# The Fascinating Realm of Infinities: A Journey through Cantor's Mind
Written on
Chapter 1: The Concept of Infinity
This narrative revolves around a brilliant individual whose obsession with a mathematical dilemma led him to madness and ultimately a solitary end in a mental institution. This piece also serves as an entry point to understanding the intriguing notion of mathematical infinity and the world of transfinite numbers.
To illustrate, consider the series of natural numbers: 1, 2, 3, 4, 5, …, represented as ℕ. This collection is evidently infinite. If we assume ℕ were finite, we could identify a maximum number and simply add one to it, creating a new natural number not included in ℕ. This contradiction confirms the infinitude of natural numbers. Yet, there are also infinitely many whole numbers. How can we compare the sizes of these infinite sets?
This inquiry was posed by the German mathematician Georg Cantor in the late 19th century. Born in 1845 in Saint Petersburg, Russia, Cantor moved to Germany at eleven. He excelled as a musician and scholar, ultimately receiving his doctorate in mathematics in 1867.
Cantor was unafraid to challenge conventional wisdom, exploring ideas considered almost heretical in his time. Notably, he devised a method for comparing the sizes (or cardinalities) of infinite sets.
To revisit the comparison of natural numbers ℕ with whole numbers ℤ, how can we assess the number of elements in these sets? The challenge is that counting elements is impractical with infinite sets.
Imagine two large piles of stones, and you need to determine which pile has more stones without counting them. One might suggest removing stones in pairs—one from each pile. The pile that depletes first has fewer stones. This elegant solution shows that we can ascertain which pile is larger without knowing the exact counts. If both piles run out simultaneously, they must contain an equal number of stones.
Cantor proposed a similar approach for infinite sets. If a one-to-one and onto mapping (termed a bijection) exists between two sets, their cardinalities are the same. A bijection is defined as a function f: A → B, where each element b in B corresponds to exactly one element a in A such that f(a) = b. This is often described as a one-to-one correspondence.
For example, bijective functions like f(x) = 2x or f(x) = x³ illustrate that the positive real numbers correspond to all real numbers. This implies that the set of positive real numbers is equivalent in size to the set of all real numbers.
It's noteworthy that there exists a bijection between ℤ and ℕ. For instance, the mapping 1 → 1, 2 → -1, 3 → 2, 4 → -2, and so on, demonstrates that these sets share the same cardinality. This may seem counterintuitive since ℕ is a subset of ℤ, yet they are indeed the same size.
If a bijection exists between natural numbers and a set A, we describe A as countably infinite. Conversely, if an infinite set is not countably infinite, it is termed uncountably infinite. Thus, the set of whole numbers is countably infinite.
Exploring Rational Numbers
How about the cardinality of rational numbers ℚ? One might assume there are more fractions than natural numbers, as all whole numbers are rational but not the other way around. Surprisingly, the answer is no; ℚ is also countably infinite. The challenge of finding a bijection f: ℕ → ℚ is left for the reader.
Cantor's Trials and Tribulations
Cantor's initial papers demonstrated that these sets have the same size. However, his groundbreaking work was met with skepticism from the mathematical community. Mathematician Leopold Kronecker labeled Cantor a "corruptor of youth" for promoting his ideas to younger mathematicians. This opposition contributed to Cantor's struggles to secure a professorship in Berlin, where Kronecker often influenced hiring decisions.
In May 1884, Cantor faced a severe depression, ready to abandon mathematics due to the criticism of his work. Fortunately, he recovered and delved into the study of yet another infinite set.
The Continuum: A New Challenge
Cantor turned his attention to the set of real numbers, which encompasses all previously discussed sets as subsets. He posed the critical question: "Is the set of real numbers countable?" In a paper from 1891, he demonstrated that the real numbers cannot be listed in a one-to-one correspondence with the natural numbers, confirming that the set of real numbers is uncountably infinite—a larger infinity than the natural numbers.
Cantor's proof is a testament to his brilliance. He hypothesized that if the real numbers were countable, one could list them using 0s and 1s as decimals. However, he illustrated that one could always construct a new number that does not appear on the list by altering the digits. This method, known as "Cantor's diagonal argument," reveals that one cannot enumerate all real numbers.
Discovering a Spectrum of Infinities
After this revelation, Cantor uncovered an entire realm of infinities, discovering even larger infinities beyond that of the real numbers. He established that for any set A, the powerset of A (the set of all subsets of A) has a greater cardinality than A itself. This finding proved the existence of infinitely many different infinities.
Cantor's work laid the groundwork for what is now known as cardinal arithmetic, demonstrating relationships between various infinities, including the intriguing equality of the powerset of natural numbers and that of the continuum.
The Continuum Hypothesis
Believing he had a divine mission to convey the significance of his discoveries, Cantor pondered whether there exists an infinity greater than the natural numbers yet smaller than that of the continuum (the real numbers). This inquiry became known as the continuum hypothesis, which posits that no cardinality exists between the naturals and the reals.
Into the Abyss
Cantor's obsession with the continuum hypothesis spiraled into a cycle of highs and lows. He frequently believed he had proven it true, only to later find errors in his reasoning. This turmoil culminated in a stay at a sanatorium in 1899, followed by a tragic loss when his youngest son, Rudolph, passed away in December of that year. This event deeply affected Cantor, leading to chronic depression.
Though he momentarily regained his strength, his fixation on the continuum hypothesis often drove him to madness, prompting thoughts of abandoning mathematics altogether. His struggles even led him to question his faith, grappling with the idea that if he were part of a divine plan, why was his path so difficult?
Cantor retired in 1913, living in poverty but achieving some recognition for his pioneering work. The esteemed mathematician David Hilbert famously remarked, "No one shall expel us from the paradise which Cantor has created for us." Unfortunately, the celebration of Cantor's 70th birthday was canceled due to the war. In June 1917, he entered a sanatorium for the last time, continuously writing to his wife, longing to return home.
Georg Cantor succumbed to a fatal heart attack on January 6, 1918, in the sanatorium where he spent his final year.
As a fitting tribute to the profound nature of his work, I conclude with a quote from the poet William Blake:
To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour.
Exploring Infinity Further
The first video titled "How many kinds of infinity are there?" delves into Cantor's revolutionary ideas on different types of infinity and how they challenge our understanding of mathematics.
The second video, "Infinity is bigger than you think - Numberphile," explores the vastness of infinity and how Cantor's work reshaped the mathematical landscape, revealing that our intuitions about size and quantity can be profoundly misleading.