History of Set

	Revision 19 . . (edit) September 30, 2001 9:46 pm by Zundark [fix link]
	Revision 18 . . (edit) August 23, 2001 4:59 am by Zundark [fix link]
	Revision 16 . . August 10, 2001 12:02 am by Zundark [imported material from Set Theory]

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Difference (from prior major revision) (minor diff, author diff)

Changed: 1c1

A set is a collection of objects such that two sets are equal if, and only if, they contain the same objects. For a discussion of the properties and axioms concerning the construction of sets, see Basic Set Theory and Set Theory. Here we give only a brief overview of the concept.

A set is a collection of objects such that two sets are equal if, and only if, they contain the same objects. For a discussion of the properties and axioms concerning the construction of sets, see Basic Set Theory and Set theory. Here we give only a brief overview of the concept.

Changed: 8,9c8,10

The notation is not self-explanatory. If every x in some set A is contained in some set B, then A is said to be a subset of B. Every set has as subsets itself, called the improper subset, and the empty set {}. The union of a collection of sets S = {S1, S2, S3, ...} is the set of all elements contained in at least one of the sets S1, S2, S3, ...}

As you see, it is possible to describe one and the same set in different ways: either by listing all its elements (best for small finite sets) or by giving a defining property of all its elements. If A and B are two sets and every x in A is also contained in B, then A is said to be a subset of B. Every set has as subsets itself, called the improper subset, and the empty set {}. The union of a collection of sets S = {S1, S2, S3, ...} is the set of all elements contained in at least one of the sets S1, S2, S3, ...}

Changed: 15c16

respectively. If you don't mind jumping ahead a bit, the subsets of a given set form a boolean algebra under these operations. The set of all subsets of X is called its powerset and is denoted 2X or P(X).

respectively. If you don't mind jumping ahead a bit, the subsets of a given set form a boolean algebra under these operations. The set of all subsets of X is called its power set and is denoted 2X or P(X).

Added: 25a27

Changed: 27,28c29,30

Care must be taken with verbal descriptions of sets. One can describe in words a set whose existence is paradoxical. If one assumes such a set exists, an apparent paradox or antinomy? may occur. For example: Suppose a barber in a town shaves only people who do not shave themselves,. Now, ask yourself, "Who shaves the barber?"

Care must be taken with verbal descriptions of sets. One can describe in words a set whose existence is paradoxical. If one assumes such a set exists, an apparent paradox or antinomy? may occur. Axiomatic set theory was created to avoid these problems.

Changed: 30c32,33

The previous example is a case of Russell's Paradox.

For example: Suppose we call a set "well-behaved" if it doesn't contain itself as an element. Now consider the set S of all well-behaved sets. Is S well-behaved? There is no consistent answer; this is Russell's Paradox. In axiomatic set theory, no set can contain itself as an element.