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Number of axioms for a theory

  1. Sep 19, 2015 #1
    I've been asking myself a few questions lately regarding the nature of a theory. It can be any type of theory. I hope someone can answer these to a degree. The questions are:
    1. Can a theory have less than three axioms? Is three the minimum for a theory to make sense?
    2. Is the statement "The less axioms, the more abstract the theory, the more facts will fit" true ?
    3. How does the number of axioms relate to the number of contradictions that may appear within the theory?

    Thanks in advance.
     
  2. jcsd
  3. Sep 19, 2015 #2
    Yes, I present "the zero theory":
    • Axiom 0: 0 is a natural number.
    I don't think you could say that the zero theory doesn't make sense, but it is not very interesting. So perhaps you want to ask "is three the minimum for a theory to be interesting?". For that question to have any meaning you would have to define "interesting", and that is a matter of philosophy not mathematics. We don't discuss philosophy in these forums.

    Again that requires a definition of "abstract"... however I think that the opposite of "the more facts will fit" is true - how many facts fit the zero theory?

    If you can find any contradictions (i.e. a statement for which both the statement and its negation can be proved true from the axioms) then the axiomatic system is inconsistent and would not be called a theory.
     
  4. Sep 19, 2015 #3
    LOL, you're a robot. I guess i had to ask the questions on a philosophy forum because I meant a philosophical theory. Besides; The statement that zero is a natural number is a matter of dispute, even in such a "precise" field like mathematics.
     
  5. Sep 19, 2015 #4

    epenguin

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    I think I heard that if you can find one contradiction then you can disprove all the statements you can prove in that theory - is this right?
     
    Last edited: Sep 20, 2015
  6. Sep 19, 2015 #5
    An axiom is a statement that is defined to be true, it is not something that can be disputed.
     
  7. Sep 20, 2015 #6

    Mark44

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    Discussions about philosophy aren't permitted at this forum.
     
  8. Sep 20, 2015 #7

    epenguin

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    There we have very nice example of reductio ad absurdam. :oldbiggrin:
     
  9. Sep 20, 2015 #8

    Ssnow

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    I think we need a Logician that is an expert of Theories & Formal Systems to answer this question ... :smile: If the question is pertinent in a math forum I don't know ...
     
  10. Sep 20, 2015 #9
    Once you have a contradiction, everything falls apart like this:
    • Let the contradiction be expressed thus: (A is a theorem of S) (Theorem 1) and ((not A) is a theorem of S) (Theorem 2).
    • Now for any well defined statement B, we have by Theorem 1 ((A or B) is a theorem of S) (Theorem 3) and also ((A or (not B) is a theorem of S) (Theorem 4).
    • By Theorem 2 and Theorem 3, (B is a theorem of S) (Theorem 5).
    • By Theorem 2 and Theorem 4 ((not B) is a theorem of S) (Theorem 6).
    So if we have a contradiction, any statement and its inverse can both be proved.
     
  11. Sep 20, 2015 #10

    Curious3141

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    Have you just proven we don't exi...(poof) :oldbiggrin:
     
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