Introduction to the Theory of Sets by Joseph Breuer

Author:Joseph Breuer
Language: eng
Format: epub
Publisher: Dover Publications

4

ORDERED SETS

XI. Ordered Sets and Order-Types

1. In grammar one is taught the difference between numbers that tell order and those that tell the size of groups (ordinal and cardinal numbers). As we learned to count, not only did we conceive of three as a plurality of three things, but we also became aware of what was meant by speaking of the third of these things (or the second, or the first).

In general, we were unable to think of the quantity three except as arranged in some order. Often, in counting, we pointed out a first, a second, and a third thing, probably with the use of our fingers. Our senses and our thinking almost always made us conceive sets of things in a definite spatial or temporal order.

The definition of the concept “set” as “a bringing together of definite distinct objects of our perception or our thought, to a whole,” in the absence of some order of the elements, places considerable demands upon our powers of abstraction. Nevertheless, the absence of order and the absence of any reference to the nature of the elements, furnished us in the case of infinite sets with the concept of transfinite cardinal number. We came to understand such a number as represented by any one of the infinite number of equivalent sets having this number. Thus

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