Everyday I take axioms for granted

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SUMMARY

This discussion centers on the importance of axioms in mathematics, particularly in relation to operations like multiplication and addition. Participants highlight that while axioms are foundational, operations are derived from them, with a focus on set theory axioms and Euclidean plane geometry. The conversation also touches on the use of Zermelo-Fraenkel set theory (ZF) combined with the hypothesis of odd perfect numbers to explore abstract concepts, such as the existence of dragons. The consensus is that the significance of axioms varies depending on the mathematical system being examined.

PREREQUISITES
  • Understanding of Zermelo-Fraenkel set theory (ZF)
  • Familiarity with Euclidean plane geometry axioms
  • Basic knowledge of mathematical operations like addition and multiplication
  • Concept of constructivism in mathematics
NEXT STEPS
  • Research the implications of Zermelo-Fraenkel set theory in modern mathematics
  • Explore the axioms of Euclidean geometry and their applications
  • Study the concept of constructivism and its role in mathematical proofs
  • Investigate the hypothesis of odd perfect numbers and its significance in number theory
USEFUL FOR

Mathematicians, educators, students of mathematics, and anyone interested in the foundational principles of mathematical systems and axioms.

J77
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Another thread got me thinking...

Everyday I take axioms for granted, eg. muliplication, addition, ordering of reals.

From the pure point of view, what axioms are the most important (most used) ones?

Wikipedia has a list: http://en.wikipedia.org/wiki/List_of_axioms

However, I'd like to know the purists opinions :smile:
 
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They're not that important, then?

:biggrin:
 
You will have to tell us what you consider to be "important".
Mathematics involves an enormous number of "systems" each of which has its own axioms. Within a specific system, from a logical viewpoint, all axioms are equally "important".

The wikipedia list you cite is essentially a list of axioms for set theory.
 
A neat system of axioms to explore are the Euclidean plane geometry axioms.
 
I'm using ZF + "odd perfect numbers exist" and trying to conclude that dragons exist.
 
It think a constructivist approach would be best for that- exhibit a dragon!
 
J77 said:
Everyday I take axioms for granted, eg. muliplication, addition, ordering of reals.
Seems to me that multiplication and addition are operations not axioms. Once you stated an axiom describing the ordering of numbers, operations like addition and multiplication would follow logically and would not require additional axioms. No?
 
CRGreathouse said:
I'm using ZF + "odd perfect numbers exist" and trying to conclude that dragons exist.

well, you just need to adopt a suitable definition for a dragon!
 
Data said:
well, you just need to adopt a suitable definition for a dragon!

Why? I'd think any old definition would do.
 

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