# Calculus by Spivak Trichotomy Law

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

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

what if a was a complex number or a quaternion?

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
Staff Emeritus
Gold Member
2021 Award
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.

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
Staff Emeritus