The concept of infinity has been fraught with paradox since antiquity. For this reason, Aristotle sought to banish it from his physics, claiming that there were no actual infinities in nature—only potential infinities. Over a millennium later, medieval scholars offered the following example when asked why infinity was forbidden.
Imagine two concentric circles. Each circle contains infinitely many points along its circumference, but since the outer circle has a greater circumference, it has more points than the inner circle. Now take any point A on the outer circle, and draw a line from A to the circle’s center. This line must intersect some point B on the circumference of the inner circle. Hence, for every point A on the outer circle, there is a corresponding point B on the inner circle, and vice versa. Therefore, both circles must have the same number of points, despite the fact that the outer circle appears to have more points than the inner circle.
Many thinkers tried to unravel this apparent paradox, including Galileo Galilei, but it wasn’t until the late 1800s that Georg Cantor brought it to a successful resolution. Cantor’s idea was simple: infinite sets of objects are so far removed from our everyday experience that they may follow rules that seem counterintuitive. For Cantor, it simply must be accepted that the inner and outer circles have the same number of points—our intuition that the outer circle has more points, according to Cantor, is just wrong.
Instead, he focused on the fact that there exists a one-to-one correspondence between the points on the outer circle and the points on the inner circle. Cantor’s insight was to use this idea as the definition for determining whether or not two sets of objects are the same size. With this definition, we no longer need to use our intuition to test whether or not sets of points have different sizes; we only have to check whether or not there exists a one-to-one correspondence between the given pair of sets. If there is no one-to-one correspondence then, according to Cantor, one of the two sets must be larger than the other.
Cantor’s definition tamed infinity. It provided a rigorous method by which infinite sets could be compared. He used this idea to prove that there are just as many integers as even integers, since the function f(x)=2x provides the required one-to-one correspondence. He showed that there are just as many lattice points on the plane as there are integers. And Cantor even proved that there are as many rational numbers as there are integers.
But not all infinite sets have the same size as the set of integers. Cantor was able to show, using his now-famous diagonal argument, that there are more real numbers than integers. This led him to introduce a distinction between countable and uncountable infinities. Sets that have the same size as the set of integers are said to be countable, while sets that are larger than the set of integers are uncountable. He also introduced the notion of a transfinite cardinal number to quantify the different magnitudes of infinite sets. The smallest transfinite cardinal is ℵ0 (pronounced “aleph-zero”). It represents the size of the set of integers (that is, a countable infinity). The next transfinite cardinal, ℵ1, represents the size of the next largest infinite set. Of course, ℵ1 is the first uncountable cardinal number. Continuing in this fashion, Cantor introduced ℵ2 to represent the size of next largest infinite set, and so on with ℵ3 and ℵ4 and ℵ5….
Using one-to-one correspondences, laws of arithmetic for transfinite cardinal numbers can be derived. Many of these laws have been implemented in Wolfram|Alpha. For example, ℵ1 + ℵ2= ℵ2.
In fact, multiplication and addition of transfinite cardinals is trivial—the result is just the larger of the two numbers.
Finite cardinal numbers (that is, non-negative integers) can also appear in formulas with transfinite cardinals.
And transfinite arithmetic isn’t just limited to addition and multiplication. Wolfram|Alpha knows rules for manipulating exponents and factorials, too.
One of the first questions that Cantor asked was the following: “Since there are more real numbers than integers, and since ℵ1 is the size of the next largest set after the set of integers, is the set of real numbers of size ℵ1?” It’s easy to prove that the set of real numbers is of size 2ℵ0, so this question is equivalent to asking, “Does 2ℵ0=ℵ1?”
Cantor hypothesized that the answer was “Yes,” and his conjecture has come to be known as the continuum hypothesis. (The word “continuum” is an archaic name for the set of real numbers.) It certainly is a plausible hypothesis, since an answer of “No” would imply that there exists a set whose size is intermediate between the set of integers and the set of reals, and no such set has ever been found.
In many cases, if the continuum hypothesis can be used to simplify a result in transfinite arithmetic, then Wolfram|Alpha will display a brief note to that effect and describe the simplified result.
Note that the cardinal number (pronounced “beth-one”) is an abbreviation for 2ℵ0. And more generally, for any non-negative integer n,
is a tower of exponents containing n many 2s.
Cantor spent much effort trying to prove his continuum hypothesis, but without success. In fact, in an address to the International Congress of Mathematicians in 1900, David Hilbert listed the continuum hypothesis as one of the greatest mathematical challenges for the twentieth century. Pursuing that challenge, Kurt Gödel showed that the known axioms of mathematics are insufficient to disprove the hypothesis. This seemed tantalizingly close to a proof of the continuum hypothesis, but then in 1963, Paul Cohen showed that the known axioms are also insufficient to prove the continuum hypothesis. In other words, the known facts of mathematics are simply not sufficient to answer Cantor’s question.
During the late 1800s, it may have seemed that Cantor’s ideas would allow us to perform transfinite cardinal arithmetic just as easily as we perform ordinary finite arithmetic. But in retrospect, we can see that Cantor’s ideas only allowed us to perform cardinal arithmetic in certain specialized cases. Many simple formulas, such as 2ℵ0 cannot be evaluated without significant new insights into the nature of infinity.
Cohen, P. J. Set Theory and the Continuum Hypothesis, W. A. Benjamin, Inc., 1966.
Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press, 1979.
Enderton, H. B. Elements of Set Theory, Academic Press, 1977.
Rucker, R. Infinity and the Mind: The Science and Philosophy of the Infinite, Birkhäuser, 1982.