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This is from Lang's "A First Couse in Calculus".
In chapter 1 he makes two statements regarding positivity:
POS 1. If ##a##, ##b## are positive, so is the product ##ab## and the sum ##a+b##.
POS 2. If ##a## is a number, then either ##a## is positive, or ##a=0##, or ##-a## is positive, and these possibilities are mutually exclusive.
("POS" meaning positive or postulate I guess.)
His proof that 1 is positive (using only the above statements) goes like this:
"By POS 2, we know that either 1 or -1 is positive. If 1 is not positive, -1 is positive. By POS 1, it must then follow that (-1)(-1) is positive. But this product is equal to 1. Consequently, it must be 1 which is positive, and not -1."
It seems like this proof hinges on that fact that (-1)(-1) is positive. However, since both (1)(1) and (-1)(-1) are positive, from this alone we really can't conclude that 1 or -1 are negative. Right? To me it'd make more sense to say that (-1)+(-1) is not positive, and by POS 1 -1 cannot be positive. By the same logic we can then claim that 1 is the positive one.
I know this is probably an elementary mistake on my part, but I'm lost.
In chapter 1 he makes two statements regarding positivity:
POS 1. If ##a##, ##b## are positive, so is the product ##ab## and the sum ##a+b##.
POS 2. If ##a## is a number, then either ##a## is positive, or ##a=0##, or ##-a## is positive, and these possibilities are mutually exclusive.
("POS" meaning positive or postulate I guess.)
His proof that 1 is positive (using only the above statements) goes like this:
"By POS 2, we know that either 1 or -1 is positive. If 1 is not positive, -1 is positive. By POS 1, it must then follow that (-1)(-1) is positive. But this product is equal to 1. Consequently, it must be 1 which is positive, and not -1."
It seems like this proof hinges on that fact that (-1)(-1) is positive. However, since both (1)(1) and (-1)(-1) are positive, from this alone we really can't conclude that 1 or -1 are negative. Right? To me it'd make more sense to say that (-1)+(-1) is not positive, and by POS 1 -1 cannot be positive. By the same logic we can then claim that 1 is the positive one.
I know this is probably an elementary mistake on my part, but I'm lost.
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