2.5. INFINITE SETS Now that we have covered the basics of elementary set theory in the previous sections, we are ready to turn to infinite sets and some more advanced concepts in this area. Shortly after Georg Cantor 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 cardinalities of infinite sets. We will explore some of these extraordinary findings, including Cantor’s eponymous theorem on power sets and his famous diagonal argument, both of which imply that infinite sets come in different “sizes.” We also present one of the grandest problems in all of mathematics – the Continuum Hypothesis, which posits that the cardinality of the continuum (i.e. the set of all points on a line) is equal to that of the power set of the set of natural numbers. Lastly, we conclude this section with a foray into transfinite arithmetic, an extension 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 theorems for the first time. Cantor himself said it best: after deducing that there are just as many points on the unit interval (0,1) as there are in n-dimensional space1, he wrote to his friend and colleague Richard Dedekind: “I see it, but I don’t believe it!” The Tricky Nature of Infinity 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 universe, 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 space 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 counting 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 number 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 finite set 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 Paradox. 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 mathematicians call an infinite geometric series. Moreover, this series has a ratio of ½ since every term 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 square 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 ad infinitum (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 Aristotle (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 metaphysics, 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 paradoxes 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 infinitesimal 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 cardinal number 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 integers 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 areas where infinite sets are to be found (number theory, algebra, geometry, analysis, logic), 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.