Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Which branch of math does that?

  1. Feb 25, 2007 #1
    Which branch of math deals with "proofs" of the axioms in arithmetic and such? I'd be interested to look into it.
  2. jcsd
  3. Feb 25, 2007 #2


    User Avatar
    Homework Helper

    You may want to investigate abstract algebra.
  4. Feb 25, 2007 #3
    I'm not sure exactly what you mean, as obviously nobody examines proofs of axioms!
  5. Feb 25, 2007 #4
    Well the Peano axioms are certainly provable... Radou, are the Peano axioms explored in abstract algebra?
  6. Feb 25, 2007 #5
    Provable from what? If they are truly axioms then they can't really be proved true or false from other statements.
  7. Feb 25, 2007 #6
    Axioms can be found logically... this is what I mean by "proof".
  8. Feb 25, 2007 #7
    A logical assumption is very different from a proof.
  9. Feb 25, 2007 #8
    If you are interested in a proof of the Peano axioms, you might want to look at an introductory set theory book, such as Halmos'.
  10. Feb 25, 2007 #9
    Herstein's, Topics in Algebra, and then there are a few unknown books that I found which surveyed selections of abstract algebra, to give you an idea of how broad the spectrum is and provide you with an illustration so that you can select areas that you enjoy. If you are curious, I can give you those as well (although, they are REALLY old and seriously, no one has heard of them but I enjoyed pieces of them). I think this is the area that you are referring to? It's beautifully elegant, you will enjoy it.

    As for axiomatic mathematics, I am under the impression that mathematicians construct axioms/assumptions/postulates to govern a particular formal logic system (such as an algebraic system) and from there, interesting properties emerge. If the system produces interesting properties, the system is studied further.

    Mathematics are deductive because if the premises are true then the conclusion must be true because the conclusion is contained in the premises. However, the premises must be assumed true.
    Last edited: Feb 25, 2007
  11. Feb 26, 2007 #10


    User Avatar
    Science Advisor

    Given any system of axioms, there always exist a simpler system in which those axioms can be proved as theorems.

    Peano's axioms assert that there exist a set of objects, N, called "natural numbers, and a function s (the "successor function"), from N to N such that
    Axiom 1: There exist a unique member of N, call "0", such that s is a one to one function from N to N\{0}.
    Axiom 2: If a subset, X, of N contains 0 and, whenever it contains n, it also contains s(n), then X= N.

    You can, however, define the natural numbers in terms of sets: 0 is the empty set, 1 is the set containing only 1 (only the empty set), 2 is the set containing only 0 and 1, and, in general, given any n s(n) is the set containing n and all of its members. From that one can show that Peano's axioms are true.
    Note: Historically, Peano's axioms included the number 0 as I have here. Nowadays, however, most people start with the number 1.

    Werg22, you might find this interesting:
    http://academic.gallaudet.edu/courses/MAT/MAT000Ivew.nsf/ID/918f9bc4dda7eb1c8525688700561c74/$file/NUMBERS.pdf [Broken]
    Last edited by a moderator: May 2, 2017
  12. Feb 26, 2007 #11
    Ok thanks all. Complex, I will try to familiarize myself with the subject before considering any book purchase, so I must put you on hold. Halls, I will check the link you gave as soon as I can.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook