Are Some Infinities Bigger Than Others?
In short: From Galileo's puzzle to Cantor's proofs and the still-debated continuum hypothesis, a short look at how mathematicians learned that some infinities are larger than others.
A 1:24 short film in the series “From Fingers to Qubits”, narrated by intellect_voice_01.
Sources used in this film: Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce); article by Piotr Błaszczyk, Pedagogical University of Cracow, “Galileo’s paradox and numerosities”; Stanford Encyclopedia of Philosophy (supplement by Henry Mendell), “Aristotle and Mathematics > The Infinite (Stanford Encyclopedia of Philosophy)”; Brigham Young University Department of Mathematics (Christopher P. Grant's English translation of Georg Cantor's 1874 paper), “cantor1874.dvi”; MacTutor History of Mathematics, University of St Andrews, “Set theory - MacTutor History of Mathematics”; Stony Brook University, Department of Mathematics (MAT 336 History of Mathematics course archive; English translation of Georg Cantor by Anthony Phillips), “Stony Brook University, Department of Mathematics (MAT 336 History of Mathematics course archive; English translation of Georg Cantor by Anthony Phillips)”; Columbia University (Haim Gaifman, "Gödel's Incompleteness Results", Chapter 0), “Diagonalization from Cantor to Gödel”; MacTutor History of Mathematics, School of Mathematics and Statistics, University of St Andrews, “Georg Cantor (1845 - 1918) - Biography - MacTutor History of Mathematics”; Anne Newstead, School of Mathematics and Statistics, University of New South Wales, “Original version of Cantor article (need corrections)”; Springer Nature (The Mathematical Intelligencer); article by David E. Rowe, “On the Origins of Cantor’s Paradox: What Hilbert Left Unsaid at the 1900 ICM in Paris”; Stanford Encyclopedia of Philosophy, “The Continuum Hypothesis”; Stanford Encyclopedia of Philosophy, “Kurt Gödel”; The Bulletin of Symbolic Logic (Association for Symbolic Logic); article by Akihiro Kanamori, hosted on Boston University's mathematics site, “14.pdf”; Scientific American (story by Natalie Wolchover, originally published by Quanta Magazine), “Dispute over Infinity Divides Mathematicians”.
Sources used in this film
- Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce); article by Piotr Błaszczyk, Pedagogical University of Cracow — Galileo’s paradox and numerosities · View →
- Stanford Encyclopedia of Philosophy (supplement by Henry Mendell) — Aristotle and Mathematics > The Infinite (Stanford Encyclopedia of Philosophy) · View →
- Brigham Young University Department of Mathematics (Christopher P. Grant's English translation of Georg Cantor's 1874 paper) — cantor1874.dvi · View →
- MacTutor History of Mathematics, University of St Andrews — Set theory - MacTutor History of Mathematics · View →
- Stony Brook University, Department of Mathematics (MAT 336 History of Mathematics course archive; English translation of Georg Cantor by Anthony Phillips) — Stony Brook University, Department of Mathematics (MAT 336 History of Mathematics course archive; English translation of Georg Cantor by Anthony Phillips) · View →
- Columbia University (Haim Gaifman, "Gödel's Incompleteness Results", Chapter 0) — Diagonalization from Cantor to Gödel · View →
- MacTutor History of Mathematics, School of Mathematics and Statistics, University of St Andrews — Georg Cantor (1845 - 1918) - Biography - MacTutor History of Mathematics · View →
- Anne Newstead, School of Mathematics and Statistics, University of New South Wales — Original version of Cantor article (need corrections) · View →
- Springer Nature (The Mathematical Intelligencer); article by David E. Rowe — On the Origins of Cantor’s Paradox: What Hilbert Left Unsaid at the 1900 ICM in Paris · View →
- Stanford Encyclopedia of Philosophy — The Continuum Hypothesis · View →
- Stanford Encyclopedia of Philosophy — Kurt Gödel · View →
- The Bulletin of Symbolic Logic (Association for Symbolic Logic); article by Akihiro Kanamori, hosted on Boston University's mathematics site — 14.pdf · View →
- Scientific American (story by Natalie Wolchover, originally published by Quanta Magazine) — Dispute over Infinity Divides Mathematicians · View →
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Sources
- Summary by Intellect · source: Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce); article by Piotr Błaszczyk, Pedagogical University: Galileo’s paradox and numerosities
- Also: Stanford Encyclopedia of Philosophy (supplement by Henry Mendell): Aristotle and Mathematics > The Infinite (Stanford Encyclopedia of Philosophy)
- Also: Brigham Young University Department of Mathematics (Christopher P. Grant's English translation of Georg Cantor's 1874 pa: cantor1874.dvi
- Also: MacTutor History of Mathematics, University of St Andrews: Set theory - MacTutor History of Mathematics
- Also: Stony Brook University, Department of Mathematics (MAT 336 History of Mathematics course archive; English translation of: Stony Brook University, Department of Mathematics (MAT 336 History of Mathematics course archive; English translation of Georg Cantor by Anthony Phillips)
- Also: Columbia University (Haim Gaifman, "Gödel's Incompleteness Results", Chapter 0): Diagonalization from Cantor to Gödel
- Also: MacTutor History of Mathematics, School of Mathematics and Statistics, University of St Andrews: Georg Cantor (1845 - 1918) - Biography - MacTutor History of Mathematics
- Also: Anne Newstead, School of Mathematics and Statistics, University of New South Wales: Original version of Cantor article (need corrections)
- Also: Springer Nature (The Mathematical Intelligencer); article by David E. Rowe: On the Origins of Cantor’s Paradox: What Hilbert Left Unsaid at the 1900 ICM in Paris
- Also: Stanford Encyclopedia of Philosophy: The Continuum Hypothesis
- Also: Stanford Encyclopedia of Philosophy: Kurt Gödel
- Also: The Bulletin of Symbolic Logic (Association for Symbolic Logic); article by Akihiro Kanamori, hosted on Boston Universit: 14.pdf
- Also: Scientific American (story by Natalie Wolchover, originally published by Quanta Magazine): Dispute over Infinity Divides Mathematicians
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