** Cardinals and Ordinals **

The cardinal number of a set is an indicator of the size of the set. It depends only on the elements of the set. Sets with the same cardinal number — or cardinality — are said to be equinumerate or (with unfortunate terminology) to be the same size. For finite sets there are no problems. Two sets, each having the same number of elements, both have cardinality . But an infinite set has the definitive property that it can be put in one-to-one correspondence with a proper subset of itself.

An obvious example is the set of natural numbers , which can be matched element for element with the set of even natural numbers by the mapping .

According to the axiom of choice, every set can be *well-ordered*. A well-ordered set is an ordered set such that every subset has a least element. Clearly, is well ordered, whereas is not.

Georg Cantor introduced the *ordinal* numbers to describe the lengths of infinite sets (strictly, we should say *ordinality* or *order type*, but length is simpler). In a sense, the cardinal number measures the “size” of a set while the ordinal measures its length. For finite sets there is essentially no difference between these: we can shuffle a finite set without changing either its size or its length.

The cardinal number of a finite set with elements is , and the ordinal number is also . Colloquially, the size and length of the set are both . While finite sets of the same size also have the same length, this is no longer true for infinite sets.

The ordinal number is the first infinite ordinal. It is the ordinal number of the set of natural numbers (with the usual ordering). Suppose we shuffle by moving the first element to the end:

Something crucial has changed: whereas has no greratest element, the set does have a greatest element, . The ordinal number of this set is . But adding an element on the left does not change the ordinality: the set has ordinal number . If we insist on indicating the additional element, we can write , but we are forced to conclude that .

Now let us move all the even numbers so that they follow all the odd numbers:

The ordinal number of is . With obvious notation, also has ordinality .

** The Hierarchy of Ordinals **

Cantor’s system was hugely controversial, and for good reasons. The non-commutative nature of the system was a cause of difficulty. Ordinals can be added, multiplied, and exponentiated. However, none of these operations is commutative. We have but . One might be tempted to subtract from each side, but this would yield the nonsensical result .

The beautiful system of surreal numbers, invented by John Conway includes the ordinal numbers and the arithmetic operations are defined in a different way. That system forms a Field, respecting all the properties of commutativity, associativity, etc. (see reference below).

**Sources**

Darling, David and Agnijo Banerjee, 2022: *The Biggest Number in the World: A Journey to the Edge of Mathematics.* Oneworld Publications. ISBN: 9-780-8615-4305-2.

ThatsMaths: The Root of Infinity: It’s Surreal.

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