# Cardinal Numbers: Simple Definition to Cantor’s Paradox

Statistics Definitions > Cardinal Numbers

A cardinal number, sometimes called a “counting number,” is used for counting, like when you count 1,2,3…
You use these numbers to answer the question “how many?”

Cardinals are always positive (or zero), as they are used to count. For example, you can have 5 loaves of bread, but having minus five loaves makes no sense (at least, in the real world).

The cardinals used in everyday language and those used in set theory are defined in different ways.For example, in set theory, cardinals can represent negative numbers. The cardinal number of this set {-5, -99, -100} is three. Infinity is also a cardinal: the cardinal number of this set {1,2,3,…} is infinity.

Trivia: In the English language, cardinals come before the noun. For example, you say “three brothers.” In American sign language (ASL), they come either before or after the noun. For example, you can say “I have brother 3” in ASL.

## Statistics and Cardinal Numbers

Many times, sets of cardinal numbers create statistics. When this happens, the cardinal numbers disappear. For example, according to the 2010 U.S. Census, the average number of people per household in the U.S. is 2.58. This number was arrived at by taking the cardinal number of people in each household and then finding the mean. Once you’ve taken that set of cardinals and found its mean (2.58), the statistic is no longer cardinal.

## Set Theory, The Largest Cardinal Number and Cantor’s Theorem

Set theory describes how many elements are in a set and tells us how many cardinal numbers exist. Cardinality in set theory forms a generalization of the natural numbers, which extends into transfinite numbers. Transfinite numbers are close to infinity, but are not exactly infinite. Infinity is itself a difficult concept to grasp on its own, because most things we can see, feel, or hear are finite. But just when you think you can wrap your head around the concept of infinity, it actually gets a lot more complicated than that; Cantor demonstrated that there are an different sizes of infinity and in fact there are an infinite amount of infinities. Cantor’s theorem sheds a little light on the idea.

Cantor’s theorem tells us that there is no largest cardinal number. The theorem also tells is that there are infinite amounts of infinite cardinal numbers. The theorem basically says a set exists that contains all cardinal numbers. This set also has a power set, which is a collection of subsets.

As a very simple example, let’s start with a small set of cardinal numbers {1,2,3}.
The power set of {1,2,3} includes the empty set { } and all of the possible combinations of sets (this is very similar to the idea of combinations in statistics):
P(S) = { {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }.
Now imagine a set containing all possible cardinals {1,2,3…∞} Even for infinite-sized sets, the power set is always larger. Mathematicians call this a paradox (a contradiction). However, it also gives rise to the idea of multiple types of infinities.

## A More Precise Definition

Cardinal numbers get their name from the literal definition, which means “chief” or “primary.” The word cardinal comes from the Latin cardin, which means “stem” or “hinge”, so these are the numbers that all others stem from, or hinge (depend) upon (Svarney & Svarney, 2012). These are the fundamental counting units, without which mathematics wouldn’t be possible. In this basic sense, cardinals are easy to use and understand. However, the precise mathematical definition is a lot more complicated and involves some pretty hefty mathematics, including set theory. Hamilton (1982) calls the concept of cardinality a “difficult notion” to grasp and many other authors agree.

Part of the reason for the difficulty in forming an easy definition is that if we say it’s what you get when you count objects (1,2,3 etc.), then the process of counting is itself ordinal. “1” is the first number, “2” is the second, and so on. In addition, the sum a + b could refer to two ordinal numbers, or two cardinal numbers, and they do not end up with the same result. Another example of cross-contamination between systems: May 10 in the Hindu-Arabic numbering system (the one in common use in the U.S.) could be read as either a cardinal (May 10) or ordinal (May 10th). The same is true for Roman Numerals, where II could be read as a cardinal two, or an ordinal second (as in Charles II, Charles the second).

If you’re confused by this, you aren’t alone. Historically, the exact definition was (an perhaps, still is), quite the convoluted topic.

## Early Definitions

Gottlob Frege (in 1884) and Bertrand Russell (1903) defined the cardinal numbers as the set of all sets equipollent to A (Moore, 1982, p.153). In English, that’s saying that the cardinal number of a particular set is the aggregate of all sets you can match with it. Or. to put it another way, it’s that unique aspect of a set you can match to another set. “Matching” implies a one-to-one correspondence. For example, let’s say you had fifty people at a Bingo game (so the set of all people equals 50). And let’s further suppose that those fifty people purchased 50 Bingo cards. As the number of people (50) exactly matches to the number of Bingo cards (50), we say there is one-to-one correspondence and the cardinality of the set is therefore 50.

Outside of mathematical philosophy, Frege and Russell’s definition didn’t stand the test of time. This may be because although the two men agreed on the wording of the definition, they did not agree on the philosophical meaning of the definition. Frege called numbers “self-subsistent objects” while Russell took it as “…enabling him to dispense with numbers as distinct from classes of equinumerous classes as unnecessary physical lumber” (Beaney, 2010). However, it was important as it set the stage for the idea that cardinals are members of a universal set made up of smaller sets of members.

## Cantor-Von Nuemann Definition

Another early definition was the Cantor-Von Nuemann definition, which is significantly more technical than the Frege-Russell definition. In brief, the theory states:

|A| is defined as the least (von Neumann) ordinal α such that A can be well-ordered with type α (Dasgupta, 2013).

To define arbitrary sets like the set of all reals (R), it requires the use of the Axiom of Choice, which has many forms. This (one of the simplest) is from Vanderbilt University’s Axiom of Choice:

Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S.

This takes us down the rabbit hole of set theory, which is beyond this rudimentary discussion of cardinal numbers. If you’re interested, I recommend Abhijat Dasgupta’s excellent book Set Theory: With an Introduction to Real Point Sets.

## Similarity to Other Numbers

Other types of numbers you’ll come across are:

• Ordinal numbers tell you where a number places relative to others (1st, 2nd, 3rd…) but — unlike cardinals — they don’t tell you anything about size (magnitude). For example, a runner in that comes in 10th place could have lost to 1st place by a fraction of a second. “10th” tells you nothing (other than they came in tenth place!).
• Natural numbers are whole, non-negative numbers (1,2,3…). Natural numbers can be ordinal or cardinal numbers.
• Integers are all the natural numbers and their opposites (i.e. it includes the negative numbers).
• Rational numbers are numbers written in a ratio, like 2:3. In other words, you can represent them with a fraction like ⅔. The whole number 10 is a rational number, because you can represent it with the fraction 10/1 (10:1 as a ratio), so are 3/11, 123/9 and 129/128. Pi (3.14…) is one example of a non-rational number as you can’t write it in fractional form.
• Real numbers are numbers you can find anywhere on a number line. It includes all of the above types of numbers.
• Complex numbers use the imaginary number “i” and involve square roots of negative numbers. The number line doesn’t contain these theoretical numbers.

References:
Beaney, M. (2010). The Analytic Turn: Analysis in Early Analytic Philosophy and Phenomenology. Routledge.
Dasgupta, A. (2013). Set Theory: With an Introduction to Real Point Sets. Springer Science & Business Media.
Hamilton, A. (1982). Numbers, Sets and Axioms: The Apparatus of Mathematics. Cambridge University Press.
Hosch, W. (2010). The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group.
Moore, G. (1982). Zermelo’s Axoim of Choice. Springer.
Russel, B. (1903). Principles of Mathematics.
Svarney and Svarney (2012). The Handy Math Answer Book. Visible Ink Press.