Table of Contents
For centuries, the concept of infinity was viewed as a philosophical abstraction or a theological absolute rather than a rigorous mathematical object. This changed radically in the late 19th century with Georg Cantor, whose discoveries did not just expand mathematics—they effectively broke it, triggering a crisis that forced humanity to rebuild the foundations of logic and truth. In a wide-ranging conversation, mathematician and philosopher Joel David Hamkins explores the landscape that emerged from this rubble: a world of multiple infinities, unprovable truths, and a multiverse of set-theoretic realities.
The following exploration dives into the technical and philosophical implications of modern set theory, challenging our intuition about numbers, computation, and the very nature of existence.
Key Takeaways
- Some infinities are strictly larger than others. While integers and fractions are "countable" infinities, real numbers represent a strictly larger, "uncountable" magnitude, proven by Cantor’s diagonalization argument.
- Mathematics cannot prove its own consistency. Gödel’s Incompleteness Theorems destroyed the hope of a purely finitary, complete mathematical system, proving that true statements exist which cannot be proven within a given system.
- The "Multiverse View" challenges the idea of one mathematical truth. The independence of the Continuum Hypothesis suggests we should not look for one "true" universe of sets, but rather accept a pluralist multiverse where axioms can be toggled like light switches.
- Computation has absolute limits. The Halting Problem proves that no algorithm can determine the outcome of all programs, linking computer science fundamentally to the logical paradoxes of set theory.
- Mathematical objects may be "more real" than physical ones. Hamkins argues that we have a clearer definition and understanding of abstract objects (like the number four) than we do of physical matter, which becomes increasingly mysterious the deeper physics investigates.
The Hierarchy of Infinities: From Hotels to Real Numbers
The journey into modern set theory begins with a counterintuitive premise: infinity is not a single size. To explain this, mathematicians often utilize "Hilbert’s Hotel," a thought experiment involving a hotel with infinite rooms (0, 1, 2...) that is completely full.
The Counterintuitive Nature of Countable Infinity
In the finite world, if a hotel is full, no new guests can check in. In the infinite world, logical paradoxes arise that violate Euclid’s principle that "the whole is always greater than the part."
- The Single Guest: If one guest arrives at a full infinite hotel, the manager moves guest $N$ to room $N+1$, freeing up room 0. The set did not grow in size, yet it accommodated more.
- The Infinite Bus: If an infinite bus arrives, the manager moves guest $N$ to room $2N$ (occupying all even rooms), leaving all odd rooms open for the new infinite arrivals.
- The Infinite Train: Even if a train with infinite cars, each containing infinite passengers, arrives, they can all fit. By mapping passengers to prime number powers (e.g., $3^C \times 5^S$), every passenger gets a unique number.
These examples illustrate countable infinity. Whether it is natural numbers, integers, or rational numbers (fractions), they all share the same "size" (cardinality) because they can be put into a one-to-one correspondence with each other.
The Uncountable Abyss
The stability of mathematics was shaken when Georg Cantor proved that not all infinities fit into Hilbert’s Hotel. Specifically, the real numbers (which include irrationals like $\pi$ and $\sqrt{2}$) form an uncountable infinity.
Cantor demonstrated this via the diagonalization argument. If you tried to list every real number between 0 and 1, you could construct a new number by ensuring its first digit differs from the first number’s first digit, its second digit from the second number’s second digit, and so on. This new number is mathematically guaranteed to be missing from your "complete" list. This proof established that the infinity of the continuum is strictly larger than the infinity of counting numbers.
Foundational Crises: Paradoxes and the Birth of ZFC
The discovery of multiple infinities led to what Hamkins describes as a "mathematical civil war." Early attempts to formalize these concepts, such as Gottlob Frege’s logicism, were dismantled by simple logical flaws, most notably Russell’s Paradox.
The Barber and the Sets
Bertrand Russell identified a fatal flaw in the idea that for any property, there exists a set of things with that property. He proposed: Consider the set of all sets that do not contain themselves.
- If this set contains itself, it violates its own definition.
- If it does not contain itself, it must be included in the set, which again violates the definition.
This is structurally identical to the "Barber Paradox" (a barber shaves everyone who does not shave themselves; who shaves the barber?). This contradiction devastated Frege, who received Russell's letter just as his magnum opus was going to print.
Rebuilding with ZFC
To salvage mathematics from these contradictions, Zermelo and Fraenkel developed the axiomatic system known as ZFC (Zermelo-Fraenkel with Choice). Rather than assuming any collection can be a set, ZFC establishes strict rules for set formation.
"In its foundational role, set theory provides a way to think of a collection of things as one thing. That's the central idea of set theory."
A crucial component of this system is the Axiom of Choice. It asserts that if you have an infinite collection of bins, you can select one item from each bin, even if you cannot specify a rule for how to choose them. While seemingly obvious, this axiom leads to non-intuitive results (like the Banach-Tarski paradox) and was historically controversial because it asserts the existence of mathematical objects that cannot be explicitly constructed.
The Limits of Knowledge: Gödel and the Halting Problem
In the early 20th century, David Hilbert proposed a program to secure the foundations of mathematics. He wanted to prove that mathematics was complete (all truths are provable) and consistent (no contradictions exist), using purely finitary reasoning.
The Incompleteness Theorems
Kurt Gödel dismantled Hilbert's dream with his Incompleteness Theorems. He proved two devastating facts about any axiomatic system strong enough to do arithmetic:
- Incompleteness: There will always be statements that are true but unprovable within the system.
- No Self-Validation: No consistent system can prove its own consistency.
This means we can never write down a list of axioms that answers every mathematical question. There will always be "independent" statements—facts that are mathematically undecidable within the current framework.
The Halting Problem
Alan Turing extended this breakdown of certainty into the realm of computation. The Halting Problem asks if there is an algorithm that can look at any computer program and determine if it will eventually stop or run forever.
Using a diagonalization argument similar to Cantor’s and Russell’s, Turing proved this is impossible. If such a "tester" program existed, one could feed it a paradoxical program designed to do the opposite of whatever the tester predicts. This proves that mathematical truth transcends algorithmic verification.
The Multiverse View: Independence as Discovery
One of the most famous independent problems is the Continuum Hypothesis (CH). Cantor spent his life trying to prove there is no infinity strictly between the countable integers and the uncountable reals. Decades later, it was proven that CH is independent of ZFC. You cannot prove it true, and you cannot prove it false.
Forcing and New Universes
To prove this independence, mathematician Paul Cohen invented a technique called Forcing. This method allows mathematicians to take a universe of sets and "expand" it, adding new sets to create a different mathematical reality.
- In one universe, the Continuum Hypothesis is true.
- In a Forcing extension of that universe, the Continuum Hypothesis is false.
Pluralism vs. The Universe View
This leads to Hamkins’ philosophical stance: the Multiverse View. Traditional "Universe" theorists argue there is one true mathematical reality, and if CH is undecidable, our axioms are just too weak. Hamkins argues for pluralism.
"When you ask a question that turns out to be independent, then you asked exactly the right question... It's carving nature at its joints."
Under this view, discovering that a question is independent isn't a failure; it is a discovery that we are standing at a fork in the road. Both paths exist, and both lead to valid, distinct mathematical universes. We do not need to choose one "true" set theory any more than we need to choose one "true" geometry between Euclidean and non-Euclidean.
The Texture of Abstract Reality
Hamkins suggests that mathematical objects—though abstract—possess a robustness that physical objects lack. While our understanding of matter shifts from atoms to quarks to wave functions, the number four remains structurally precise.
Surreal Numbers
One of the most beautiful structures in this abstract realm is the system of Surreal Numbers, discovered by John Conway. Generated by a simple recursive rule of "left sets" and "right sets," this system contains:
- All real numbers (integers, fractions, irrationals).
- Infinite ordinal numbers (like $\omega$).
- Infinitesimal numbers (numbers smaller than any positive fraction but greater than zero).
The Surreals form a "Real Closed Field," meaning they support all standard algebra and calculus operations, yet they encompass a richness of infinity that standard real numbers lack.
Infinite Chess
Hamkins also applies set-theoretic tools to Infinite Chess—a game played on an unbounded grid with no edges. This variant introduces bizarre tactical realities:
- Mate-in-$\omega$: A position where White can force a win, but Black can delay the loss for any arbitrary finite number of moves ($N$). White wins, but Black chooses how long it takes.
- Decidability: Despite the infinite board, the "mate-in-$N$" problem is computable. However, determining if a position is a win in the abstract "infinite play" sense requires advanced set theory.
Conclusion
The crisis of infinity that began with Cantor did not destroy mathematics; it liberated it. By accepting that intuition is an unreliable guide to the infinite, mathematicians have constructed a vast, rigorous landscape where truth is not monolithic. From the unprovable depths of Gödel’s theorems to the expanding horizons of the set-theoretic multiverse, we learn that mathematics is not just a tool for describing the physical world—it is a distinct, infinite reality of its own, waiting to be explored.
