By Grasberger
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This book meets that need. A naive approach, which depends upon observation of the concrete world for its development and meaning, is a natural way to introduce the subject, and this procedure is used in the following exposition. Little by little, certain properties and principles are developed, which in turn are used to prove further theorems concerning sets as collections of abstract entities. Thus one is led from concrete finite sets, to cardinal numbers, to infinite cardinals, and thence to ordinals via the use of ordinal-types.
In the fruit bowl there is a set of seven pieces of fruit. We can also say: the bowl contains a set of four apples and set of three pears. Notice that the elements belonging to a set are determined by the distinguishing characteristics of the set. For each thing considered, one must be able to say whether or not it is an element of the set. The set of all male members of family A hence contains the elements, father A, son Fred A, and son Peter A. The set of female members of family A contains only one element—mother A.
2 FINITE SETS I. Set, Element, Equality of Sets 1. What is a set? It is not that which we usually refer to in our everyday speech, when we speak of a large 3 of people, of ships, or of things. Rather: A set is a collection of definite distinct objects of our perception or of our thought, which are called elements of the set* 2. The following are examples of sets: (a) In Figure 1, the four persons sitting at the table form a set of four persons because they are four definite distinct objects of our perception.
Fractal attractors by Grasberger
by George
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