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Math Help - doubts about axioms

  1. #1
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    Question doubts about axioms

    I wonder if an axiom can be proved. I was checking a website: this
    In this site addition's Commutative Law is said to be an axiom. However, in this site there's a proof to the addition's commutative law. How can this be? What is the difference between axiom and definition?
    Thanks in advance!
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  2. #2
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    Axioms are different from definitions in the sense that they "look" like Theorems. But an Axiom is assumed to be true, whereas a theorem needs to be proved. A statement can be an axiom in one context and a theorem in a different context. For example, associativity is one of the axioms for a group, but it is a theorem that the real numbers are associative.
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  3. #3
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    Further, there can be many "equivalent" statements such that any one can be taken as an "axiom" and the other proved. There are, for example, a number of statements that distinguish the real numbers from the rational numbers:
    Monotone Convergence
    Cauchy Criterion,
    Bolzano-Weierstrasse,
    Heine-Borel,
    Connectedness,
    Least Upper Bound Property.

    Any one of those can be taken as an axiom and then the others proved from that.

    Further we can define the real numbers in terms of the rational numbers, and prove those statements using the axioms for the rational numbers. Defining the real numbers as "Dedekind Cuts", subsets of the rational numbers, makes it easy to prove the Least Upper Bound Property while defining the real numbers as equivalence classes of monotone or Cauchy sequences makes it easy to prove those statements.
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  4. #4
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    thanks!
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