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MATHEMATICS CONCEPTS

What is a Number?
The Fregé Concept

Fregé's concept [2] of number rests on the notion of equivalent sets.

Equivalent Sets
Consider two sets A and B. Let A = {Bob, Mary, Marg} and B = {Brian, chair, Mississipi river} . Consider a correspondence between A and B:
Bob --> chair
Mary --> Brian
Marg --> Mississipi river
In fact, we can write down five more such correspondences! Let us use the one above. Such a correspondence is called a one-to-one (often written 1-1) correspondence from A on to (often written "onto")B. A member of each set is accounted for exactly once in a 1-1 onto correspondence. A 1-1 onto correspondence from A to B has the property that you can reverse the arrows and get a 1-1 onto correspondence from B to A.

It is not always easy to show that a correspondence is onto, especially when dealing with infinite sets. But if we can show a 1-1 correspondence from set A to set B and another 1-1 correspondence from B to A, then there exists a correspondence from A to B that is 1-1 and onto; this is the Cantor-Bernstein Theorem.

Mathematicians call a 1-1 onto correpondence also a "bijection." They say A is "bijective" with B. This is really a fancy way of saying that there is a way of matching up the elements of A with those in B so that each object in A corresponds to exactly one object in B, and vice versa. Now, there are all kinds of sets that are bijective with A. So we might as well continue with A and B. The sets A and B share a quality. That quality is called the "cardinal" number. Cardinal means important. The cardinal number of our A or B is written as "3". We will call this important number just "number."

So what is a number? It is that property of sets which is common to all sets that are bijective with each other. Two sets that are bijective with each other are also called "equivalent" sets. So our A and B are equivalent sets. A number is the property common to equivalent sets. This is known as Hume's Principle after David Hume.

Zero
0 is the quality shared by all sets that contain nothing. Mathematicians call such sets, the empty sets. So 0 is not nothing; it is a numeral or let us be lazy like everybody else and call it a number! So if you are asked if zero is a number, you could say "well, it is." "Well" because it is a number in the sense of being a numeral. Well, it should be really called a numeral. But number will do.

Why use equivalent sets to define number?
First, it makes sense that we consider these objects called sets; for, afterall, a use of numbers is for counting the elements in a set. Second, bijection does make sense. Consider this. Suppose there was a shepherd in 5000 B.C. who had a large flock of sheep. How does he make sure that at the end of the day, all the sheep are back? Since, probably, counting to numbers as large as the cardinal number of the large herd did not exist at that time, he could still account for all the sheep as follows. He will collect a bunch of stones and, when the herd goes out in the morning, pass on a stone for each sheep to form a heap of stones - and throw away the stones left over. The heap is his SET of stones. When the herd returns in the evening, he can take out one stone from the heap for each sheep that comes in. If when the last stone is taken a sheep has indeed gone in to correspond to that stone, the SET of sheep is equivalent to the set of stones in the heap and all the sheep are indeed back.

A third reason for not abandoning the concept of equivalent sets to define number is that it enables us to also handle sets that are not finite. Georg Cantor (1845-1918) is the creator[1] of the theory of sets and the use of equivalence to define number. Cantor went farther with equivalence. He used it to introduce a notion called the "power set" of a set, and discovered infinitely many distinct and successively larger infinities!

A more precise definition of 1-1 correspondence and Number
Let us revisit the definition of a 1-1 correspondence from set A to set B. There is a problem with our earlier definition, namely, that the numeral 1 is used in it. Notice that we used the 1-1 correspondence to define Number. So, using the numeral 1 to lead to a definition of number is circular - or self-referential - and needs to be avoided.

Here then is a more precise definition. A correspondence from A to B is "one-one" if an element x in A corresponds to an element y in B and no other element x' in A corresponds to y, and x does not correspond to any element y' other than y. When only the first of these conditions holds, the correspondence is called "one-many"; when only the second holds, it is called "many-one." You might still object that we have used the word "one" which is representing the numeral 1. This is not a problem, for you could use another word, say, "banana' and the definition is not affected. The use of the word "one" only serves as a pnemonic; that is, the word "one" reminds us what we need to accomplish.

We can make the definition even more precise as follows: A correspondence from A to B is "one-one" if when elements x and x' in A correspond to an element y in B then x = x', and if x also corresponds to an element y' then y = y'.

Let us now revisit our definition of number. I stated: It is that property of sets which is common to equivalent sets. The problem with this definition is that we have not proved that equivalent sets have a common property and that there is no other common property. In the language of mathematics, we need to show that there exists a common property and that it is unique. A proof of this will take us too far into the field of mathematical logic. Therefore I will give you the precise definition of number that does not explicitly mention the common property. Here is the definition. A number is the set of all equivalent sets. While this definition can be shown to be precise, it lacks the feel one would get referring to the definition using the common property of equivalent sets - which was why we resorted to that definition earlier.

A fundamental question might occur to you: How do you add two numbers? The answer is this. Let A be the first number; that is, A is a set of equivalent sets. Let B be the second number. Then the sum of the two numbers is the union of the distinct sets A and B. Think about it! The multiplication of A and B is accomplished by taking the set A X B, which is called the Cartesian Product of And B; its elements being (a,b), where a is in A and b is in B. Subtraction and division are also accomplished, but laborious; we omit them here.

Cardinal Numbers and Ordinal Numbers
There are mainly two kinds of numbers: cardinals and ordinals. Cardinal numbers tell how many things there are in a set, as in "There are four people in my family." If you consider a set and a copy of it, you get two identical sets which are clearly equivalent. The number here is often referred to as the "cardinality" of the set; it is still the cardinal number of the set. The cardinality of a set is the size of the set; it is the number of things - or elements - in the set.

Ordinal numbers have to do with the order or priority of the elements in a set that has such order or priority for all its elements. So ordinal numbers give the positions of elements in an ordered set. For example, "I am fourth in the queue for lunch." The word "fourth" denotes an ordinal number.

On first meeting this classification, one would question the point of this. The reason that such questioning is natural is that the two classifications do not make any difference for the natural numbers we are used to. That is, in natural numbers, each ordinal is also a cardinal and vice versa under the natural ordering. It is in the realm of infinite numbers that you see the differing aspects of the two classifications play out.

A defect in Fregé's concept of Number
We have worked hard to make our treatment of the definition of number precise. Yet Fregé's approach suffers from the logical blemish of being self-referential. The self-reference is when we use the concept of equivalent sets, for equivalent sets use the concept of "one" which is a number. So we are using a number to explain what a number is! It is the reasoning error called Circular Reasoning or Petitio Principii. We would wonder then why we keep the Fregé concept. The answer is it is the best we have so far.

Arithmetic - The mathematics of counting
There is a third approach to the concept of number. That is to just say "let it be." In other words, leave it as an undefined or "primitive" concept. This is the approach taken in the method called the AXIOMATIC method. In this method, it is best to leave the basic objects undefined, or problems might arise later when proofs are sought. This is exemplified by Euclid. Having the undefined concepts also settles questions about the concept embodied in the undefined term.

Mathematically, this third approach is more satisfactory since it does not have any logical blemish. Of'course, it does not tell us what a number is.

Giuseppe Peano[4] started with the undefined concepts of "set," "natural number," "successor," and "belongs to", and proceeded to provide a system of axioms for arithmetic. His first five axioms deal with the natural numbers:

(1) 1 is a natural number.
(2) 1 is not a successor of any other natural number.
(3) Each natural number a has a successor.
(4) If the successors of a and b are equal then so are a and b.
(5) If a set S of natural numbers contains 1 and if when S contains any number a it also contains the successor of a, then S contains all the natural numbers.

This last axiom is the axiom of mathematical induction which is a method of proof used for proving statements directly or indirectly involving natural numbers. Peano wrote down a few more axioms to compare numbers and to add and multiply them, and proved the properties we take for granted of numbers - like 1 + 1 = 2. From these, others defined and established the properties of positive and negative integers and rational numbers in terms of pairs of natural numbers. Then the logical foundation of all FINITE numbers including irrational numbers - which are precisely the nonperiodic decimals - was attained.

Fregé's concept of the number helps us to visualize the abstract; Peano's axioms helped to provide the foundation for calculating with the finite ones of those numbers. Leopold Kronecker has said, "God made the integers, all else is the work of man."

HISTORICAL SOURCES
[1] Georg Cantor, Mathematische Annalen (1874)
[2] Gottlob Fregé, Die Grundlagen der Arithmetik (1884)
J.L. Austin (1953 English trans.) The Foundations of Arithmetic: A Logico-Mathematical Enquiry in to the Concept of Number
[3] David Hilbert, Grundlagen der Geometrie (Seventh Edition), 1930 - Foundations of Geometry
[4] Giuseppe Peano, Arithmetices Principia Nova Methodo Exposita (1889)

(G.R.T.)

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