Joel David Hamkins on Infinity, Gödel Incompleteness and the Mathematical Multiverse
Speaker: Joel David Hamkins Source: Lex Fridman Podcast #488 URL: https://lexfridman.com/joel-david-hamkins-transcript
Key ideas
- Diagonalisation is the master argument of mathematical logic. Cantor’s uncountability proof, Russell’s paradox, Gödel’s incompleteness theorems, and Turing’s halting problem all use the same abstract move: given a list, construct an object that differs from every listed item at a corresponding position.
- Truth and proof are categorically different. Proof is syntactic (symbol manipulation following rules). Truth is semantic (what is the case in a mathematical structure). Gödel’s incompleteness shows these come apart: there are true statements not provable within any sufficiently strong consistent theory.
- Gödel decisively destroyed Hilbert’s programme — both goals. First Incompleteness Theorem: no consistent computably axiomatisable theory including arithmetic is complete. Second: no such theory proves its own consistency. Hilbert imagined mathematics as rote computation; incompleteness says this is structurally impossible.
- The mathematical multiverse. Hamkins’ pluralist position: there is no single true model of set theory. Multiple non-isomorphic ZFC universes are all equally mathematically real. The Continuum Hypothesis is true in some, false in others — exactly as the parallel postulate distinguishes Euclidean from hyperbolic geometry.
- The union of countably many countable sets is still countable. Hilbert’s Hotel, Hilbert’s bus, Hilbert’s infinite train — all can be accommodated using prime factorisation or diagonal paths. Euclid’s principle (the whole exceeds the part) fails systematically for infinite sets.
Infinity: from potential to actual
Aristotle through Galileo: infinity was potential — always continuable, never complete. Galileo’s paradox: perfect squares can be put in bijection with all natural numbers (n ↔ n²), yet seem fewer. Galileo threw up his hands; Cantor resolved it.
Cantor-Hume principle: two sets have the same size iff there is a one-to-one correspondence between them. This is the correct definition of equinumerosity for infinite sets — Euclid’s principle (whole > part) fails, and this is not a problem but a theorem.
Hilbert’s Hotel: countably infinite rooms, full. New guest: shift everyone up one room. Infinite bus: double all room numbers (even), put passengers in odd rooms. Infinite train (infinite cars, infinite seats): assign room number 3^C × 5^S — unique by unique prime factorisation, always odd. Result: the union of countably many countable sets is countable.
The rationals are countable (each fraction = two integers = Hilbert train argument). The algebraic numbers are countable. Most real numbers are transcendental.
Cantor’s diagonal and uncountability
Assume the real numbers are countable — they can be listed R₁, R₂, … Construct Z: the Nth decimal digit differs from the Nth digit of Rₙ (using only digits 1–8 to avoid dual representations). Then Z ≠ Rₙ for all n — contradiction. The reals are uncountable.
The power set generalisation: for any set X, P(X) is strictly larger than X. Proof: suppose a bijection f: X → P(X). Define D = {x ∈ X : x ∉ f(x)}. Then D must be f(d) for some d ∈ X. But d ∈ D ↔ d ∉ f(d) = D — contradiction. The committee version: there are more committees than members, even with infinitely many members.
This argument structure — define the diagonal object, ask if it is in its own image, reach contradiction — is the backbone of Russell’s paradox, Gödel’s theorems, and Turing’s halting problem.
Russell’s paradox
Frege’s logicism assumed general comprehension: for any property, form the set of objects with that property. Russell’s letter (while Frege’s second volume was at press): R = {x : x ∉ x} leads to R ∈ R ↔ R ∉ R. Frege: “Hardly anything more unwelcome can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished.”
Hamkins’ reframing: this is Russell’s theorem — there is no universal set. In ZFC, the restricted comprehension (separation) axiom prevents the paradox: you can only form subsets of existing sets. No paradox, just a theorem.
ZFC and the Axiom of Choice
Zermelo developed his axioms in 1908 to formalise his (controversial) 1904 proof that every set can be well-ordered, given the Axiom of Choice. Many objectors were using AC implicitly in their own proofs.
Russell’s shoes/socks: for infinite collections of shoes, you can always pick the left shoe — a rule exists. For infinite collections of indistinguishable socks, no rule can specify which to pick. AC asserts that a choice function exists anyway. Whether you accept AC depends on your attitude toward mathematical ontology: does mathematical reality contain objects that cannot be explicitly specified?
ZFC is now the standard foundation. Gödel (1938) proved AC is consistent with ZFC; Cohen (1963) proved ¬AC is also consistent. AC is therefore independent of ZF.
Gödel’s incompleteness theorems
Hilbert’s programme assumed mathematics could be:
- Founded on a complete, consistent, computably axiomatisable theory
- Proved consistent by purely finitistic means
In Hilbert’s imagined success, a theorem-enumeration machine answers every mathematical question — creative thinking becomes rote computation.
First Incompleteness Theorem: any consistent computably axiomatisable theory T including basic arithmetic is incomplete — there are statements undecidable in T. This destroys goal (1).
Second Incompleteness Theorem: no such T can prove its own consistency. This destroys goal (2). Moreover, a theory claiming its own consistency is like a used-car salesman claiming to be trustworthy — even an inconsistent theory proves its own consistency.
Consequence: mathematical reality is permanently pluralistic. Independence (statements neither provable nor disprovable from the axioms) is not an anomaly but a permanent structural feature of any sufficiently strong formal system.
Truth vs proof
Pre-Gödel, “true” and “provable” were conflated (even in Bourbaki). Gödel and Tarski separated them decisively.
Proof (syntactic): a finite sequence of formal sentences following specified rules of inference (modus ponens etc.). Proof is about symbol manipulation.
Truth (semantic, Tarski): “Snow is white” is true iff snow is white. Truth is defined recursively over a formal language with respect to a mathematical structure. To ask whether a statement is true requires specifying which structure.
Soundness: provable → true (proofs preserve truth). Completeness: true → provable. First-order logic is complete (Gödel’s Completeness Theorem, 1930). But specific theories like Peano Arithmetic are incomplete (Gödel’s Incompleteness Theorems, 1931) — there are statements true in the standard model that are not provable.
Truth in the structure can outrun the formal theory about it. The formal theory is a finite syntactic object; truth in a structure is a richer semantic notion.
The mathematical multiverse (Hamkins’ pluralism)
Hamkins’ signature contribution: set-theoretic pluralism. The Continuum Hypothesis (CH) — is there a set strictly between ℕ and ℝ in size? — is independent of ZFC (Gödel + Cohen). Rather than seeking new axioms to settle CH, Hamkins argues CH simply has different truth values in different, equally valid models of set theory.
Analogy: non-Euclidean geometry. The parallel postulate is true in Euclidean geometry, false in hyperbolic geometry. We do not ask which is “really” true — we ask which fits which application. The same pluralism should apply to set theory.
The multiverse contains all non-isomorphic models of ZFC. Each is mathematically real. Set-theoretic questions like CH are not objectively determinate — they are relative to a choice of universe. This contradicts the unificationist programme (seeking new axioms to pin down a single canonical model).