vol 4: Glossary
Set
Site map
Directory
Search this site
Home
1: About
2: Synopsis
3: Development
Next:
Previous: Glossary: Toc
... to restore theology to the mainstream of science
Set
[Hazewinkel, Set] [Borowski, Set]
Here and elsewhere in science, as expressed not least by Henri Poincare, that view is out of date that used to say "define your terms before you proceed." All the laws and theories of physics ... have this deep and subtle character, that they both define the concepts they use ... and make statements about these concepts. Contrariwise, the absence of some body of theory, law and principle deprives one of the means properly to define or even to use concepts. Any forward step in human knowledge is truly creative in this sense: that theory, concepts, law and meathod of measurement - forever inseparable - are born into the world in union. Misner, Thorne & Wheeler, 71
Without doubt, the birth of the mathematical concept 'set' marks a great divide in our understanding of our selves and our environment. The founder of set theory, Georg Cantor, defined a set as
any collection into a whole M of definite and separate objects m of our intuition or our thought. Cantor, 85.
Cantor's set theory replaced the old mathematical definition of point with the new definition of element. What was something with position but no magnitude became a definite and separate object. At a stroke he broadened the compass of mathematics from points and lines to anything thinkable.
This definition is psychological, since it talks about objects of our intuition or our thought. We may say that this is equivalent to everything that crosses our minds. Our minds are private, but we share our thoughts through language. In the public domain, every object of our intuition or our thought is represented by a string of actions (words, gestures) with a certain meaning.
Thus the set of all dogs, written D = {all x's with the property that x is a dog}, collects into a whole all the things we mean by the word dog. The natural way to define sets is through properties and meanings. But this can lead to trouble, as Russell's paradox illustrates. We can get into trouble when the properties defining a set are self referential, such as the set whose members are 'all sets that are not members of themselves.
The existence of paradoxes in set theory caused anguish to more people than Russel. The answer came slowly, in the guise of formalism, a program devised by Hilbert to strip mathematical symbols af meaning. The result is well described by the phrase 'what you see is what you get'. Formalism treated symbols as a builder treats bricks, arranging them in space without asking too much about what they mean. A brick wall is a brick wall, and that is that.
The foundation of formalism is axiomatisation. Sets and set theory have been axiomatised in a number of different ways since Cantor's time. Here we shall use the axiom system called ZF, for Zermelo and
Naive set theory
Naive set theory can lead to inconsistency. Naive set theory can be represented in a most general way by a system of two axioms:
- axiom of extensionality: if sets x and y contain the same elements that are equal.
- axiom of comprehension: given an arbitrary proposition P, there exists a set y containing only elements x for which P is true.
The axiom of comprehension is in fact a very large number of axioms (an axiom scheme), one for each proposition A.
This system is self contradictory ('Russel's Paradox') . Consider the proposition P = the set x is not a member of itself, and the set y of elements for which P is true. y is then the set of all sets that are not members of themselves. Is y a member of itself? If it is, then by the proposition P it is not. If y is not a member of itself, on the other hand, P is true, so it is a member of itself.
Axiomatic set theory
So, like many very powerful things, naive set theory can self destruct. Mathematicians saught to eliminate this problem by axiomatisation. Hilbert had demonstrated the power of the method by his axiomatisation of geometry. Hilbert. One, Zermelo-Franklin set theory, will serve whenever these pages try to be very formal.
Click on an "Amazon" link in the booklist at the foot of the page to buy the book, see more details or search for similar items
Related sites:
Concordat Watch
Revealing Vatican attempts to propagate its religion by international treaty
