1. Cantor’s six-part treatise on set theory appears in the German journal Mathematische Annalen.

2. In mathematical terminology, a pairing of the objects in two sets is referred to as either a “one-to-one correspondence” or a “bijection.”

3. Cardinality refers to the number of elements in a set; conceptually, the set’s magnitude or size.

4. Using the most fundamental of all measures in Measure Theory—the counting measure—yields the same measurement for the set of positive integers as it does for the set of positive even integers. Hence, we may say, “a part can be as large as the whole.”

5. This follows from the Fundamental Counting Principle.

6. For the intrepid reader, I give an informal proof that the power set of {1, 2, 3, 4, . . .} is larger than {1, 2, 3, 4, . . .} in cardinality. Before doing so, it is customary (for the sake of notational simplicity) to identify the power set of {1, 2, 3, 4, . . .} with the collection of infinite strings—called sequences—of zeros and ones. We can represent any subset of {1, 2, 3, 4, . . .} by a sequence of zeros and ones. A “one” in the sequence denotes the presence of a number in the subset and a “zero” denotes its absence. For example, the subset {1, 2, 3} is represented by 1, 1, 1, 0, . . . (zeros thereafter); the subset {1, 3, 4} is represented by 1, 0, 1, 1, 0, . . . (zeros thereafter); and the subset {2, 4, 5, 7} is represented by 0, 1, 0, 1, 1, 0, 1, 0, . . . (zeros thereafter). The process also works in reverse: given any sequence of zeros and ones we can reproduce the subset of {1, 2, 3, 4, . . .} that corresponds to it. To prove that the power set of {1, 2, 3, 4, . . .} is larger than {1, 2, 3, 4, . . .} in cardinality, we need only show that every pairing between the positive integers and sequences of zeros and ones will leave some sequence without a partner. Given any pairing between the positive integers and sequences of zeros and ones, we can construct a sequence that has no partner. To illustrate this, consider a particular pairing between the positive integers and sequences of zeros and ones:

1	is paired with	1, 0, 1, 1, 0, 0, . . .
2	is paired with	0, 0, 1, 0, 1, 1, . . .
3	is paired with	1, 1, 0, 1, 1, 0, . . .
4	is paired with	0, 1, 0, 1, 1, 1, . . .
.	.	.
.	.	.
.	.	.

We can construct a sequence of zeros and ones that is different from the sequences listed above. For instance, consider the sequence 0, 1, 1, 0, . . . ; it differs from the first sequence in its first digit; it differs from the second sequence in its second digit; it differs from the third sequence in its third digit, and so on (see bolded digits in the diagram below). This means the sequence 0, 1, 1, 0, . . . is not the same as any of the sequences listed; hence, it is not paired with any positive integer.

1	is paired with	1, 0, 1, 1, 0, 0, . . .
2	is paired with	0, 0, 1, 0, 1, 1, . . .
3	is paired with	1, 1, 0, 1, 1, 0, . . .
4	is paired with	0, 1, 0, 1, 1, 1, . . .
.	.	.
.	.	.
.	.	.

No matter what pairing we are given between the positive integers and sequences of zeros and ones, we can always find sequences without partners. Therefore, the set of sequences of zeros and ones is larger in cardinality than the set of positive integers. Consequently, the power set of {1, 2, 3, 4, . . .} is larger than {1, 2, 3, 4, . . .} in cardinality.

7. In mathematical parlance, countably infinite refers to a set that has the same cardinality as the set of positive integers {1, 2, 3, 4, . . .}. Whereas a countable set can be either a finite set or a countably infinite set. For this reason, we do not shorten countably infinite to countable. There is no such ambiguity in the uncountable case.

8. The common belief among mathematicians that there is no infinite set whose cardinality is greater than that of the positive integers but less than that of the power set of the positive integers is formalized in The Continuum Hypothesis.

9. Assuming time is sufficiently divisible.

10. (For advanced mathematics readers only.) Let R represent the set of real numbers. To prove that a line and two-dimensional space have the same cardinality as sets of points, it suffices to show that there is a bijection from R onto the Cartesian product RxR. This can be accomplished by “unzipping” the real numbers. To illustrate this idea, consider the decimal expansion for a real number between 0 and 999: d₁ d₂ d₃ . d₄ d₅ d₆ d₇ d₈ d₉ d₁₀ d₁₁ . . . , where d_j is a digit between 0 and 9. Assign this real number to (d₂ . d₄ d₆ d₈ d₁₀ . . . , d₁ d₃ . d₅ d₇ d₉ d₁₁ . . .) in the Cartesian product. Extending this assignment to all of R, while handling non-unique expansions with care, gives a bijection from R onto RxR. Hence, a line and two-dimensional space have the same cardinality as sets of points.

11. Assuming matter is sufficiently divisible.

12. In 1884, Edwin A. Abbott wrote a delightful story about life in a two-dimensional world. His book Flatland is still in print, published by Dover Publications.

13. Orthogonal (perpendicular) directions.

Kent A. Bessey

BYU Studies Publications