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Home , Ad infinitum, Cardinality, Continuum (set theory), Equinumerosity, Georg Cantor, Infinite set

2.5. INFINITE SETS

Now that we have covered the basics of elementary theory in the previous sections, we are ready to turn to infinite sets and some more advanced concepts in this . Shortly after laid out the core principles of his new theory of sets in the late 19th century, his work led him to a trove of controversial and groundbreaking results related to the of infinite sets. We will explore some of these extraordinary findings, including Cantor’s eponymous on power sets and his famous diagonal , both of which imply that infinite sets come in different “sizes.” We also present one of the grandest problems in all of – the Hypothesis, which posits that the of the continuum (i.e. the set of all points on a line) is equal to that of the of the set of natural . Lastly, we conclude this section with a foray into transfinite , an of the usual arithmetic with finite numbers that includes operations with so-called aleph numbers – the cardinal numbers of infinite sets.

If all of this sounds rather outlandish at the moment, don’t be surprised. The properties of infinite sets can be highly counter-intuitive and you may likely be in total disbelief after encountering some of Cantor’s for the first time. Cantor himself said it best: after deducing that there are just as many points on the unit (0,1) as there are in n-dimensional space1, he wrote to his friend and colleague : “I see it, but I don’t believe it!”

The Tricky Nature of

Throughout the ages, human beings have always wondered about infinity and the notion of uncountability. Gazing at a star-dotted night sky and imagining the different worlds beyond our cosmic neighborhood, or thinking about all the irreducible bits of matter that make up our , is an easy way to attest of our intrinsically limited nature as humans. These quantities – though unimaginably large to any one of us – are nevertheless finite and have been estimated today to within a few orders of magnitude (there are about 1024 stars and somewhere between 1078 and 1082 atoms in the observable

1 n-dimensional Euclidean is an extension (and generalization) of the usual two- and three-dimensional Cartesian spaces where points have n coordinates. universe). Real difficulties arise when trying to understand quantities in the infinite realm. Suppose you conceive of becoming immortal. Then how many days will you have to live through? Indeed, how could we possibly count forever and hope to study numbers that we can’t even enumerate? Not surprisingly, it is precisely at this epistemological edge that mankind so often veered into convenient concepts of theology that obscured the difficult – yet fascinating – questions underlying the true nature of infinity.

The notion of infinity isn’t just limited to numbers endlessly. It may actually be hard-wired in our brain with our ability to understand language. Think of the following strings of words: “A turtle is an animal,” “An animal is a turtle,” “Turtle animal is a,” “Animal a is turtle.” Clearly the first two sentences are grammatical while the last two aren’t (those are just word salads). So how do we manage to distinguish actual sentences from meaningless ones if there are, potentially, an infinite of sentences (both grammatical and ungrammatical) that can be constructed with words? The influential American linguist Noam Chomsky (born 1928) argued that we could do so because we are innately equipped to understand the rules of language – a theory he called Universal Grammar. He pointed out that “a small set of rules operating on a large but of words generates an infinite number of sentences.” This particular view of language is still debated among linguists and philosophers. If true, though, it would confirm that we are innately equipped as human beings to grasp sets that are infinite.

To get a sense of how tricky it is to think about infinity, let us revisit one of the greatest problems in Ancient Greek philosophy – The Dichotomy . Zeno of Elea (5th century B.C.E.) put forth this famous paradox (along with other ones) to address head-on the problem of dealing with infinite quantities. He argues in the paradox that motion is an illusion since a person who wishes to travel a fixed distance must first travel half that distance, then half of the remaining half of that distance, then half the remaining half of the remaining half of that distance, and so on. So how could this person possibly travel an infinite number of distances in a finite amount of time? The conclusion that motion is just an illusion evidently runs counter to reality. Surprisingly, though, no one for more than 2,000 years could explain the flaw in Zeno’s argument! Ultimately, this deeply challenging problem was resolved with tools of calculus using concepts that were only validated in the early 19th century. Suppose the person travels a distance of one mile. Then he or she would have to travel half a mile first, then a quarter of a mile, and then an eighth of a mile, and so on. To confirm that the person actually travels the full mile (thus resolving the paradox) amounts to showing that the infinite sum of all these half distances would equal 1, or

1 1 1 1 + + + + ⋯ = 1. 2 4 8 16

A good calculus student can verify this statement rather quickly. The left-hand side is what call an infinite geometric series. Moreover, this series has a ratio of ½ since every in the series is generated by multiplying the previous one by ½. Because this ratio is less than 1, the series is known to converge – meaning that it eventually reaches a finite number. Applying the formula for convergent geometric series, which is easily established, yields the number 1 on the right-hand side. Here is another much less technical, yet quite convincing argument of the same result: suppose you had a blank page with a surface area equal to one square foot. Now color half the page, and then a quarter of the page, and then an eighth of the page, and then a sixteenth of the page, and so on. Figure 11 below gives an idea on how to complete this coloring task. Imagine repeating this process of coloring half of each remaining, but diminishing, blank portion of the page (i.e. without end). Eventually the entire page would be colored completely.

Figure 11: A Visual Solution to Zeno’s Dichotomy Paradox The famous Greek philosopher (4th century B.C.E.) tried to solve Zeno’s problem by differentiating between an actual and a potential infinity. His work, while influential in its , did not provide the mathematical tools required for a final resolution. Eudoxus and Archimedes (respectively, 4th and 3rd century B.C.E.), the two greatest mathematicians of antiquity, would have probably come closest to solving the paradox. Both used a method called exhaustion in their geometric constructions that explained how processes, such as the one with the colored page described above, could produce finite numbers in an infinite number of steps. Archimedes famously used this method in approximating the value of 휋, the constant used in measuring the area of a circle from its radius (퐴 = 휋푟2), and also in relating the area of a parabolic segment to an inscribed triangular area (this is known as the quadrature of the parabola). Their approaches would, in fact, predate ideas underlying the modern theory of integral calculus developed in the 18th century.

The Elephant in the Room

Zeno’s of motion posed fascinating questions that, while childishly simple to comprehend, were surprisingly difficult to explain. A common feature to all of these paradoxes is that they revolve around so-called processes (another famous one is Achilles and the Tortoise, which pits the great warrior Achilles in a losing race against the slow tortoise). In each problem, there are infinitely many quantities that keep on diminishing in size until they become infinitesimally small. With the modern tools of integral calculus, these problems can be quickly resolved. However, there are still many fundamental questions about infinity that do not involve infinitesimal processes and, consequently, cannot be handled with the machinery of modern calculus. These problems require, instead, a proper understanding of the relative “sizes” of infinity – i.e. the cardinalities of infinite sets. Suppose you knew the of ℕ, the set consisting of all the natural numbers: 1, 2 3, 4, ... Is this cardinal number the same as the one corresponding to all the fractions that can be placed between 0 and 1? Does this answer change if we extend the set to include all real numbers between 0 and 1? Should we distinguish, say, between the cardinality of the set of all and the cardinality of the set of all integers that are divisible by 6? These are all tricky questions that can only be answered within a precise mathematical framework for infinite sets. Despite the tremendous advances made by 19th century mathematicians in virtually all where infinite sets are to be found (, , , analysis, ), no rigorous methods had yet been developed to study these kinds of cardinalities by the 1870’s. The truth is that this state of affairs did not directly impact the daily work of mathematicians, so most were content with ignoring the elephant in the room. Eventually Cantor put forth his revolutionary theory of infinite sets and – with much audacity – was able to answer some fundamental questions related to the nature of infinity, including the ones posed earlier. In the process, he transformed mathematics forever.

Galileo’s Paradox

The great Italian and astronomer (1564 – 1642) was one of the first to try to solve the riddle of infinity. In particular, he was puzzled by a seeming that is now known as Galileo’s paradox. Consider the set of natural numbers ℕ and the set of perfect squares 푆. These two infinite sets can be described using the roster method as follows:

ℕ = {1, 2, 3, 4, 5, … , 푛, … } 푆 = {1, 4, 9, 16, 25, … , 푛2, … }

In 1630, Galileo came to the following conclusions:

1. The set of perfect squares 푆 is clearly a proper of ℕ since every perfect square is a , but there are infinitely many naturals that are not perfect squares (such as 2, 5, 8, 20, etc.) Thus, the cardinality of 푆 must be less than the cardinality of ℕ:

|푆| < |ℕ|

2. Since every perfect square corresponds to a unique natural number (its square root) and every natural number has a unique square, the numbers in ℕ and 푆 can be matched one by one as follows:

1 ⟷ 1 2 ⟷ 4 3 ⟷ 9 ⋮ ⋮ 푛 ⟷ 푛2 ⋮ ⋮

As a result, both sets have the same cardinality:

|푆| < |ℕ| Clearly, both statements are contradictory since the cardinality of 푆 cannot be both less than and equal to the cardinality of ℕ. However, Galileo could not find a fault with either of these two conclusions. So how did he reconcile these statements and solve the paradox? He finally settled by concluding that the symbols related to the well-ordering of numbers (>, <, =) could only be applied to finite numbers (i.e. the cardinalities of finite sets).

Galileo’s line of inquiry stopped short of explaining why the cardinalities of infinite sets seemed to violate the usual rules of well-ordering. Cantor would be the first to explain that cardinalities of infinite sets do not, in fact, violate these rules. The problem in Galileo’s argument lies with his first conclusion. Though it is clear that 푆 ⊂ ℕ, it does not follow from this premise that |푆| < |ℕ|. His second conclusion, however, was correct – it is the one-to-one match between the natural numbers and the perfect squares that implies the of both sets. Unbeknownst to Galileo, he had found the key that Cantor would later use to unlock the remarkable properties of infinite sets.

Infinite Sets

Exactly what makes a set infinite? The easy answer, of course, is the obvious one: an is not finite. Number sets such as ℕ or ℤ are infinite because they consist of all natural numbers or all integers, respectively. In other words, these sets cannot be described by the roster method without the use of ellipses (…) Cantor realized that this was a weak definition for infinite sets; it did not shed any light on their actual infinite nature. Soon Cantor came to one of his first great insights: infinite sets are those sets that remain infinite after any finite number of elements is removed from them. To make this clearer, we have to first define the concept of a 1-1 correspondence between sets to have a for comparing cardinalities of infinite sets.

DEFINITION: A 1-1 correspondence between two sets 퐴 and 퐵 is a rule that pairs each of 퐴 with exactly one element of 퐵 and vice-versa. The elements in both sets are thus ‘paired up,” so that each element in either set is matched with a single corresponding element in the other set. If there is a 1-1 correspondence between the sets 퐴 and 퐵, then both sets are equinumerous and their cardinal numbers are, as a result, equal: |퐴| = |퐵|.

Here is a classic analogy for understanding 1-1 correspondences. Suppose you walk into an empty theater after a sold-out performance. If you assume no one was standing or sitting on someone’s lap, then how could you deduce how many people attended the performance? A simple way to answer this is to count the number of seats in the theater. Better yet, just look at the number on the “Maximum Occupancy” sign and there is your answer! This would clearly work based on the principle that each seat 푥 in the theater would have been paired with a single attendee holding the ticket numbered 푥. Another example is matching the first five natural numbers (1, 2, 3, 4, 5) with each of our fingers (pinkie, , middle, index, thumb). In fact, the process we know as counting is nothing more than placing whichever objects we are counting in a 1-1 correspondence with a subset of ℕ. For instance, if you found 27 DVDs in your movie collection at home, then you surely arrived at that number by pairing each DVD with one of the natural numbers 1 through 27. So you deduced that you had 27 DVDs in your collection because this set of DVDs can be placed in a 1-1 correspondence with the set {1, 2, 3, …, 26, 27}.

So it is clear how to determine cardinalities for a finite set 퐴: just place the elements of 퐴 in a 1-1 correspondence with the set {1, 2, 3, …, 푛} and the cardinality of 퐴 is then given by |퐴| = 푛. Suppose you take any proper subset of 퐴. This set now has a cardinality that must be less than 푛. But what happens if we consider infinite sets instead? Going back to Galileo’s paradox, we see that 푆 is a proper subset of ℕ that can be placed in a 1-1 correspondence with ℕ. What Cantor realized is that the nature of infinite sets is tied to this very property. In other words, infinite sets are those sets that can be placed in a 1-1 correspondence with a proper subset of itself. What Galileo had effectively done was proving that ℕ is, well, infinite! We are now ready to introduce Cantor’s strong definition of an infinite set.

DEFINITION: A set 퐴 is said to be infinite if it has a proper subset 퐵 (i.e. 퐵 ⊂ 퐴) that can be placed in a 1-1 correspondence with itself. In other words, 퐴 and 퐵 are equinumerous, or |퐴| = |퐵|.

While it may seem obvious that certain sets are infinite (like ℕ or ℤ or ℝ) without referring to the definition above, it is now clear what truly determines the infinite nature of a set.

Denumerable Sets

At the beginning of this section we asked the following question: ”Is the cardinality of the set of all integers different from the set of all the integers that are divisible by 6.” It turns out that the answer to this question is “no”. To see why, think of any 푥. This number can be paired with the integer 6푥, which clearly belongs to the set of integers divisible by 6. Conversely, each integer 6푥 can be divided by 6 to be paired with the integer 푥. By our previous definition, we conclude that both sets are equinumerous and, therefore, have the same cardinality. It turns out that both sets can also be placed in a 1- 1 correspondence with the positive integers. We shall prove this result shortly. In such cases we call these sets denumerable, or countably infinite.

DEFINITION: An infinite set 퐴 is denumerable, or countably infinite, if it can be placed in a 1-1 correspondence with ℕ. The cardinality of such sets is given by

|퐴| = |ℕ| = ℵ0. 2 While Cantor denoted the cardinality of denumerable sets by the ℵ0 , he understood it to be something altogether different than a finite number. Later we will discuss briefly the arithmetic with such numbers, which Cantor labeled as transfinite.

Much of Cantorian is centered on one simple question: “Are all number sets denumerable?” Apparently Cantor was initially convinced for a few years that the answer to this question was “yes” until he actually found a way to construct a non- denumerable set of numbers via his famous diagonal argument. The process of verifying the denumerability of number sets resulted in some of Cantor’s most famous theorems. We now present these results below and outline the ideas behind his proofs.

THEOREM: The set of whole numbers 푊 is denumerable.

Proof. Pair each whole number 푤 = 0,1,2,3, … with the corresponding natural number 푛 = 푤 + 1 = 1,2,3,4, … Enacting this simple shift clearly produces a 1-1 correspondence between the sets ℕ and 푊. Therefore, The set of whole numbers is denumerable.

THEOREM: The set of integers ℤ is denumerable.

Proof. For this proof we use the roster method to write ℤ, the set of integers, alternating between positive and negative integers with increasing magnitude as follows:

ℤ = {0,1, −1,2, −2,3, −3, … , 푛, −푛, 푛 + 1, −(푛 + 1), … }

The 1-1 correspondence between ℕ and ℤ is then given as follows:

- Pair the integer 0 with the natural number 1. - Pair each positive integer 푛 with the natural number 2푛. - Pair each negative integer −푛 with the natural number 2푛 + 1.

It is easy to verify that this assignment pairs every natural number with a single corresponding integer and vice-versa. For example, the integer 27 would be paired with the natural number 54 and the natural number 2017 would be paired with the integer - 1008. Therefore, the set of integers is denumerable.

THEOREM: The set of rational numbers ℚ is denumerable.

Proof. To produce a 1-1 correspondence between the rational numbers (i.e. all fractions) and the natural numbers is a bit trickier. However, Cantor came up with a

2 The symbol ℵ0, which reads “aleph-null,” uses the first letter in the Hebrew alphabet. 푝 clever way to visualize this proof. First, start by placing all fractions of the form , where 푞 푝, 푞 are integers, as points (푝, 푞) on the 푥푦 plane. Since 푞 cannot be zero (otherwise the is undefined), omit all points (푝, 0) on the 푥-axis whose 푦-coordinate is zero. Now, imagine placing an infinite outward square spiral starting at the origin that rotates counter-clockwise through all grid points (푝, 푞) as shown in figure 1.

Figure 12: Square Spiral Going Through Points Representing Rational Numbers

As the spiral goes through all the grid points, pair each point that does not lie on the 푥-axis with an increasing natural number: 1, 2, 3, etc. If a point represents a fraction that has already been enumerated (such as (−1, −1) for (1,1))), skip it. Following this assignment, we get the 1-1 correspondence between the natural and rational numbers shown in the table below.

Natural number 1 2 3 4 5 6 7 8 … Grid point (푝, 푞) (1,1) (0,1) (−1,1) (2, −1) (2,1) (1,2) (−1,2) (3, −2) … 푝 Rational number 1 0 -1 -2 2 1 1 3 … 푞 − − 2 2 2

The set of rational numbers is, therefore, denumerable. A more that makes the 1-1 correspondence above explicit is omitted here.

So we have now seen that |ℕ| = |푊| = |ℤ| = |ℚ| = ℵ0. In other words, the sets of natural numbers, whole numbers, integers, and rational numbers are all countably infinite. In fact, Cantor showed that many other sets that would seem to be much “bigger” than Q are also denumerable. Most of these sets involve technical objects, so we omit them here. One notable example is the set of discrete points (i.e. points whose coordinates are integers) in n-dimensional . In 1924, the great German mathematician (1862 – 1943) introduced in a lecture his paradox of the Grand Hotel, a marvelous thought experiment that explained some of these technical results. Imagine a hotel that has a countably infinite number of rooms, all of which are occupied. Hilbert asked: can the hotel still accommodate a coach coming in with a countably infinite number of guests? In fact, it can! The manager would simply need to ask each guest in Room 푛 (where 푛 = 1, 2, 3, 4, 5, …) to move to Room 2푛, thus freeing up all the odd-numbered rooms for the new guests. So the guest in Room 1 would move to Room 2 freeing up Room1, the guest in Room 2 would move to Room 4 freeing up Room 2, and so forth. Next, Hilbert asked whether the hotel could accommodate a countably infinite number of coaches, each filled with a countably infinite number of guests. Incredibly, this can also be done! One solution asks current occupants of Room 푛 to move to Room 2푛, and 푛 푛 assigns the new guests in coaches 푐1, 푐2, 푐3, … to, respectively, Rooms 3 , Rooms 5 , Rooms 7푛, ... given by powers of new prime numbers. This method works since all these rooms with prime power numbers are unique (the Fundamental Theorem of Arithmetic guarantees a unique prime for all natural numbers greater than 2).

Following Cantor’s findings and the remarkable properties of Hilbert’s Grand Hotel, does it follow then that all infinite number sets are denumerable? Is ℵ0 destined to be the only “size” for infinity and close this discussion? As we mentioned earlier, this is not to be the case. In 1874, Cantor shocked everyone by proving that the power set of an infinite set 퐴 must have a larger cardinality than 퐴. This is knows as Cantor’s Theorem.

Cantor’s Paradise

CANTORS THEOREM: Let 퐴 be a set (finite or infinite). Then the cardinality of the power set of 퐴 is strictly greater than the cardinality of 퐴:

|푃퐴| > |퐴|.

The Diagonal Argument

THEOREM: The set of real numbers is non-denumerable.

The

Transfinite Arithmetic