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Everyday I take axioms for granted

  1. Aug 30, 2006 #1

    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:
     
  2. jcsd
  3. Sep 5, 2006 #2

    J77

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    They're not that important, then?

    :biggrin:
     
  4. Sep 5, 2006 #3

    HallsofIvy

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    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.
     
  5. Sep 5, 2006 #4

    radou

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    A neat system of axioms to explore are the Euclidean plane geometry axioms.
     
  6. Sep 5, 2006 #5

    CRGreathouse

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    I'm using ZF + "odd perfect numbers exist" and trying to conclude that dragons exist.
     
  7. Sep 5, 2006 #6

    HallsofIvy

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    It think a constructivist approach would be best for that- exhibit a dragon!
     
  8. Sep 5, 2006 #7
    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?
     
  9. Sep 5, 2006 #8
    well, you just need to adopt a suitable definition for a dragon!
     
  10. Sep 5, 2006 #9

    CRGreathouse

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