A general theory of linear sets

Abstract

Section I of the following paper, though using the postulational method, is motivated by the consideration of classes of vectors, on a finite range PI (1, 2, ...* n), whose elements belong to a general division or, as we shall say, number system. Section II deals only with vectors oni a finite range. Section I is also of use as giviing a general basis preliminiary to tlle more intensive study of (a) classes of vectors on a general range, (b) number systems over a division numbersystem; that is, to the initiation of a theory analogous to that of an algebra over a field, where the field is replaced by an associative division number system. Notation. Throughout the paper certain logical notationst will be used as follows: logical idenitity t logical diversity definitional identity definitional identity between statements implies is equivalent to .3. such that 31 there exists is unique, used before the element which is unique: thus, I a means a is unique. and U or not . : .: :: etc. punctuation signs; the principal implication of a sentence has its signi accompanied by the largest number of periods, thus a :): b .). c is a statement that a implies that (b implies c) whereas a.). b :): c states that the implication a implies b, implies the fact c. We may also use punctuation to show continued implication, thus a.). b .). c means a.). b and b .). c.