Speaker
Joel David Hamkins
Appearances over time
1 episodes
Episodes
1
Podcasts
Quotes & moments
Putting together countably many countably infinite sets still yields only a countable infinity — you can fit them all in Hilbert's Hotel.
Via his diagonal argument, Cantor showed the real numbers cannot be put in one-to-one correspondence with the natural numbers, proving there is more than one size of infinity.
No consistent, computably axiomatizable theory that includes basic arithmetic can be complete — there will always be true statements it cannot prove or refute.
A consistent theory cannot prove its own consistency — not even the infinitary set theory can vouch for itself.
There is no computable procedure that correctly decides for all programs whether they will halt — proven via a diagonal argument identical in structure to Russell's paradox and Cantor's theorem.
Roughly 1/e² ≈ 13.5% of Turing machine programs never transition to a halt state — so their non-halting can be detected by inspection alone, with no deep analysis needed.
Using the Pólya recurrence theorem for random walks, Hamkins and collaborators showed the halting problem can be computably solved for asymptotically 100% of programs (those whose head falls off the tape).
Gödel (1938) showed CH is consistent with ZFC; Cohen (1963) showed its negation is also consistent — making CH neither provable nor refutable from the ZFC axioms.
Practically every non-trivial statement about infinite combinatorics turns out to be independent of ZFC — neither provable nor refutable.
Bertrand Russell discovered a one-line contradiction in Frege's monumental logicist program just as the book was going to press, forcing Frege to append a response acknowledging the collapse of his system.
Whether a specific cell in Conway's Game of Life will ever become alive is computably undecidable — equivalent to the halting problem.
For any model of set theory, there exists a forcing extension where CH is true and another where it is false — you can toggle it at will in closely related mathematical universes.
A proof by contradiction: if any boring numbers existed, the smallest boring number would itself be interesting — contradicting the assumption.
Cantor proved that for any set, its power set (collection of all subsets) is strictly larger — the number of committees exceeds the number of people, even for infinite groups.
A hotel with infinitely many rooms, all full, can still accommodate a new guest — just shift everyone up one room. This violates Euclid's principle that the whole is always greater than the part, and it's the cleanest illustration of why infinite sets behave nothing like finite ones.
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