Calculus by Spivak Trichotomy Law

[Bashyboy]

In the third edition, on page 9, I am reading about the Trichotomy law.

It says, for every number a, one and only one of the following properties holds

(i) a = 0
(ii) a is in the collection P,
(iii) -a is in the collection

Before stating this, though, the author said that P is the collection of all positive numbers (set of positive numbers). If P is the set of positive numbers, how can negative a be in P?

 

[dextercioby]

What if -a is positive ? As I can see, there's no restriction on a.

 

[Bashyboy]

Oh, so, for instance, if a = - 2, then -(-2) would be part of the set P.

 

[fakecop]

what if a was a complex number or a quaternion?

 

[Bashyboy]

This still seems odd to me. One of the following three properties will always be met, meaning that all numbers are in the set of positive numbers, P. Why is this description of the law of trichotomy so different from others I have seen?

 

[Office_Shredder]

Bashy, it says that for each number, only one of those is true. It does not say that for each number, if one is true then the number is in P.

For the number 2:
Either 2=0 (nope), 2 is in P (yup) or -2 is in P (nope).
For -2:
-2=0(nope), -2 is in P (nope) or -(-2) is in P (yup).

So we see that even though the property is satisfied for both -2 and 2, in both cases it's only saying that 2 is positive, and not -2.

 

[Bashyboy]

Ah, I see. Thank you very much. One more question, would you agree that the law of trichotomy is not generally stated in this way?

 

[Office_Shredder]

No, I have seen it stated that way (word for word basically) in several other places, though I don't know if it was explicitly called the law of trichotomy.

 

[mathwonk]

you are making the mistake of reading the sign - as "negative" rather than minus. a number is negative if it is less than zero. but minus a number is negative or positive if and only if the original number is respectively positive or negative. Thus: do not read "-" as "negative", but as 'minus". Unfortunately I will never live long enough to make this point.