Types of infinity

I understand that there are different types of infinity: one can (even intuitively) understand that the infinity of the reals is different from the infinity of the natural numbers. Or that the infinity of the even numbers is the same as that of the natural numbers. How many types of infinity are there? Or is this infinite itself?

Solution 1:

A very nice introduction to the many different notions of infinity in mathematics is Rudy Rucker's book: Infinity and the Mind. Unlike many other popularizations, this is written by someone who did a Ph.D. on the topic. Moreover, Rucker has gone to great lengths to make the presentation faithful to the mathematics but still accessible to an educated layperson. Below is an excerpt on the Alephs.

Solution 2:

Not only infinite - it's "so big" that there is no infinite set so large as the collection of all types of infinity (in Set Theoretic terms, the collection of all types of infinity is a class, not a set).

You can easily see that there are infinite types of infinity via Cantor's theorem which shows that given a set A, its power set P(A) is strictly larger in terms of infinite size (the technical term is "cardinality").

http://en.wikipedia.org/wiki/Cantor%27s_theorem

Solution 3:

I should point out that your examples are all the same type of infinite number: they are all cardinal numbers, measuring an aspect of sets that extends the notion of "how many?" There is a related notion of ordinal numbers which extend the notion of counting.

But there are other types of infinite numbers. One family of related concepts includes the extended real numbers $+\infty$ and $-\infty$ that you might see in calculus. These two infinite numbers quantify something very different than what cardinal numbers quantify. The geometric idea they express is that $+\infty$ marks one "end" of the real number line, and $-\infty$ marks the other "end". There are a variety of other examples in this family, such as the projective real line, and the geometric concept of points at infinity (e.g. see the projective plane or read about elliptic curves).

Another type of infinite number is probably better expressed in relation to infinitesimal numbers: objects that behave as if they were smaller in magnitude than any non-zero real number, but yet were non-zero themselves. In some such number systems (e.g. the hyperreal numbers), one can take the reciprocal of a non-zero infinitesimal number, and the result is an infinite number.

All three of these types of infinite numbers behave very differently and are used for very different purposes.

Solution 4:

Well, in my opinion, there is no definite answer.

Some mathematicians contend that the concept of (actual) infinite itself is ill-defined, and you may have only a potential infinite, the one you have when you say that a straight line segment may be extended at will, but you do not have a whole straight line.

If you accept Cantor theory, the cardinal numbers (the ones which count how many objects there are in a set) are infinite; you start from ℵ₀ which corresponds to the integers, and go on. With the usual axioms, you cannot decide whether there are or not infinite cardinal numbers strictly greater than ℵ₀ and strictly less than the cardinality of the real numbers, but anyway the infinite numbers are infinite.

You may also consider ordinal numbers (the ones which consider the ordered elements of a set); they are many more than the cardinal numbers ;-)