Mathematical Logic by Stephen Cole Kleene

Author:Stephen Cole Kleene
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 1967-06-26T16:00:00+00:00

* § 39.Some other formal systems. In this section, we give further examples [2]-[50] of formal systems (where [1] is the system N of §38).

We now describe [2] a system G, which formalizes the elementary theory of an unspecified “group” (explanation follows).

The formal symbols shall be

Variables are to be constructed as in § 38 for N. (The only difference in the formation rules between N .and G is that + ,·, ′ , 0 are replaced by · -1 , 1 .)

DEFINITION OF “TERM”. 1. 1 is a term. 2. The variables . . . are terms. 3-4. If r is a term, so are (r)·(s) and (r)-1. 5. The only terms are those given by 1-4.

Given this definition of “term”, the definition of “formula” from “term” reads as in § 38.

Parentheses are omitted under the same conventions as there. Furthermore (as we could also have done in § 38), we shall abbreviate r·s (i.e. (r).(s)) as “rs” (omitting the dot).

As postulates, we take those of the predicate calculus, with the present notions of “term” and “formula” (§§ 21,28); just as for N, the r for the ∀-schema and the ∃-schema can be any term free for x in A(x). In addition, there shall be the following six particular axioms:

El.. . .

Gl. (Associative law.)

G2.(Right identity.)

G3.(Right inverse of .)

We do not intend a single interpretation of this formal system G, as we did of N. The system G may be interpreted by any “group” G, as we shall explain and illustrate in a moment. For any choice of a “group” G, the terms are interpreted as naming members (specified or unspecified) of G (or more precisely of G0, below).

Axioms E1-E3 give what we need to postulate of the properties of equality (identity).158

The axioms G1-G3 for groups will be familiar to many students. A group G is briefly any “system” of objects which “satisfies” these three axioms (with the variables in the generality interpretation §§ 20,38, and with = expressing “identity” or “equality”§ 29).

That is, a group G consists of a nonempty set G0, a 2-place function a.b with arguments and values in G0, a 1-place function a-1 with arguments and values in G0, and a member of G0 (or 0-place function) 1, such that the closures of G1-G3 are all t in our model theory of the predicate calculus with equality § 29 when G0 is the domain and a·b, a-1, 1 are the values of · , -1, l. 159

To give only a few particular interpretations, G can be (1) the positive rational numbers with ·, -1, 1 having their usual meanings. (The student should have no trouble verifying that G1-G3 are satisfied by each of our interpretations.) Alternatively, G can be (2) the rational numbers except 0, (3) the positive real numbers, (4) the real numbers except 0, or (5) the complex numbers except 0, with ·, -1, 1 in their usual meanings. Still again, G can be (6) all the integers, (7) all the rational numbers, (8) all

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