The sequence that grows remarkably large, then drops to zero!

Comments · 358

• FU

  @fuzzy-pillar · 3 years ago

  This is like a contract you would write with an all-powerful demon to get unlimited power while weaseling out with fine print.

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• MA

  @matteofalduto766 · 3 years ago (edited)

  How mathematicians manage to (mostly) remain sane despite spending their careers figuring out things of this kind, really amazes me 😮

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• NE

  @nerdboy628 · 1 year ago

  I thought I would be able to understand this then I heard &quot;let&apos;s attempt to define numbers&quot; (<a href="https://www.youtube.com/watch?v=Fa5MQTDIhrY&amp;t=425">7:05</a>) and I knew I had no chance

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• CA

  @Caramelldanson · 3 years ago

  oh it&apos;s like the &quot;ant crawling along a stretching rubber band&quot; problem

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• BE

  @benjaminpedersen9548 · 3 years ago

  Cool video. You do say it, but I think it is important to stress the fact that the new sequence is strictly decreasing until it hits zero rather than just terminating. Otherwise it could terminate at a larger ordinal, even an infinite one, which would not limit the Goodstein sequence at all.

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• RE

  @Real_Potato_Man · 1 year ago

  Goodstien when badstien comes in:

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• OM

  @omegaminusfour · 3 years ago

  YouTube just asked whether this video was a good recommendation.<br>5 stars. Absolutely.

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• CA

  @cartatowegs5080 · 4 years ago

  Deserves more views 100%

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• BE

  @benjaminshropshire2900 · 3 years ago

  I think just the left have of the image at <a href="https://www.youtube.com/watch?v=Fa5MQTDIhrY&amp;t=754">12:34</a> makes for a much simpler intuitive argument for why this works: The -1 &quot;captures&quot; the right most non-zero term in the sum (it results in a trivial number term with no n&apos;s) and eventual eats it away to a zero term. Then the next term starts being removed. As long as nothing creates terms faster than they are removed, eventually there will be no non-zero terms. As there are no operations (other than the -1) that can <b>add</b> to the count of terms and only terms containing the number n change, the only thing that remains is to argue that the subtracting 1 from a term on average results in fewer terms containing n, which intuitively seems like it should be true as n grows.<br><br>Not a formal proof, but it shows why the proven result is not unreasonable.

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• ST

  @stephenhousman6975 · 3 years ago

  What I like to visualize here is a timer ticking its way to zero. &nbsp;The base doesn&apos;t really matter until you need to need to tick down from zero in the ones place.

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• HY

  @hymnodyhands · 3 years ago (edited)

  If the power of subtracting one is that big there in Goodstein&apos;s Theorem... that explains a lot about the Collatz Conjecture, although said conjecture is much harder to prove... meanwhile, this is beautiful in every way, and greatly appreciated!

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• AD

  @adamschulte5777 · 3 years ago

  Roger Penrose&apos;s &quot;The Emperor&apos;s New Mind&quot; brought me here, and man... you&apos;ve provided such a great way to understand this theorem. My set theory was just rusty enough to benefit from your synopsis. This deserves more views.

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