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I am reading this book "Math Proofs Demystified" by Stan Gibilisco in which the author treats all of the congruence criteria/theorems of triangles as axioms instead of treating of one of them as axioms as using it to prove others as theorems, not only that he also defines sum, difference n product of fractions as axioms. Is this justified ?

max_zorn
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MathDumbo
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    Axiomatizations need not be minimal. Many (most?) textbooks introduce groups with very redundant axioms, in particular using two-sided neutrals and two-sided inverses, for example. Just as often you will find the existence of the empty set or the Axiom schema of Comprehension as axioms of set theory, even though they can be derived from the rest as theorems. – Hagen von Eitzen Jul 03 '18 at 05:31
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    What do you mean by "justified"? – Eric Wofsey Jul 03 '18 at 05:57
  • I mean ...I am just starting out on my math journey and these kinda things stop me in my tracks . And the author has not provided as to why it's being done that way . I mean he could have very well said that all of these criterions are not usually taken as axioms and I am doing this since I don't want to bother with the proofs here . But the author goes straight goes to list the three criterions as the three axioms of congruence . Isn't the aim of mathematics to keep the axioms as small in number as possible and as self-evident/simple as possible . – MathDumbo Jul 03 '18 at 14:23
  • See e.g. Edwin Moise, Elementary Geometry from an Advanced Standpoint, Addison-Wesley (3rd ed 1990) page 100-on for the proof of other congruence criteria on the basis of SAS and for the proof of the independence of the SAS postulate from the remaining ones. – Mauro ALLEGRANZA Jul 03 '18 at 20:36

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