Choice Functions and Length: Why you can't measure everything you choose скачать в хорошем качестве

Choice Functions and Length: Why you can't measure everything you choose

I haven't talked about the axiom of choice in a while, and the relationship between choice functions and length (or size) and why you can't measure everything you choose seemed like a good way to do so. The interplay between choice functions and how one can construct sets for which measure is not an adequate descriptor is one of the handful of paradoxes out there that leads mathematicians to believe that the axiom of choice is not a great thing to postulate when proving mathematical statements. The video can be broken down into the following sections. 00:00 Intro 00:45 Choice functions We haven't talked about choice functions in a long time, so it seemed like a good place to start! 01:01 Axiom of Choice The axiom of choice is introduced in relation to the existence of choice functions on non-empty collections of non-empty sets 02:17 An Example of a Choice Function 03:09 Equivalence Relations A review of equivalence relations, both the symmetric, reflexive, transitive and the partition approach is given. The latter is more easily applied to the idea we are addressing in this video. 04:39 Example of an equivalence relation, Rational Equivalence 07:27 Vitali Sets An argument for the existence of Vitali sets, or non-measurable sets is given. This is an important technical result in measure theory/real analysis when operating with the axiom of choice, but it is fairly accessible from an intuitive approach which we mostly appeal to here. 12:08 Conclusion ===================================================== Other Videos relating to Measure Theory and Mathematical Size: ===================================================== Measure Zero versus Meager:    • Some Small Ideas in Math: A Set of Measure...   Some Cantor Set shenanigans:    • I Used the Cantor Set Construction to Make...   Hausdorff Distance:    • What is the distance between two sets of p...   ===================================================== Other (very old) Videos relating to the axiom of choice: ===================================================== The last time I talked about choice functions:    • Choice Functions & The Axiom of Choice | N...   Zorn's Lemma video:    • Zorn's Lemma, The Well-Ordering Theorem, a...   _____________________ Last PhD Update:    • Math PhD Update: My 1st week of graduate s...   Last Time:    • What is the distance between two sets of p...   The CHALKboard:    / chalkboard   Find the CHALKboard on Facebook: http://bit.ly/CHALKboard _____________________ BOOKS AND RESOURCES RELATED TO THIS VIDEO: -Real Analysis by H.L. Royden and P.M. Fitzpatrick: https://amzn.to/3amPONI -A User-friendly Introduction to Lebesgue Measure and Integration by Gail S. Nelson: https://amzn.to/2LuE9Cb WHAT GEAR I USED FOR THIS VIDEO: Camera - https://amzn.to/30bJRho Lens - https://amzn.to/3fgDaPa SD card - https://amzn.to/2PbTiag Microphone - https://amzn.to/2D3KYXE Lighting Kit - https://amzn.to/3ffibMH Editing Hard Drive - https://amzn.to/2P7hSsM DISCLAIMER: Links included in this description may be affiliate links. If you purchase a product or service with the links that I include here, I may receive a small commission. There is no additional charge to you! Thank you for supporting my channel and for helping me create these free videos! Subscriber Count as of release: 6562(+3,327 😱That has over doubled since last time I uploaded 😱) #CHALK #MeasureTheory #ChoiceFunctions