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Euclid's 13th Proposition goes as :

Proposition 13. If a straight-line stood on a(nother) straight-line makes angles, it will certainly either make two rightangles, or (angles whose sum is) equal to two rightangles.

I am annoyed because I find some text book treats this Proposition as an AXIOM.

It confused me a lot. I need an expert comment.

saibal
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1 Answers1

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Axioms are pedagogy: a means of exposition. The same theory can be presented in many different forms. Axiomness isn't an intrinsic quality of a statement, so some presentations may have different axioms than others.

(axioms also have technical value, but I believe that's unrelated to this question: e.g. you can show that something is a model of a theory by showing it satisfies a choice of axioms for the theory)

  • My question is simple. Whether Proposition 13 of Euclid is a proposition or an axiom ? – saibal Jul 02 '14 at 08:12
  • @saibal: It's an axiom in and only if you decide to include it in an axiomatization. It's always a theorem. (note that axioms are always theorems: as they are trivial to prove a statement using itself) –  Jul 02 '14 at 08:46