Countable Sets: Naturals, Integers, Rationals

For much of history, many mathematicians—following thinkers like Aristotle—viewed infinity as a never-ending process rather than a completed object. In the late 19th century, Georg Cantor revolutionized this view by treating infinite sets as mathematical objects that could be compared and studied. His work showed that not all infinities are equal, and that there are infinitely many different sizes of infinity. While his ideas are foundational in modern mathematics, some philosophical schools, such as finitism and ultrafinitism, continue to question whether infinite objects meaningfully exist. 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 FOLLOW: - Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e - Substack: https://curtjaimungal.substack.com/subscribe - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs - Crypto: https://commerce.coinbase.com/checkout/de803625-87d3-4300-ab6d-85d4258834a9 - PayPal: https://www.paypal.com/donate?hosted_button_id=XUBHNMFXUX5S4 TIMESTAMPS: - 00:00 - Potential vs. Actual Infinity - 03:12 - Cardinality and Aleph-Null - 06:12 - Diagonalization and Uncountability - 09:21 - ZFC and Logical Independence - 12:23 - Finitism and Ultrafinitism - 15:26 - Continuum Hypothesis Paradoxes - 16:00 - Foundational Mathematical Crisis LINKS MENTIONED: - The Most Abused Theorem in Math [TOE]: https://www.youtube.com/watch?v=OH-ybecvuEo - Dror Bar Natan [TOE]: https://youtu.be/rJz_Badd43c - Hilbert’s Problems: https://mathworld.wolfram.com/HilbertsProblems.html - The Independence of the Continuum Hypothesis [paper]: https://www.pnas.org/doi/pdf/10.1073/pnas.50.6.1143 - Piano arithmetic: https://ncatlab.org/nlab/show/Peano+arithmetic - Cantor’s Diagonal Argument: https://www.researchgate.net/publication/335364685_A_Translation_of_G_Cantor's_Ueber_eine_elementare_Frage_der_Mannigfaltigkeitslehre - Hartog’s Construction: paultaylor.eu/trans/HartogsF-wellord.pdf - Cohen’s Forcing Method: https://timothychow.net/forcing.pdf - Norman Wildberger [TOE]: https://youtu.be/l7LvgvunVCM - Woodin’s lecture: https://youtu.be/nVF4N1Ix5WI In Search of Ultimate-L [paper]: https://www.jstor.org/stable/44164514 - Emily Riehl [TOE]: https://youtu.be/mTwvecBthpQ - Sir Roger Penrose [TOE]: https://youtu.be/sGm505TFMbU - Why Write? [article]: https://curtjaimungal.substack.com/p/why-write ASSETS USED: - Infinity display: https://youtu.be/osa4ptG5lMg - Number counter: https://youtu.be/HRL5uNGXh9U Guests do not pay to appear. #science

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