how we define integers and addition? For example, if we define integers in two different ways, does this lead to two different number theories? I mean this in the same way that different axioms lead to different geometries.
 
ChuckleFox said:
We didn't define anything. We just used our intuitive understanding of integers and addition. I assume this makes what we were doing uselessly vague?

No, it's not uselessly vague. It is probably fine for what your course/book needs. It's not always possibly to do things starting from the axioms, and it would probably be overkill here. However, it does mean that you'll have to take some things for granted, for example that the sum of two integers is an integer. But since that is very intuitive anyway, it shouldn't be a problem.

Your answer makes me think that integers and addition can be defined in different ways.

Yes. Do different theories exist depending on how we define integers and addition? For example, if we define integers in two different ways, does this lead to two different number theories? I mean this in the same way that different axioms lead to different geometries.

Now, we can prove that all theories are "isomorphic" (this means that they are the same for all practical purposes). We can alter the axioms a bit and get a new theory, but we don't call this the integers anymore.
 

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