The Genius Who Invented Reverse Mathematics
5/18/20261 hr 39 min
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- I subscribe to The Economist for their science and tech coverage. As a TOE listener, get 35% off! No other podcast has this: https://economist.com/TOE Harvey Friedman — the youngest professor in Stanford's history, founder of reverse mathematics, and the mathematician Kurt Gödel personally chose to sponsor his final paper — has spent 60 years on a single, audacious question: can ordinary, finite math be trusted? His theorems suggest otherwise, showing that even the most concrete and natural mathematical statements — involving nothing more exotic than rational numbers — cannot be proved or refuted within the gold standard of mathematical foundations, ZFC. The foundations of mathematics, Friedman argues, are not settled bedrock but something far more vertiginous: totally up in the air, and made more mysterious, not less, by his own work. FOLLOW:
- Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e
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- PayPal: https://www.paypal.com/donate?hosted_button_id=XUBHNMFXUX5S4 TIMESTAMPS:
- 00:00:00 - Gödel’s Incompleteness Misinterpretations
- 00:09:48 - Woodin vs. Friedman Foundations
- 00:17:28 - Category Theory vs. Logic
- 00:24:30 - Borel Determinacy Paradoxes
- 00:31:23 - Embedded Maximality Principles
- 00:41:18 - Tree(3) and Kruskal’s Theorem
- 00:47:40 - Finitism and Large Cardinals
- 00:53:11 - Divine Consistency and Angels
- 01:03:25 - Reverse Mathematics Origins
- 01:11:14 - Constructive Logic and Intuitionism
- 01:21:17 - Theology and AI Immortality LINKS MENTIONED:
- Harvey Friedman Papers: https://u.osu.edu/friedman.8/foundational-adventures/publications/
- Harvey Friedman YouTube: https://www.youtube.com/@harveyfriedman4465/videos
- Harvey Friedman Chess Club: https://cclchess.com/
- This Man Is About to Blow Up Mathematics [Article]: https://nautil.us/this-man-is-about-to-blow-up-mathematics-236446
- Harvey Lecture at OSU: https://youtu.be/NAGQD-bSXok
- Most Abused Theorem in Math [TOE]: https://youtu.be/OH-ybecvuEo
- John Norton [TOE]: https://youtu.be/Tghl6aS5A3M
- Emily Riehl [TOE]: https://youtu.be/mTwvecBthpQ
- What Is Infinity? [TOE]: https://youtu.be/rHtqGrtcB1w
- Norman Wildberger [TOE]: https://youtu.be/l7LvgvunVCM
- Wolfgang Smith [TOE]: https://youtu.be/lF4S_P_o-g0
- Scott Aaronson [TOE]: https://youtu.be/1ZpGCQoL2Rk
- Consciousness Iceberg [TOE]: https://youtu.be/65yjqIDghEk
- Edward Frenkel [TOE]: https://youtu.be/n_oPMcvHbAc
- Elan Barenholtz [TOE]: https://youtu.be/A36OumnSrWY
- Michael Levin [TOE]: https://youtu.be/c8iFtaltX-s
- Godel Incompleteness Theorems: https://plato.stanford.edu/entries/goedel-incompleteness/
- Consistency of Axiom of Choice [Book]: https://archive.org/details/dli.ernet.469796/page/18/mode/2up
- Independence of Continuum Hypothesis [Paper]: https://www.jstor.org/stable/71858
- Borel Determinacy [Paper]: https://www.jstor.org/stable/1971035
- Paris-Harrington Theorem: https://mathworld.wolfram.com/Paris-HarringtonTheorem.html
- The God Letter: https://uncertaintist.wordpress.com/wp-content/uploads/2012/10/einstein-letter-gutkind-excerpts.pdf
- Undecidable Propositions of Principia Mathematica [Book]: https://amazon.com/dp/0486669807?tag=toe08-20
- Categories for the Working Mathematician [Book]: https://amazon.com/dp/1441931236?tag=toe08-20
- On Necessary Use of Abstract Set Theory [Paper]: https://www.sciencedirect.com/science/article/pii/0001870881900219
- Borel Set: https://en.wikipedia.org/wiki/Borel_set More links: https://curtjaimungal.substack.com Guests do not pay to appear. #science Learn more about your ad choices. Visit megaphone.fm/adchoices
Clips
Transcript preview
First 90 secondsHarvey Friedman· Guest0:00
Pretty outrageous idea. In other words, all this real number stuff, all this partial differential equations, even all this set theory stuff, large cardinals, it's all fundamentally finite. This is my crazy hat.
Curt Jaimungal· Host0:10
[instrumental music] This is Professor Harvey Friedman's first podcast. At 18, he was given not only a PhD, which is outstanding, but the title of Professor at Stanford University for his work in mathematical logic. The Guinness Book of World Records even listed him as the youngest professor ever. Kurt Gödel, while alive, personally sponsored his last paper for the proceedings of the National Academy of Sciences. And Professor Friedman founded the field of reverse mathematics. Gödel's incompleteness theorems are the most celebrated results in modern logic. The textbook examples are recondite, self-referential curiosities that no working mathematician tends to meet in practice. However, Friedman says they're pointing at the wrong target. The question is, can ordinary finite math be trusted? His theorems suggest otherwise.
Harvey Friedman· Guest1:06
So now it's harder for the mathematical community to ignore foundations.
Curt Jaimungal· Host1:12
On this channel, I, Curt Jaimungal, interview researchers regarding their theories of reality with rigor and technical depth, and probe at the foundations. Today, we discuss Tree Three, reverse mathematics, and the divine consistency proof, where- An angel is a weak form of God