Can an Uncountable Sum Ever Be Finite-Valued? | Why Measure Infinity?

Comments · 145

• DA

  @darreljones8645 · 4 years ago

  Yes, quite easily. The sum of all values of the function f(x) = 0 is zero.

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• MI

  @mikikaboom9084 · 2 years ago

  Me, a physics student: Look, I’ve had enough analysis for today, so I’m gonna take a rest and do the assignment tomorrow.<br><b>proceeds to watch a video about analysis</b>

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• EX

  @exampleDOTslangwhanger · 3 years ago

  i never have any idea what these videos are talking about but i watch them all the way through every time because <b>graph</b>

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• DR

  @DrBarker · 4 years ago

  This was a fun question, and nicely motivated through wanting to generalise the dot product. I love your use of animations!

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• NI

  @NigelMelanisticSmith · 4 years ago

  Just realized that this channel is brand new, and not years old. Great work!

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• RU

  @ruzgar1372 · 5 months ago

  Just had this thought in the back of my mind a few days ago. Feels good to know that no matter how weird of a topic you&apos;re thinking about in math, there&apos;s always some other person who&apos;d thought about it before you.

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• RO

  @robharwood3538 · 4 years ago

  Wow, dude! You finally helped me break through a mental barrier that left me stumped by things like measure theory. I never could really &apos;get&apos; why it was important, or, more importantly, how it could actually be used or how it works in general. It just seemed like it was shrouded in waaayy too much jargon and &apos;symbology&apos; (for lack of a better word), without much in the way of intuition or motivation.<br><br>Thank you! Cheers!

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• IS

  @isavenewspapers8890 · 2 years ago

  <a href="https://www.youtube.com/watch?v=uLja-yAwuCI&amp;t=686">11:26</a> YOU SAID IT! YOU SAID THE THING! WOOOOOO

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• UN

  @Unemployed-Math-Major · 3 years ago

  What a gold channel I just came across

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• AL

  @alberto3071 · 4 years ago

  Breathtaking proof! I was having this doubt some time ago, I am amazed that there&apos;s even a video about it.

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• SH

  @shimrrashai-rc8fq · 3 years ago (edited)

  Or a better way to think about it is that an integral <i>is</i> an uncountable sum . . . where all the terms have been weighted down small enough that the sum can be finite, which means they have to be so small they cannot be understood as real numbers. And the thing that does that? dx.<br><br>That&apos;s <i>really</i> why &quot;dx&quot; is always at the end of an integral. In fact, I advocate we should <i>not</i> write integrals as<br> &nbsp;<br>int_{a...b} f(x) dx<br> &nbsp;<br>but, like sums, we should write the variable of integration with the indices:<br> &nbsp;<br>int_{x=a...b} f(x) dx.<br> &nbsp;<br>This may seem redundant, but the &quot;&apos;dx&apos; indicates the variable of integration&quot; is actually something that won&apos;t go far when you get into something like physics, and run into this:<br> &nbsp;<br>Moment of Inertia = int r^2 dm<br> &nbsp;<br>which is most assuredly NOT integration &quot;with respect to mass&quot; in most cases, but rather integration over all <i>points of the object&apos;s volume.</i> It <i>really</i> is<br> &nbsp;<br>Moment of Inertia = int_{P e R^3} [r(P)]^2 dm(P)<br> &nbsp;<br>where r(P) is the distance to the point P from the rotation axis, and dm(P) is the infinitesimal mass located at point P, which also equals rho(P) dV, where dV is the infinitesimal volume unit and rho(P) is the mass density at P. The former notation is worse than useless; it doesn&apos;t even make sense at all!

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• HA

  @Hamboarding · 3 years ago

  I really like the א‎ &amp; ב‎ numbers but I ❤ large cardinals! I wish there would be more videos about them that explain the connection to axioms and how the strength of an axiomatic system relates to large cardinals and how large cardinals are constructed going way further than all the Powersets etc…

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