From: veritasium
Georg Cantor was a pivotal German mathematician who profoundly reshaped our understanding of infinity. His work laid the groundwork for modern set theory, though it was met with significant controversy during his lifetime [00:02:31].
The Challenge of Choice in Mathematics
A fundamental challenge in mathematics is the ability to choose an element from a set [00:00:29]. While humans can “pluck” a random number, mathematical processes cannot truly pick things at random without a defined rule [00:00:40]. Formulas always yield the same result, which is why computers use algorithms based on factors like local time to simulate randomness [00:00:48].
The only way to select something mathematically is to follow a rule, such as “always choose the smallest thing” [00:01:07]. This works for sets like positive integers (smallest is one) or prime numbers (smallest is two) [00:01:14]. However, for real numbers—which include positive, negative, whole, fractional, and irrational numbers—choosing the smallest is impossible as they stretch to negative infinity [00:01:22]. Even attempting to specify a rule like “choose the smallest number after one” fails due to the infinite density of real numbers (e.g., 1.01, 1.0001, and so on) [00:01:38]. This difficulty in defining order within the real numbers highlighted a significant problem in mathematics [00:02:01].
Cantor’s Exploration of Infinite Sets
Cantor began his mission to address this issue around 1870, aiming to definitively order the real numbers [00:02:16]. At the age of 29, he published a paper that challenged centuries-old notions of infinity [00:02:37].
Prior to Cantor, Galileo’s 1638 book had heavily influenced the understanding of infinity [00:02:42]. Galileo posed the question of whether there are more natural numbers or more square numbers [00:02:48]. Although square numbers appear more sparse, Galileo realized a one-to-one mapping could be made between every natural number and its square, leading him to conclude that the two sets must be the same size [00:03:05]. From this, Galileo concluded that terms like “more than” or “less than” do not apply to infinity in the usual sense; it was seen as one “big concept of foreverness” [00:03:25].
Different Sizes of Infinity: Cantor’s Diagonalization Proof
Cantor, however, was not satisfied with this prevailing view [00:03:45]. In 1874, he questioned if there could be two infinite sets that couldn’t be perfectly mapped to each other, implying different “infinities” [00:03:50]. He set out to compare the natural numbers with the real numbers between zero and one [00:04:00].
Cantor began by assuming a perfect one-to-one mapping was possible [00:04:06]. He imagined an infinite list pairing natural numbers with real numbers between zero and one [00:04:11]. Since there’s no smallest real number, they could be listed in any order [00:04:19].
Then, using his Diagonalization Proof, Cantor constructed a new real number by taking the first digit of the first number in the list and adding one, the second digit of the second number and adding one, and so on [00:04:29]. (If the digit was 8 or 9, he subtracted one to avoid duplicates) [00:04:43]. This newly constructed number, by design, differed from every number on the infinite list in at least one decimal place (the digit on the diagonal) [00:05:00].
This proof demonstrated that the original assumption of a perfect mapping was false [00:05:07]. There must be more real numbers between zero and one than there are natural numbers [00:05:17]. Cantor had revealed that infinity does not come in just one size [00:05:26]. He categorized infinities:
- Countable Infinities: Sets like square numbers, integers, or rational numbers that can be perfectly paired with natural numbers, meaning they can be “counted” [00:05:32].
- Uncountable Infinities: Bigger infinities, like the set of all real numbers or complex numbers, which cannot be matched one-to-one with the natural numbers [00:05:47].
Cantor’s results were revolutionary and met with strong opposition, labeled as “a horror and a grave disease” by some mathematicians [00:06:00].
The Well-Ordering Theorem and Controversy
Undeterred by the criticism, Cantor pursued an even grander goal: to show that even uncountably infinite sets could be placed in a definitive order, which he called a “well-order” [00:06:17]. For a set to be well-ordered, it must meet two conditions [00:06:29]:
- The set must have a clear starting point [00:06:33].
- Every subset of that set must also have a clear starting point [00:06:36].
Natural numbers are well-ordered (starting point 1, any subset has a clear start) [00:06:44]. Cantor also successfully well-ordered integers (which stretch to infinity in both directions) by picking zero as a starting point and ordering them by absolute value (0, 1, -1, 2, -2, etc.), demonstrating that it’s possible to count “beyond infinity” in terms of ordering [00:07:01].
In his next book, Cantor published his “well-ordering theorem,” claiming that every set—even uncountably infinite ones like the real numbers—could be well-ordered [00:08:09]. The major problem was that he couldn’t actually prove this, as every method he tried failed [00:08:21]. Despite the lack of mathematical proof, Cantor was confident in his theorem, believing his work was divinely inspired [00:08:30].
The mathematical community reacted by attacking and ostracizing Cantor for a second time [00:09:08]. Leopold Kronecker, head of mathematics at the University of Berlin and Cantor’s former teacher, completely dismissed Cantor’s work, labeling him a “scientific charlatan” [00:09:15]. This professional and personal attack deeply affected Cantor, leading to his first of many nervous breakdowns and confinements to a sanitarium [00:09:42].
At the 1904 International Congress of Mathematicians, Julius König announced he had a proof that Cantor’s well-ordering theorem was wrong, publicly humiliating Cantor [00:10:18].
Zermelo’s Breakthrough: The Axiom of Choice
Among the attendees at the 1904 congress was Ernst Zermelo, a German mathematician interested in Cantor’s work [00:10:42]. Within 24 hours of König’s presentation, Zermelo found a contradiction in König’s proof [00:10:57]. Within a month, Zermelo published a flawless three-page article titled “Proof That Every Set Can Be Well-Ordered,” validating Cantor’s theorem [00:11:05].
Zermelo’s breakthrough identified an implicit assumption Cantor had been making: the ability to make an infinite number of choices at once from any set, including uncountable infinite sets like the real numbers [00:11:15]. This assumption, though intuitively used by mathematicians for decades, was not explicitly permitted in the formal rules (axioms) of mathematics [00:11:40].
Zermelo formalized this assumption into a new axiom: the Axiom of Choice [00:11:52]. The Axiom of Choice states that “if you have infinitely many sets, and each set is not empty, then there is a way to choose one element from each of the sets” [00:12:07]. For finite sets, this is obvious [00:12:16]. For infinite sets, it’s easy if a clear rule exists (e.g., “choose the smallest thing”) [00:12:22]. However, when there is no natural rule, especially for uncountable infinities, the Axiom of Choice allows for these choices to be made all at once, even if it doesn’t specify which element is chosen [00:12:31].
Zermelo used the Axiom of Choice to well-order the real numbers by choosing a number (X1) from the set of all real numbers, then choosing another (X2) from the remaining reals, and so on [00:12:58]. The choices are made from all possible subsets simultaneously [00:13:26]. To account for the uncountably infinite nature of real numbers, he introduced “omega numbers” (omega, omega plus one, etc.) to serve as labels beyond the natural numbers, allowing every real number to be placed in an ordered set [00:13:47]. This process proves that a well-ordering exists for the real numbers, even if the resulting order is unfamiliar (e.g., a billion could come before 0.2) [00:14:43]. Thus, Cantor’s well-ordering theorem and Zermelo’s Axiom of Choice were shown to be equivalent [00:15:11].
Consequences and Acceptance
Zermelo’s discovery revealed that many mathematicians, including those who had criticized Cantor, had unknowingly relied on the Axiom of Choice in their own work [00:17:02].
However, playing with the Axiom of Choice led to “disturbing results” that challenged common intuition:
- Vitali Set (1905): Giuseppe Vitali used the Axiom of Choice to construct a set of numbers that shattered the idea of length. This “Vitali set” is “unmeasurable,” meaning it has no consistent definition of size, length, area, or probability, contradicting the fundamental mathematical idea that everything can be quantified [00:17:55].
- Banach-Tarski Paradox (1924): Stefan Banach and Alfred Tarski demonstrated that a single solid ball could be split into just five pieces, which then, by careful rotation and movement (without stretching or distorting), could be reassembled into two identical balls, each the same size as the original [00:23:19]. This process could be repeated to create an infinite number of balls from one [00:23:41]. This paradox arises because the pieces involved are non-measurable, similar to the Vitali set [00:28:06].
These counterintuitive results plunged mathematics into a crisis for over 30 years [00:29:20]. The question became whether the Axiom of Choice was a fundamental axiom or something that could be proven [00:29:27].
The resolution came with two groundbreaking proofs:
- Kurt Godel (1938): Proved that in a world where all other accepted axioms of set theory hold true, the Axiom of Choice also holds true [00:30:32].
- Paul Cohen (1963): Proved that there is also a world where all other axioms of set theory hold true, but the Axiom of Choice does not [00:29:50].
This established that the Axiom of Choice can neither be proven nor disproven from the other axioms of set theory [00:30:38]. It is analogous to the parallel postulate in geometry: choosing whether it holds or not defines the specific “universe” of mathematics one operates in (e.g., spherical, flat, or hyperbolic geometry) [00:29:57].
After Godel and Cohen’s work, debates about the Axiom of Choice largely subsided [00:30:58]. Today, despite its counterintuitive consequences, the Axiom of Choice is almost universally accepted [00:31:14]. It is incredibly useful, allowing mathematicians to replace lengthy proofs with more concise arguments, extending finite proofs to infinite cases in a single step [00:31:24]. Many theorems cannot be proven in their general case without it [00:31:45]. While some mathematicians still prefer proofs without it for the additional information they provide, the Axiom of Choice is considered essential for modern mathematical progress [00:32:05]. The question is not whether it is “right,” but whether it is “right for what you want to do” [00:32:36].