What Defines Group Axioms in Mathematics?

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Discussion Overview

The discussion revolves around the nature of group axioms in mathematics, specifically questioning why certain properties are termed axioms rather than definitions. Participants explore the implications of these axioms and their relationship to models of groups.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions why the properties of groups (associativity, neutral element, inverse element, closure) are called axioms instead of definitions, suggesting that axioms are typically starting points from which truths are deduced.
  • Another participant asserts that the properties are indeed definitions of a group, emphasizing that axioms are not inherently 'defined to be true' but are true within any model that satisfies them.
  • A participant suggests that considering examples, such as the isometries of a cube, can provide clarity on the application of group axioms.
  • There is a challenge regarding the nature of axioms, with one participant stating that axioms cannot be proven, while the properties can be proven or disproven for specific sets.
  • Another participant reiterates that proving properties for a set does not equate to proving the axioms themselves, indicating a distinction between axioms and their application in models.
  • A participant expresses understanding of the distinction between axioms and models after the discussion.
  • One participant seeks recommendations for accessible introductions to mathematical logic.
  • Another participant recommends Robert R. Stoll's book on Set Theory and Logic as a useful resource for both set theory and logic.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the classification of group properties as axioms versus definitions, with no consensus reached on the terminology or implications of this distinction.

Contextual Notes

The discussion highlights the ambiguity in the definitions and roles of axioms and models in mathematics, as well as the potential confusion surrounding these concepts.

broegger
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Hi.

I'm reading a simple introduction to groups. A group is said to be a set satisfying the following axioms (called the 'group axioms'):

1) Associativity.

2) There is a neutral element.

3) Every element has an inverse element.

4) Closure.

My questions is simply: why are they called axioms? I thought an axiom was something we take as a starting point, defining it to be true and then deduce something from it (possibly together with other axioms). Why are 1-4 not just the definition of a group?
 
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They are the definition of a a group (modulo the fact that you've omitted to mention the binary operation). A group is something that satisfies these axioms (a model). Note, axioms are not things that are 'defined to be true' . They are just 'things' and in any model of the axioms they are true.

It just depends on how you like to label these things.
 
Last edited:
more useful is to think about an example, like the isometries of a cube, possibly orientation preserving, i.e. rotations carrying a cube into itself.
 
But you can't prove an axiom, and 1-4 can be proved (or disproved) for a given set?
 
In that case you are just proving that whatever set with whatever binary operation satisfies those axioms. You are not proving the axioms themselves.
 
Cincinnatus said:
In that case you are just proving that whatever set with whatever binary operation satisfies those axioms. You are not proving the axioms themselves.

Oh, I think I get it now. I guess I was confused about the distinction between the axioms themselves and 'the model' to which they are applied. Thanks, everyone.
 
By the way, does anybody know of a good, relatively accessible, introduction to the subject of mathematical logic?
 
Robert R. Stoll's Set Theory and Logic is an okay intro set theory text (although it only looks at naive set theory), but an excellent intro logic text. It's also put out by Dover so it's cheap.

edit: Link to book.
 
Last edited:
Thanks, I think I'll pick that one up.
 

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