1 - Near-linear spaces
Published online by Cambridge University Press: 26 March 2010
Summary
I suppose you know that students of geometry and arithmetic and so forth begin by taking for granted odd and even, and the usual figures, and the three kinds of angles, and things akin to these, in every branch of study; they take them as granted and make them assumptions or postulates, and they think it unnecessary to give any further account of them to themselves or others, as being clear to everybody. Then, starting from these, go on through the rest by logical steps until they end at the object which they set out to consider.
Plato The Republic Book VISome basic concepts: consistency and dependence
The understanding throughout this book is that we work with a set P whose elements are called points, and a set L of certain subsets of P, whose elements are called lines. We remind the reader that by definition a set has distinct elements.
A space S = (P, L) is a system of points P and lines L such that certain conditions or axioms are satisfied. We can then consider two points of view: given a system of axioms about points and lines, can we find any spaces which satisfy it, or, given a familiar space (for example, real 3-space), what system or systems of axioms can be used to define it? We are only interested here in the former of these two questions.
As we shall be working a great deal with axiom systems, we discuss some of their properties in this section.
An axiom system is said to be consistent if it is possible to construct an example of a structure satisfying all the axioms.
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- Combinatorics of Finite Geometries , pp. 1 - 22Publisher: Cambridge University PressPrint publication year: 1997