continuum hypothesis – Quanta Magazine https://www.quantamagazine.org Illuminating science Mon, 16 Sep 2024 14:10:47 -0400 en-US hourly 1 https://wordpress.org/?v=6.2 How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer. https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/ https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/#respond Thu, 15 Jul 2021 14:58:50 +0000 https://www.quantamagazine.org/?p=106602 The post How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer. first appeared on Quanta Magazine

]]> In October 2018, David Asperó was on holiday in Italy, gazing out a car window as his girlfriend drove them to their bed-and-breakfast, when it came to him: the missing step of what’s now a landmark new proof about the sizes of infinity. “It was this flash experience,” he said. Asperó, a mathematician at the University of East Anglia in the United Kingdom, contacted the collaborator with whom he’d...

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]]> https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/feed/ 0 How Gödel’s Proof Works https://www.quantamagazine.org/how-godels-proof-works-20200714/ https://www.quantamagazine.org/how-godels-proof-works-20200714/#respond Tue, 14 Jul 2020 14:00:54 +0000 https://www.quantamagazine.org/?p=88930 The post How Gödel’s Proof Works first appeared on Quanta Magazine

]]> In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history. Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths. But Gödel’s shocking...

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]]> https://www.quantamagazine.org/how-godels-proof-works-20200714/feed/ 0 Mathematicians Measure Infinities and Find They’re Equal https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/ https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/#respond Tue, 12 Sep 2017 12:32:02 +0000 https://www.quantamagazine.org/?p=50901 The post Mathematicians Measure Infinities and Find They’re Equal first appeared on Quanta Magazine

]]> In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers.

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]]> https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/feed/ 0 To Settle Infinity Dispute, a New Law of Logic https://www.quantamagazine.org/to-settle-infinity-question-a-new-law-of-mathematics-20131126/ https://www.quantamagazine.org/to-settle-infinity-question-a-new-law-of-mathematics-20131126/#comments Tue, 26 Nov 2013 14:00:58 +0000 https://www.quantamagazine.org/?p=7973 The post To Settle Infinity Dispute, a New Law of Logic first appeared on Quanta Magazine

]]> In the course of exploring their universe, mathematicians have occasionally stumbled across holes: statements that can be neither proved nor refuted with the nine axioms, collectively called “ZFC,” that serve as the fundamental laws of mathematics. Most mathematicians simply ignore the holes, which lie in abstract realms with few practical or scientific ramifications. But for the stewards of math’...

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