Difficulties with Dedekind cuts | Real numbers and limits Math Foundations 116 | N J Wildberger

Comments · 70

• AV

  @avieus · 2 years ago

  This is great.. finally an effective and clear explanation on this.....

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• FR

  @frankcarr6207 · 11 years ago

  I am pretty certain that Dedekind&apos;s own book contained only that example, of the square root of 2, itself.<br><br>When we did this in my analysis class, I remember that they proved the least upper bound property of the reals before abandoning the topic of &apos;constructions&apos; of the reals: But basically, they ignored this stuff even at UChicago.

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• ME

  @methatis3013 · 8 months ago

  At around <a href="https://www.youtube.com/watch?v=jlnBo3APRlU&amp;t=1140">19:00</a> you&apos;re still making an infinite amount of choices. You take each number and check whether the algorithm picks it up. You still do an infinite amount of actions, even though they are governed by a &quot;simple&quot; rule

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• TO

  @tommasoc.2207 · 2 years ago

  Best explaination

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• PM

  @pmascaros · 11 years ago

  min <a href="https://www.youtube.com/watch?v=jlnBo3APRlU&amp;t=1525">25:25</a> The difference between choice and algorithm, it is the key point. Yes it is.<br>I remember professor Franklin said in a commentary (pertaining to the infinity debate video) that &nbsp;the Axiom of Choice is a pure existence statement, that&apos;s right but we need an algorithm in order to be sure the elements exists , as I see.

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• KY

  @kyaume21 · 11 years ago

  Brilliant video. I always told my students that the reals are the ugliest objects in mathematics (I had a careful look at the Rudin book -- I took it from my shelf again while I watched the video-- and that was my conclusion). Btw there was a paper at some point by the Nobel laureate Gerard &apos;t Hooft disputing the role of the reals in physics. &nbsp;I can&apos;t find it any longer but at some point I am sure I had a (preprint) copy. As far as I recall, he used quantum mechanics to redefine the reals.

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• GO

  @GodlessPhilosopher · 5 years ago

  Very interesting! This reminds me of some of Wittgenstein&apos;s remarks on infinity and set theory. Have you read his work in the philosophy of mathematics? I&apos;m very curious what you think of Gödel&apos;s theorems and Wittgenstein&apos;s comments on them.

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• HE

  @helloitsme7553 · 1 month ago

  <a href="https://www.youtube.com/watch?v=jlnBo3APRlU&amp;t=1870">31:10</a> The problem that there are infinitely many conditions can easily be resolved using quantifiers

• JO

  @josephavant8250 · 9 years ago

  Great presentation - THANK YOU for sharing

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• IN

  @insanity7538 · 5 years ago

  Really enjoying your videos. As someone who stopped studying mathematics in favor of computer science, this really resonates with me. During my studies, I found mathematics to be really &quot;esoteric&quot; in its use of real numbers, infinite sets and infinite sequences. I used to think I&apos;m just too dumb to get it, but perhaps I was the only person in the room being completely honest with myself.<br><br>Your thoughts about viewing mathematics from a more computational perspective is a very pragmatic and sensible approach in my opinion. There is absolutely no practical use for supposed real numbers or infinite sequences when you cannot map them to anything in the real world. I think they are a convenient abstraction over very large numbers and operations performed on them. Contrary to you, I also think that rational numbers aren&apos;t really numbers. I believe both the rational numbers and the real numbers to be &quot;intermediate forms of numbers&quot;, or more precisely, algorithms operating on natural numbers, which I believe to be the only numbers that actually exist in the real world. If we believe that our concrete universe cannot be made up of magical waves, but must be constructed with some sort of smallest building block, then really the count of all of these building blocks in our universe must be the biggest natural numbers to REALLY exist.<br><br>In that sense, I think rational numbers are helper constructs for when we cannot perform arithmetic with such fantastically large numbers, but want to &quot;zoom out&quot; a little. If we regard a lenght as 1 meter, then that is just an abstraction over an &quot;unknown, but very large amount of smallest building blocks, whose number is impractically large to do arithmetic with&quot;. Since half a meter obviously exists and is a practical thing to consider, we have constructed the division operation (which means building ratios of natural numbers: the very large number of smallest building blocks contained in a meter divided by 2). I see the &quot;division&quot; as a binary operation (algorithm) that is performed on two natural numbers.<br><br>In that sense, I don&apos;t believe that rational division is an actual thing, but just a helpful abstraction. Integer division (with remainder) is the only applicable division in the real world. The square root of any number is just an iterative algorithm performed on natural numbers because we cannot count the length of a diagonal in terms of its smallest, discrete building blocks. Since we cannot possibly run such an algorithm infinitely long (limited time and space in our universe), we should think of the square root as an algorithm that does not terminate (WHILE-program). If we want a practical number out of such an algorithm, we&apos;d need to have the algorithm terminate at some arbitrary point and yield an arbitrarily accurate result (FOR-program).<br><br>Doing arithmetic with real numbers should not be seen as proper arithmetic. It&apos;s more like combining different WHILE-programs for so long that, at some point, you can hopefully reduce the resulting algorithm to an actual (natural) number again. Similar to how sqrt(2) - which is an algorithm - times sqrt(2) - another algorithm - is 2 - an actual number - again. This also applies to rational numbers in my opinion.<br><br>I haven&apos;t really fleshed this out properly, but wanted to share my thoughts while I had the time to do so. Maybe you find some obvious flaws!<br><br>If you should ever read this, then please know that your work is important and that you are a very precious individual for inquiring deeper than others. You represent what is greatest in us humans: insatiable curiosity. Thank you.

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• VE

  @Verschlungen · 5 years ago

  Beautiful presentation! Thank you for being the voice of reason!<br>Here is another way, quite different from yours, that leads to the same view of the Dedekind cut:<br>The term &apos;irrational number&apos; (as applied to root 2, pi, etc.) I regard as a kind of carnival huckster&apos;s trick in that it focuses one&apos;s attention on the adjective &apos;irrational&apos; when our attention should be on &apos;number&apos;: I.e., IS root 2 a number? IS pi a number? No. Rather, there exists an algorithm to produce digits of root 2 or digits of pi For Ever, but the output from an algorithm that runs For Ever is not a number, it is (again) the output from an algorithm that runs For Ever. In other words, since root 2 and pi, etc. are not numbers, they have no place on the number line, so Dedekind need not have wasted his time trying to find a clever way to put them there. Being non-numbers, they live in a separate, non-number space of their own, a space where computer algorithms run. (And those algorithms run not &apos;to infinity&apos;, which is an infantile babble-phrase, but FOR EVER, which is a grown-up concept that actually works.)

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• TU

  @tullioaeb6284 · 1 year ago

  time <a href="https://www.youtube.com/watch?v=jlnBo3APRlU&amp;t=1869">31:09</a>: Dear Professor, in the Dedekind&apos;s cut set definition there is not the equal, whether when he se it for pi greek of Euler, is udìsed the equal. Why? Thank you Aebischer (Italy)