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In chapter 1 of Spivak's Calculus text he lays out some fundamental axioms of the integers. For instance that: $a \cdot 1 = a$, for all $a$. However he doesn't list an axiom that for instance says: $a \cdot 0 = 0$, for all $a$. This seems a bit arbitrary. Can we derive $a \cdot 0 = 0$ from Spivak's other axioms? On page $6$ he just says that $a \cdot 0 = 0$, for all $a$, without explanation.

Also he seems to be taking for granted that if $a = b$, then $a + c = b + c$. Another implicit axiom.

Why doesn't he mention these “implicit axioms” explicitly?

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    Your "if a + b, then a + c = b + c" is part of the definition of "operator". I agree that it's a bit confusing.2012-12-22
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    Actually, it's part of what constitutes *identity*. Whenever $a=b$, we also have $\phi(a)\leftrightarrow\phi(b)$ for all predicates $\phi$. If we take $\phi(x)\equiv x+c=a+c$ then this means $a+c=a+c\leftrightarrow b+c=a+c$. And $a+c=a+c$ is of course also true )by refelxivity of $=$). Such is sometimes taken for granted ("FOL with identity")2012-12-22
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    I'd note that Spivak does actually give an explanation of a * 0 = 0 on the next page. Sorry about that2012-12-22

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