The Nature of Infinity: From Ancient Greek Paradoxes to Cantor's Uncountable Sets and Hilbert's Hotel

This conversation with mathematician and philosopher Joel David Hamkins explores the profound and often counter-intuitive concept of infinity, tracing its understanding from ancient Greek thought to modern set theory. The discussion begins by highlighting Georg Cantor's revolutionary late 19th-century discovery that \"some infinities are bigger than others,\" a notion that triggered theological crises, mathematical civil wars, and personal struggles for Cantor himself. Early philosophical views, such as Aristotle's emphasis on potential infinity and Galileo's pioneering observations of equinumerosity, are presented as crucial precursors to Cantor's work, setting the stage for the eventual breakdown of traditional mathematical intuitions.\n\nA central theme is the tension between the Cantor-Hume principle, which defines equinumerosity by one-to-one correspondence, and Euclid's principle, which states that the whole is always greater than its part. Galileo's paradoxes, demonstrating that sets like natural numbers and perfect squares, or line segments of different lengths, can be put into one-to-one correspondence, illustrate this conflict. The concept of countable infinity is vividly explained through Hilbert's Hotel, a thought experiment showing how an infinite hotel, even when full, can always accommodate more guests—whether one, twenty, or even an infinite number (like Hilbert's bus or train)—by simply re-arranging existing occupants. This is achieved through clever mapping techniques, such as doubling room numbers or using prime factorization (e.g., 3^C * 5^S for train cars and seats), demonstrating that the union of countably many countable sets remains countable.\n\nWhile the podcast doesn't offer direct practical advice, it provides a deep conceptual framework for understanding abstract mathematical ideas. The "zigzag" path across an integer lattice offers a visual algorithm for comprehending how a countably infinite set of countably infinite sets can still be enumerated. The discussion also clarifies that rational numbers, despite their dense ordering, are still only countably infinite. A playful yet insightful proof by contradiction demonstrates that "every number is interesting," highlighting the elegance of mathematical reasoning.\n\nUltimately, the conversation culminates in Cantor's diagonal argument, a groundbreaking proof demonstrating the existence of uncountable infinities. This argument definitively shows that the set of real numbers is strictly larger than the set of natural numbers, thereby establishing that there is more than one size of infinity. The existence of transcendental numbers (like pi and e), which are not algebraic and constitute the vast majority of real numbers, further underscores this distinction. The episode illuminates how these foundational mathematical discoveries challenged long-held intuitions, reshaped our understanding of numbers and sets, and continue to provoke philosophical questions about the nature of reality and mathematical truth.

Some infinities are bigger than others.

Cantor was deeply religious himself. Second, there's a kind of mathematical civil war. The leading German mathematician Kronecker called Cantor a corrupter of youth and tried to block his career.

Many fascinating paradoxes emerged from this, like Russell's paradox, about the set of all sets that don't contain themselves, and those threatened to make all of mathematics inconsistent.

Almost all mathematicians were potentialists only and thought that it was incoherent to speak of an actual infinity at all.

Galileo was quite troubled by this observation because he took it to cause a kind of incoherence in the comparison of infinite quantities.

The tension between the Cantor-Hume principle and what could be called Euclid's principle, which is that the whole is always greater than the part, is a principle that Euclid appealed to in the Elements many times.

It's a property of infinity that sometimes when you add an element to a set, it doesn't get larger.

We've proved that if you have countably many countable sets, then the union of those sets, putting all those sets together into one giant set, is still countable.

It's not true, and that's the profound achievement that Cantor made is proving that the set of real numbers is not a countable infinity. It's a strictly larger infinity, and therefore there's more than one concept of infinity, more than one size of infinity.

Most real numbers are transcendental.

Suppose, toward contradiction, that there were some boring numbers... But that's a contradiction, because the smallest uninteresting number is a super interesting property to have.