The Number System (Dover Books on Mathematics) by H. A. Thurston

Author:H. A. Thurston [Thurston, H. A.]
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2012-10-22T22:00:00+00:00

Therefore (i) is true when x = 0. And

Therefore (ii) is true when x = 0. Therefore 0 ∈ M.

If z ∈ M, let

(θ(z, y) is defined, because z ∈ M.) Then

because z ∈ M. Therefore (i) is true when x = z′. And

Therefore (ii) is true when x = z′ and so z′ ∈ M.

Therefore, by 1c, M contains every whole number, and so (i) and (ii) are true for every x and y.

We may now talk about the operation for which (i) and (ii) are true; (v) and (vi) are also true for it.

Definition (of sum of whole numbers): The number θ(x, y) just defined is the sum of x and y, and will be written as x + y. (i), (ii), (v) , and (vi) become

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