What A General Diagonal Argument Looks Like (Category Theory)

Comments · 268

• TH

  @Thricery · 4 years ago (edited) · pinned

  ====Timestamps====<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc">00:00</a> Introduction<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=59">00:59</a> A first look at uncountability<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=304">05:04</a> Why generalise?<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=413">06:53</a> Mathematical patterns<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=460">07:40</a> Working with functions and sets<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=700">11:40</a> Second version of Cantor&apos;s Proof<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=820">13:40</a> Powersets and Cantor&apos;s theorem in its generality<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=938">15:38</a> Proof template of Diagonal Argument<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=1000">16:40</a> The world of Computers<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=1265">21:05</a> Gödel numbering<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=1385">23:05</a> An amazing program (setup of the Halting Problem)<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=1505">25:05</a> Solution to the Halting Problem<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=1789">29:49</a> Comparing two diagonal arguments<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=1873">31:13</a> Lawvere&apos;s theorem<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=1969">32:49</a> Diagonal function as a way for encoding self-reference<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=2111">35:11</a> Summary of video<br><a href="https://www.youtube.com/watch?v=dwNxVpbEVcc&amp;t=2144">35:44</a> Bonus treat - Russell&apos;s Paradox

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• VE

  @venividi2940 · 3 years ago

  Bruh, how can you make one of the very best math explainer videos on youtube and then just stop!? Can you imagine how disappointed I was to finish this video and go to your page to start watching them all?

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• CA

  @Castlepod · 1 year ago

  bro dropped ONE(1) amazing video then dipped

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• PA

  @pashi47 · 4 months ago

  I can&apos;t believe I only just found this video, 3+ years after SoME2. I don&apos;t know why I didn&apos;t watch it back then, maybe I was still not exposed to enough abstract math back then yet, but this is one of my favorite math videos ever. I had individiually heard about each of the proofs / ideas you mentioned in the intro, but had never thought to make the connection that they were all somehow using the same underlying diagonal argument. <br><br>I&apos;m quite sad to see that you said you&apos;d make a follow-up video on Gödel&apos;s Incompleteness Theorems, but that still hasn&apos;t been made yet. I don&apos;t know if you still read comments all these years later or if this will get through to you, but if there&apos;s a small chance you see this, then please know that this video is amazing. I&apos;m sure that your next ones will be amazing too.

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• KE

  @kesleta7697 · 3 years ago

  This is the best introduction to category theory I&apos;ve ever seen. It managed to actually motivate category theory for me. Calling it the &quot;linguistics of mathematics&quot; was a really clever metaphor.

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• AL

  @alvarez110 · 3 years ago

  This YouTuber discovered this one trick, see why mathematicians hate him.

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• FN

  @fniunior · 3 years ago

  Thinking about category theory as the linguistics of mathematics is a big breakthrough for me. I think this might be the best introduction to category theory I have seen so far. Could you make videos exploring more of category theory from this perspective?

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• MA

  @marcopaviotti7917 · 1 day ago

  Great video, thanks! Although the follow up on Godel&apos;s incompleteness theorem though never came up, we&apos;re still waiting :)

• TH

  @TheDoh007 · 3 years ago

  That second version of Cantor&apos;s proof made it finally make sense to me, i&apos;ve seen it explained countless times before but i&apos;ve never felt it was rigorous enough. Awesome video!

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• LO

  @lojack5 · 3 years ago

  This was a really cool view of diagonal arguments &nbsp;that I&apos;d never seen before. &nbsp;I really like the (new to me) view that diagonal functions enable self reference! &nbsp;And everyone knows self-reference opens to door to all sorts of contradictions, so it makes perfect sense in hindsight!

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• KA

  @kartiksunaad · 3 years ago

  Oh man, I liked this video so much that I hit a like from all my google accounts. Really great work!

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• JA

  @jamesmstern · 3 years ago

  This video is a shining example of clarity.

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