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Cantor's Theorem

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The tacit and erroneous assumptions underlying the mathematical concept of set are so rooted in the text, teachings and the very collective souls of mathematicians that they may never be completely extracted. When introducing sets, many mathematical texts do so apologetically, as if the concept is so simple that it requires little discussion � a mere formality that must be endured on the way to more exciting topics. For example, at the end of the Preface to his introduction to set theory, P. R. Halmos says, "�general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it." Yet in Section 1, Halmos, makes the rather profound statement, "The mathematical concept of a set can be used as the foundation for all known mathematics" (Halmos 1). Halmos compares sets to the points and lines of Euclidean geometry and "�assumes that the reader has the ordinary, human, intuitive (and frequently erroneous) understanding of what sets are�", so he sees no need to define set (nor does he see fit to tell his reader exactly what the erroneous understanding is) (Halmos 1). Throughout this text, the author�s discomfort is evident: For example, at the end of Section 2, after stating the conclusion that "nothing contains everything" and "there is no universe", Halmos states, "To specify a set, it is not enough to pronounce some magic words � it is necessary also to have at hand a set to whose elements the magic words apply" (Halmos 7). General set theory is rather trivial, but in its current state it is fraught with nagging inconsistencies, such as the Russell Paradox: Let N denote the set of all sets which are not members of themselves. Let X be any set; then if X is a member of N, X is not a member of itself. Since X is any set, taking X to be N results in the contradiction that N is a member of N and not a member of N. (Eves 475) Modern approaches to set theory, such as the one outlined by Halmos, ignore the Russell Paradox (Halmos 7) and other inconsistencies and simply move on. Because, after all, mathematics has successfully proceeded on without worrying that its entire foundation may be unsound. The goal of this writing is to address the Russell Paradox.

The concept of set is so "na�ve" and intuitive because it is directly related to the fundamental trait of human cognitive functioning to group and count in order to make sense of the constant bombardment of sensory stimuli. We group these stimuli based on common properties, and their distinctness allows us to count and relate how many items with common properties we encountered. What mathematicians have done is given this grouping concept "life" and the name of "set" and shifted focus away from the items with common properties that make up this mental grouping. Mathematicians have abstracted the concept and gone off to play with it without stopping to consider, at the most basic level, what it should and should not be.

The first order of business is to establish what "exists" and what does not. Based on the general human cognitive ability to distinguish between and group "objects of thought", we can safely say that such objects of thought "exist". We generally agree upon the existence of objects since we can discuss the properties of objects among ourselves and conclude that we are discussing the same objects. And we can assume reasonably that there are many distinct objects. We can also reasonably assume that there are objects with common properties because the world would otherwise be in complete chaos. (Some might argue that the world is just that!) Philosophical questions aside, our first Axiom (1) is:

There exist distinct objects that can be grouped according to a shared property (or properties).

This is an axiom, so no attempt is made to define object, grouping, property nor how to determine properties. This is the na�ve, intuitive part that makes general set theory appear so simple. Obviously, the mathematical abstraction of this grouping process is the set. But before we jump to this abstraction, we must establish whether or not the grouping process "creates" additional objects of thought. In one sense, this is the purpose for grouping objects together. It allows us to discuss the objects as a whole and to make valuable generalizations. This is of course one of the hallmarks of mathematics. Mathematics makes this process precise and symbolic and is indeed what mathematics is built upon. But stepping back from the abstraction for a moment, let us look at the simple example of a group of teacups. Once "shown" an example of a teacup, we can then group each new teacup we encounter with all others we have or have encountered. But this group itself is obviously not a teacup. A group of trees can be called woods, but grouping the trees together as woods does not make for one object of thought. Mentioning woods evokes the image of a group of trees in close proximity. Though technically a tree is an organism, we basically think of a tree as a "primary" object of thought and of woods as a "secondary" object of thought. Woods would not exit without trees. Woods is a convenient cognitive grouping we have established. Likewise in mathematics, the abstraction of grouping into sets should not allow mathematics to create "primary" objects of thought. Stated more succinctly, we have our second and third axioms. Axiom (2):

For any mathematical discussion, there are agreed upon primary objects with all groupings of these objects constituting "secondary" objects of thought, or "sets".

And Axiom (3):

All non-empty sets can be "ungrouped" into primary objects.

Axiom 3 implies that no set is ever a primary object. Sets are so intuitive and basic to mathematics that they are always tacitly assumed to be primary objects of thought. They are the symbols that allow mathematicians to develop and convey their ideas, so it is easy to see how the mistake is made. A single alphanumeric character, such as "X", is generally used to denote a set since listing all members (the primary objects that make up the set) is either impractical or impossible. This practice leads to sets mistakenly being considered single, "whole" objects. But what Axiom 3 implies is that there is always a separation between a set and its members. A set is not a "��collection into a whole��" (Eves 476) as Georg Cantor defined it. Obviously, referring to a set with shorthand notation such as a letter is still necessary and correct. Axiom 3 states that sets "can be" ungrouped into primary objects, but it does not say anything about if or when this must occur. However, because it "can" occur, there is always a separation between any given set and its members.

So far, we have established primary objects and their groupings into the secondary objects of thought known as sets. But can sets be members of sets? Let us begin this discussion with our next Axiom (4):

Set membership is determined by properties of an object that do not depend upon any other object.

Axiom 4 implies that if a distinct object is a member of a given set based upon shared properties with another separate, distinct object, then those two objects must be one, inseparable object as a member of that set. Further, based upon Axiom 3, there must be a primary object that corresponds to these two distinct objects when they are considered together. We know that there is always a separation between a set and its members (and among the members themselves), so if a set X is to be a member of a set Y, then one of two conditions must be true. Either each member of X separately is a member of Y or there is a primary object that corresponds to all the members of X taken together which is a member of Y. In the former case, X is irrelevant when discussing Y, since the members of X, not X itself, meet the definition of Y. In the latter case, since X corresponds to a primary object, we can refer to the member of Y either way � as X or the primary object � but we are referring to one primary object. Thus, the only sets that can be members of sets are those that correspond to primary objects.

The remaining question is whether a set can be a member of itself. If X is a set, then in order for X to be a member of itself, it must be a primary object. If X is a member of itself, then X is both a primary object and a set consisting of two members. Since this is contradictory, there are no sets that are members of themselves.

Let us now consider the Russell Paradox. If N denotes the set of all sets that are not members of themselves, then exactly what is N? We know that if a set is a member of another set, it must correspond to a primary object, and we know that all sets are not members of themselves. Thus N is nothing more than the universe of primary objects of thought for a given mathematical discussion. In order for X to be a member of N, it cannot be any set � it must be a set corresponding to a primary object. And since we have made the assumption that there is more than one primary object, X cannot be N. Therefore, with the axioms and arguments set forth in this writing, the Russell Paradox is avoided.

� Copyright 1999, 2001 David B. Lowe

Last Modified: May 29, 2001

REFERENCES

Eves, Howard. An Introduction to the History of Mathematics. 5th Ed. Philadelphia: CBS College Publishing, 1983.

Halmos, Paul R. Naive Set Theory. New York: Springer-Verlag, 1974.