Note: A much more thorough and precise discussion of the topics illustrated here is the article Set Theory in the Macropedia of the Encyclopedia Britannica (1992 edition).
How could one generalize the concept of a natural number beyond infinity? It turns out that there is a natural way that leads to surprising discoveries.
It's based on the concept of a set. According to George Cantor (1845-1918), the founder of set theory),
The individual objects of the set are its elements. A set may have no elements, in which case it is called the empty set and denoted by
There is only one empty set. Sets can be defined in words, or by listing the elements between curly braces separated by commas, or between curly braces containing some other defining symbols.
A set is finite if it's empty or it contains a finite number of elements. It is infinite otherwise.
A set S is a subset of a set T, denoted by
if every member of S is also a member of T . The empty set is a subset of every set, and every set is a subset of itself.
We will use the following sets based on numbers and prime numbers.
Obviously these sets are related. For example:
Two finite sets are equivalent if they contain the same number of elements.
Next we take a key step: to define equivalence in such a way that it also works for infinite sets. Think of two finite equivalent sets S and T as being ordered. Thus they each have a first, second, third, and so on element. Clearly we can pair the first element of S with the first element of T, and so on, until each element of S and T is a member of a unique pair. It is also clear that two finite sets are equivalent if we can so pair them. This idea can be generalized to infinite sets. Two sets S and T are equivalent, denoted by
if we can pair their members such that every element of S and T occurs in exactly one pair. You may want a more technical definition .
We say a set S is larger than a set T if T is equivalent to a subset of S, but S is not equivalent to any subset of T .
For finite sets S we denote the number of elements of S by
The power set of a set S is the set of all subsets of S, denoted by
.This definition can be clarified by some examples:
We can now understand the following statements which were first proved by Cantor. To see a proof click on the appropriate statement.
In other words, any finite set of N elements has 2 to the power N subsets.
In other words, there are as many rational numbers as there are natural numbers, or prime numbers, or even numbers, or odd numbers, or integers. That's amazing! All natural numbers are rational numbers, but intuitively most rational numbers are not natural numbers. So how can there be as many natural numbers as there are rational numbers? Put differently: How can any set be equivalent to a "much smaller" subset?
For any infinite set S we can consider the property that it has in common with all equivalent sets. That's called the cardinal of S. For finite sets, the cardinal is simply the number of elements. For infinite sets that property is harder to define. However, it seems clear what we mean. We can think of the cardinal as some measure of the size of the set.
The cardinal of some sets have been given names. The cardinal of the sets of natural numbers is denoted by
where the symbol on the right of the equations is pronounced aleph-null. Similarly, the cardinal of the set of real numbers is denoted by
Cardinals are also called transfinite numbers. Clearly one can obtain a hierarchy by repeatedly forming the powerset of a powerset as follows:
It's natural to ask if there is a set that's larger than the set of natural numbers, and smaller than the set of real numbers. The continuum hypothesis states that such is not the case. Whether this is true or false is not known, but it's unknown in a more subtle sense than that we just can't figure it out!
It turns out that a naive application of the concept of sets leads to contradictions. The simplest example goes as follows: Consider the set of all sets. Since it's a set it contains itself as an element. So it makes sense to define a set A which is the set of all sets that do not contain itself as an element. Now, if A is an element of A then, by the definition, A is not an element of A. On the other hand, if A is not an element of A, then, by the definition, A is an element of A. In either case, we have a contradiction.
To avoid contradictions one has to build a system of Axioms that are consistent and that allow all relevant statements to be derived. Now, reasonable looking systems of axioms can be built that contain the continuum hypothesis as an axiom, and other reasonable looking systems can be built that contain its opposite. So in a sense whether the continuum hypothesis is true or not depends on the taste of the mathematician. There are mathematicians who are happy with this state of affairs. On the other hand, looking at the problem naively, it seems clear that it is either true or false that there is a subset of the real numbers that's smaller than the set of real numbers and larger than the set of natural numbers. In other words, it seems that there should be metamathematical concept of sets that makes the continuum hypothesis either true or false.
The Encyclopedia Britannica article concludes on this note:
Although there is little supporting evidence, the optimists hope that the status of the continuum hypothesis will eventually be settled.
Fine print, your comments, more links, Peter Alfeld, PA1UM