Computability and problems with Set theory | Math History | NJ Wildberger

We look at the difficulties and controversy surrounding Cantor's Set theory at the turn of the 20th century, and the Formalist approach to resolving these difficulties. This program of Hilbert was seriously disrupted by Godel's conclusions about Inconsistency of formal systems. Nevertheless, it went on to support the Zermelo-Fraenkel axiomatic approach to sets which we have a quick look at. Then we introduce Alan Turing's ideas of computability via Turing machines and some of the consequences. The lecture closes with a review of historical positions on the contentious idea of completed infinite sets, quoting illustrious mathematicians from Aristotle to A. Robinson, along with G. Cantor himself. In summary, it appears that this is not a closed chapter in the History of Mathematics. For those interested in a more in depth discussion of these and other interesting issues, see my MathFoundations series of YouTube videos--also at this channel. Video Contents: 00:00 Computability & problems with set theory 01:00 Cantor's definition of a "set" 04:47 K. Godel (1906-1978) 11:28 Zermelo - Fraenkel Axioms for "set theory" 16:31 Computability 25:20 Consequences; countable numbers of computable sequences 32:31 E.Borel (1871-1956)- founder of Measure theory ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated.