Order

Theorem 9 (Landau 10). For any given A and B, exactly one of the following is true: A = B, A > B, A < B.

This follows from Definition 2 and Axiom 1; it is also seen to be true from Theorem 8 and Definitions 5 and 6.

Theorem 10 (Landau 11 to 12). If A > B, then B < A; and, if A < B, then B > A.

This follows from the meaning of greater and lesser for numbers. But it is also clear from Definitions 5 and 6: A > B means A = B + D, and B < A means A = B + D and vice versa.

Theorem 11 (Landau 13 to 14). If A ≧ B, then B ≦ A; and, if A ≦ B , then B ≧ A.

This theorem follows from the meaning of greater, lesser, and equal for numbers. It also follows from Theorem 10.

Theorem 12 (Landau 15 to 17). If A > B and B > C, then A > C. (Transitivity of Ordering)

This follows from the meaning of greater and lesser for numbers. But it can be seen as follows:

Noting that this theorem also entails the reverse, for A < B, etc., we see that A = B + D and B = C + E, for some D and E, and A = (C + E) + D = C + (E + D), and we therefore have that A > C. Obviously, also, this entails the transitivity of A ≧ B and A ≦ B and since A ≧ B, etc. entails A > B, it also entails transitivity of ≧ and > across each other in the appropriate way. We prefer to preserve whenever possible the minimum relation given, e.g. if A ≧ B, B > C, then A > C, rather than A ≧ C, which can be substituted for A > C even if B ≠ C. Relations of greater and lesser can also have more than two terms, e.g. a > b > c ≧ d, etc.

Theorem 13 (Landau 19). If A > B, A = B, or A < B, then A + C > B + C, A + C = B + C, or A + C < B + C, respectively.

This follows from the definition of number and Common Notion 2. But it can also be seen as follows:

If A > B, then A = B + D, A + C = (B + D) + C = (D + B) + C = D + (B + C) = (B + C) + D and so A + C > B + C.

When A = B, this is simply a symbolic statement of Common Notion 2.

If A < B, then B = A + D, A + C = (A + D) + C = (D + A) + C = D + (A + C) = (A + C) + D, and therefore A + C < B + C—or, by 1., B > A, and the rest follows with B replacing A.

Theorem 14. If A ≠ B, then either A > B or A < B.

Proof: 1. Fix A as 1 and let B be any number, and therefore A < B, since by definition 1 and 2 the unit produces all numbers and so is less than any number.

2. Fix A and B and consider B’. If A ≠ B’, then A ≠ B + 1; since A ≠ B, either A = B + D, where D > 1, or not; if so, then by definition, A > B; and if there is no D such that A = B + D and A ≠ B, then A < B by Theorem 9.

Theorem 15 (Landau 20). If A + C > B + C or A + C = B + C or A + C < B + C, then respectively A > B, A = B, or A < B.

This follows from the definition of number and Common Notion 2. But it can also be seen from Theorem 13, since these three are mutually exclusive and exhaust all possibilities.

Theorem 16 (Landau 21). If A > B and C > D, then A + C > B + D.

Proof: By Theorem 13 we have A + C > B + C, and

B + C = C + B > D + B = B + D, and so, A + C > B + D.

Theorem 17 (Landau 22). If A ≧ B, C > D or A > B, C ≧ D, then A + C > B + D.

This follows from Theorem 13 if an equality sign holds, and from Theorem 16 if >.

Theorem 18 (Landau 23). If A ≧ B, C ≧ D, then, A + C ≧ B + D.

This obvious in the case of equalities, otherwise it follows from Theorem 17.

Theorem 19 (Landau 24). A ≧ 1.

This follows from the definition of number and the unit, together with our practice of letting variables stand for anything numeric, which are numbers and their principle. But there is a tidy little way to see it too: either A = 1 or A = D’ = D + 1 > 1.

Theorem 20 (Landau 25). If B > A, then B ≧ A + 1.

Proof: B = A + D, when D ≧ 1, and B ≧ A + 1.

Theorem 21 (Landau 26). If B < A + 1, then B ≦ A.

Proof: If this is not the case, then the only other option for B is B > A, and so B ≧ A + 1, which is clearly false.

Theorem 22 (Landau 27). If some number not greater than A has a given property, there is a least number that has it. (So numbers cannot be taken each less than the one before without end: Theorem 70.)

Proof: By Axiom 3. I. Let A be 1. The only number not greater than 1 is 1 (Theorem 19), and it is the least.

II. Let it hold for A, and let some number not greater than A’ have the property. Either some number not greater than A has it, and then there is a least by hypothesis. Or none does, and then A’ has it (Theorem 21) and is the least.