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Spivak, Calculus, prologue, page 6 says:

Moreover, $1\ne 0$. (The assertion may seem a strange fact to list, but we have to list it, because there is no way it could possibly be proved on the basis of the other properties listed - these properties would all hold if there were only one number, namely, $0$).

I did not understand what would go wrong without that assertion. $a*1 = 1*a = a$ couldn't hold true if $1 = 0$.

J. W. Tanner
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Bruno Bartolomasi
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    If you assume that $1 =0$, then you see that every element is $0$ because $a\cdot 1 = a \cdot 0 = 0$. This is fairly noninteresting. – rubikscube09 Mar 13 '19 at 17:23
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    You don't get a contradiction. But you get something utterly trivial. $a = 1a = 0a = 0$. So your field consists of only one element: $0$ with the property $0+0 =0$ and $0*0 = 0$. That's too boring to even bother with. – fleablood Mar 13 '19 at 17:41
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    @Chickenmancer: multiplication by $1$ is perfectly well defined, because the field only contains one element: $1\times 1 = 1 = 0$. – TonyK Mar 13 '19 at 17:45
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    @Chickenmancer. It's perfectly well-defined! $0$ (and therefore $1$) don't have multiplicative inverses. But the result is a trivial field with only one element. And there's utterly nothing wrong with $1=2$. EVERYTHING is equal to $1$ which is equal to $0$. It's just very boring. – fleablood Mar 13 '19 at 17:45
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    @fleablood: $1$ certainly does have a multiplicative inverse: $1\times 1=1$. And $0$ has a multiplicative inverse too, but this is not required by the field axioms :-) – TonyK Mar 13 '19 at 17:46
  • @TonyK good point. I mean and should have said "don't need to have multiplicative inverses". My point being $\frac 10$ need not be defined (although it is). The conclusion that $1 = 2$ is not a contradiction however. – fleablood Mar 13 '19 at 19:01
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    " there is no way it could possibly be proved on the basis of the other properties listed" I think what spivak is saying is that we know that $1\ne 0$. But we must state that explicitly; it can not be proven simply by the field axioms (as Spivak lists them). A field (with one element) where $1 = 0$ is consistent with the axioms as Spivak lists them. – fleablood Mar 13 '19 at 19:04

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