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I have just begun reading "Mathematical Analysis", $2^{nd}$ edition by Apostol. In the beginning of chapter $1$ we are introduced to $9$ axioms ($+1$ later). They are the field axioms and the order axioms (the later one which I haven't gotten to is the "completeness axiom" or "axiom of continuity"). However, on this website I found more, such as "There exists a unique number $0$ such that $a + 0 = a$ for any real number $a$."

Do the axioms outside of field axioms, order axioms, axiom of continuity have a special name? Why where they not even mentioned in the book?

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  • $\begingroup$ Those would be called "basic properties" I think. You should be able to derive existence of additive identity $0$ from the axioms in your book. $\endgroup$
    – MathCrackExchange
    Dec 18, 2015 at 2:17
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    $\begingroup$ Related. $\endgroup$
    – Cameron Buie
    Dec 18, 2015 at 2:34

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This is the axiom of identity for addition, which states that there is an identity element (adding it does not change the value) for addition.

See if you can figure out the identity element for multiplication.

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  • $\begingroup$ Well the multiplicative identity would be x$1$, but I am wondering if these are really axioms or if they can be derived from the 10 axioms in my book, because I don't understand why the author wouldn't bother to include the identity of addition since it is so crucial $\endgroup$
    – Ovi
    Dec 18, 2015 at 2:20
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    $\begingroup$ The way to show that an axiom is independent of the other axioms is to construct a system in which the other axioms are true and the axiom is false. The way to show that an axiom is not independent of the other axioms is to derive it as a consequence of them. Have at it! $\endgroup$
    – marty cohen
    Dec 18, 2015 at 2:39
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    $\begingroup$ @Ovi There is no such thing as something being "really" an axiom. A set of axioms is just a starting point. Different authors may use different sets of axioms, and one author's axiom might be another author's theorem. Apparently the author of your textbook chose not to include $x+0=x$ as an axiom, but it should be derivable from the axioms that are included. $\endgroup$
    – Ted
    Dec 18, 2015 at 3:41
  • $\begingroup$ @Ted I'd love to be able to start working on some proofs, but I don't really know what I'm allowed to work with. For example, axiom $4$ says that "there is always a z such that $x + z = y$, and x - x is denoted by $0$ (it can be proven that $0$ is independent of $x$.)" So I'm not even sure if at this point I'm allowed to work with zero as a real number, or if it's just a symbol/placeholder. And if it is a number, I don't see why you need a proof, that $0$ is independent of $x$, if by definition $x - x$ = $0$ and $x$ could be anything $\endgroup$
    – Ovi
    Dec 18, 2015 at 9:21
  • $\begingroup$ @martycohen too^^ sorry can't put 2 @ user's in the same comment $\endgroup$
    – Ovi
    Dec 18, 2015 at 9:22

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