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  • What is exactly the difference between a definition and an axiom?
    We like to have an axiom which says "there is an empty set", even if it's really a theorem that can be derived from the other axioms, because it's just a bit neater To summarise, the line is not always very clear-cut, and it's not always clear whether something should be an axiom or a definition
  • real analysis - What is the difference between lemma, axiom, definition . . .
    In learning basic analysis I am encountering a lot of language that seems somewhat tough to define: Axiom, proposition, definition, lemma, theorem, law, corollary Are there clear separations in
  • What is the definition of axiom (mathematically speaking)
    An axiom can simply be a definition or it can be a theorem A definition only identifies something, gives it a name, and does contain any real information about it (as in the definition of a point) A theorem, on the other hand, says what something can can't do (eg parallel lines can never cross)
  • Difference between axioms, theorems, postulates, corollaries, and . . .
    An axiom is some property of your undefined object So even though you never define your undefined terms you have rules about them The rules that govern them are the axioms One does not prove an axiom, in fact one can choose it to be anything he wishes (of course, if it is done mindlessly it will lead to something trivial)
  • Definition of an axiom? - Mathematics Stack Exchange
    An axiom is any unproven mathematical assumption we make, so we can regard any well-formed mathematical assertion as an axiom Whether a statement is good axiom is a different story For instance, we should avoid assuming things if we can prove them from our other assumptions, we should try to make the things we assume well-motivated and
  • elementary set theory - Difference between Axiom and Definition . . .
    It's an axiom that's needed as part of the definition of the Reals So there's basic definitions, then axioms, from which you compose a definition Perhaps another difference between a definition and an axiom is definitions are always bidirectional logical statements Axioms are typically unidirectional logical statements
  • What is an axiom in laymans terms? - Mathematics Stack Exchange
    Nevertheless, a "good" axiom system is one that allows us to apply it to interesting things As among those interesting things are the theory of natural numbers, geometry, and set theory, certain axiom systems (Peano, Euklid, Zermelo-Frenkel) relating to these theories have become somewhat standard
  • Use of word axiom in definition of vector spaces
    The definition bit can be reinterpreted as if you prove the axioms about your arbitrary thing, you also get the entire theory of vector spaces "for free" $\endgroup$ – Yakk Commented Sep 28, 2016 at 18:25
  • Definition of Natural Numbers which gives rigorous formulation to . . .
    So I know the first order theory that is Peano Arithmetic (just to ensure we are on the same page, it is a first order theory with signature $\{0, S, +, \cdot, = \}$ with the axioms: (i) All axioms from theory of pure equality, (ii) Zero and Successor Axiom (iii) 2 axioms on defining $+$ (iv) 2 axioms on defining $\cdot$ and (v) Induction
  • Why isnt the axiom of extensionality considered a definition of equality?
    The answer in the linked post seems to address what would happen if the axiom were rejected altogether, but doesn't discuss whether there is good reason to consider it an axiom rather than a definition If axioms and definitions are just two different words for the same thing, then that's fine—but I get the feeling I'm missing something





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