Comments by "MC116" (@angelmendez-rivera351) on "Numberphile2" channel.

  1. Φ is pronounced /fi/ in every single major language. English speakers are literally the only people in the planet who refuse to pronounce it correctly.
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  2. Trying to solve the Collatz conjecture simplifies to answering the question, "Can you get to 1 if we start from any odd number?", simply because that if we start with an even number, it will go to 1 if it is a power of 2, and it will go to an odd number otherwise. Furthermore, we know odd numbers are of the form m = 2n - 1 for all natural n. Hence 3(2n - 1) + 1 = 6n - 2 is always an even number, so we can divide by 2, obtaining 3n - 1 for all n. Then, every odd number of the form 2m - 1 = (2^p + 1)/3 is guaranteed to be go to 1 eventually since 3m - 1 = 2^p, and powers of 2 are guaranteed to go to 1. Thus, now we are interested in the question, "Starting from any odd number, can we always get to a number of the form 2(2^n + 1)/3 - 1 = 2^(n + 1)/3 + 2/3 - 1 = (2^(n + 1) - 1)/3 for some n?" An affirmative answer would prove the Collatz conjecture. Now, is this not beautiful?
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  3. He never claimed that was the largest number to store. He said this number expresses a cap on the data in terms of the area of the universe.
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  4. Mike Wagner TREE(n) for all n can be proven to be finite for all n in transfinite arithmetic, but not with finite arithmetic. However, for each individual value of n = k, a theorem stating that TREE(k) is finite exists in finite arithemetic, but this proof would be impossible to complete, it would take "too long" in a rigorous sense.
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  5. Cat Life I have. English is a terrible language IMO.
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  6. Ian moseley That is not atomic physics per se, and the uncertainty principle has nothing to do with infinitesimal numbers. In fact, the statistics of the uncertainty principle come from real-valued mathematics with no infinitesimal numbers in it.
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  7. No, the amount of seconds will be even less than g(64). Actually, it's smaller than g(1). You are REALLY underestimating how stupidly large g(1). This number cannot be expressed using exponentiation alone.
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  8. kazedcat TM No. First of all, it has never been confirmed a multiverse exists. Second, that multiverse you describe is a type II multiverse, but again, there is no guarantee of its existence.
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  9. Generic Internetter Not at all. There are no infinitesimals in sub-atomic physics specifically.
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  10. abj So it is not a number. Cool
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  11. maciej53 The hyperreal numbers are OBJECTIVELY SIMPLER than the ε-δ definition of limits. The set of axioms and definitions contains less quantifiers and the conditionals contain less connectives, and most proofs are deductively shorter. It literally is simpler by definition. The hyperreal numbers are an objectively better and more useful way of doing mathematics with exactly the same level of formal axiomatization and rigor.
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  12. PiaNoONE They are not going back to a messy definition. If anything, limits are messier than infinitesimals, AND at best, only equally as rigorous as the hyperreal numbers. Abraham Robinson proved this. The proofs are simpler, the axioms are simpler as they require less connectives and quantifiers, and they are objectively more useful as there is more that can be done with infinitesimals.
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  13. Guy Yes, you can. It's called calculus.
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  14. Ethan Smoller The surreal numbers are a proper class, not a set.
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  15. Jamie Den Adel Mmmmm... no, it is not.
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  16. Zathras Yes That is not how physical existence works.
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  17. Michoss9 TREE(3), not tree(3). tree(3) is a different but related quantity.
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  18. Also, if you really want to talk about physics here, the consensus is that it is not meaningful to talk about the universe beyond the observable universe. For scientific purposes, it does not exist. We can make hypothesis all the time, but hypothesis is not science.
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  19. Kelly Stratton 10^10^10^10^10 is an upper bound to the Poincaré recurrence time of what in physics is called an "empty, vast universe," which represents an universe bubble several orders of magnitude larger than our own observable universe. This is to say, if we had to consider the amount of time it would take for the universe to reset itself and traverse every single quantum microstate possible, then 10^10^10^10^10 planck units of time is an upper bound. This number must obviously be bigger than the number of possible microstates of the universe, which is in itself bigger than the biggest possible number that can be encoded in the universe.
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  20. erik2000 There i no such a thing as "the whole universe," that's ill defined.
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  21. Alazoral Also, we can almost be certain Everett's interpretation is not true. Sure, it makes sense as an interpretation of QM, but once you delve into the more accurate description of reality, QFT, the concept of the Everett multiverse ceases to be coherent. People need to abandon this whole notion of "interpretations of quantum mechanics," as they are extremely inaccurate and outdated.
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  22. Alazoral I'm not sure why you are making a philosophical argument on a topic that is strictly physical. Also, bold of you to claim that angles cannot be quantized without proof. If that is the game you want to play, then nobody has any reason to give proof to anything.
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  23. Luis Eduardo HD That is more complicated as there are too many combinations to consider. Exhaustive case proofs are worse than analysis proofs such as this one.
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  24. Tim Williams Because mathematicians are the worst at nomenclature.
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  25. FBI Wrong. 1/k does have a a monad, and the monad is given by the set of all numbers of the form 1/k + r/k^2, where r is a real standard number.
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  26. Fabrice Nonez No, that is wrong. The hierarchy of infinitesimals in the hyperreal numbers ALREADY is infinite by definition. The monad of 1/K is 1/K + r/K^2 where r is real, and this is different from the monad of 0 in that it is strictly a subset.
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  27. Thomas Bridge The power set of the countable infinite cardinal K is, to be precise, 2^K, and this is strictly bigger than K. However, it is not known and it is furthermore not provable that 2^K is equal to the cardinality of the real numbers.
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  28. Jacob Marburger R* is the set of hyperreal numbers, and N* the set of hypernatural numbers. The construction can be found very easily if you search “hyperreal numbers” on Google. The theory is super rigorous and expansive.
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  29. Jaap Versteegh Yes.
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  30. aman makhija Incorrect description. The fact that two numbers share the same monad does not disprove the idea of both numbers still having a monad.
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  31. particle No. Incorrect. The monad of 1/e is simply the set of numbers of the form 1/e + 1/e^2*r, where r is a real number.
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  32. Grugknuckle It is not useless for everyday life. Maybe get more educated on the subject?
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  33. Matti Kauppinen No, 0.(9) and 1 are THE SAME NUMBER in every Euclidean number system, even in the surreal numbers, which contain the hyperreal numbers. This is true by definition. There is no describable or effectively constructible and formally axiomatizable number system which preserves the definition of rational numbers and addition such that 0.(9) is infinitely close to 1 yet not equal to 1. There exists a number which is infinitely close to 1 of the form 1 - ε, but whatever this number is, it cannot be 0.(9), because infinitesimal numbers, by construction, have no finite or infinite decimal expansion.
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  34. Martin Savc The real numbers are not limited to fractional notation. This is why 1/K cannot be real.
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  35. Fabrice Nonez 0.(9) = 1 in the real numbers, in the hyperreal numbers, and in the surreal numbers, because 0.(9) by construction is merely a real number whose infinitesimal part is 0. There are multiple ways of dealing with irrational numbers and infinite decimal expansions for real numbers without ever needing to invoke the limit. In fact, it would be somewhat of a mistake to invoke it. You can appeal to the Dedekind construction of the real numbers and prove that the number is 1, not because the infinitesimal part of it cannot be described, but because THERE IS NO INFINITESIMAL PART.
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  36. Tau being the better circle constant is completely debatable.
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  37. notoriouswhitemoth Between any two hyperreal numbers, there are infinitely many surreal numbers. Between any two surreal numbers, there are infinitely many surreal numbers, because the set of surreal numbers has the biggest cardinality of any ordered field and the longest measure. The cardinality of the surreal numbers is the same as that of the universe of set theory.
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  38. Giannhs Polychronopoulos You are asking if it is possible? Possible it is, but very difficult. I think the key to the fact is that the surreal numbers technically do not form a set. They’re too much to be able to form one. Instead, they form what is called a class. Though you cannot have a set of all sets, you can have a class of all ZFC sets, for instance. These classes form a more general type of object than a set. It is in the study of clases that you can prove a statement such as the cardinality of the class of surreal numbers is that of the set theoretic universe. Edit: I know I did use the term set in my first comment, which I realize is misleading. I did not want to delve into the technicalities initially, and using the word “collection” instead does not feel better, though.
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  39. Giannhs Polychronopoulos ah, I don’t the know the specifics, to be honest. I watched John Conway’s lecture about games and how his theory of games led to his discovery of the surreal numbers. However, the class of games is a super-class of the surreal numbers and is said to be “bigger” in some sense. In this lecture, he did say specifically the cardinal number describing the class of surreal numbers is that of the set theoretic universe. He did not specify an explanation of proof of this, which I imagine he intended for us to take for granted in the lecture, since the lecture is not about classes and cardinalities in general. I have read online that the surreal numbers form a class, not a set, but I don’t know of any specific books to read. Actually, I’m looking for books of my own myself to read about it too. I can link you his lecture, though, if it helps
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  40. rotwang2000 Actually, the universe will just reset itself before this happens, making this never happen.
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  41. Alexander Sanchez Now, you're just claiming the best one could hope to do is 3d -2 rows without any proof, inmediately after declaring that 7 MIGHT be possible. How am I supposed to take that contradiction seriously?
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  42. There is no faster speed of light, though.
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  43. Andrew Cunningham In, fact, even a number such as 10^10^10^10^10 is unfathomable. It is beyond the realm of comprehensibility of anyone, depending on how you comprehensibility.
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  44. Archit Siby Since the fifth row is being excluded, the sum of the four rows in those cases is just less, and the sum below the line is bigger.
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  45. Arnaz87 Because the idea of "the universe" is completely meaningless, it's like talking about 1/0. The only thing that is known to exist is the observable universe, so the only thing we are logically allowed to pretend exists is the observable universe.
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  46. Belsaros Well not quite, because it is nothing like Zeno's paradox, and they also neglected other conditions. Zeno's paradox didn't involve actions that REQUIRED supertasks. This game REQUIRES a supertask. Hence the difference. And games by definition cannot be played with supertasks. That is the problem. In Zeno's paradox, the amount of segments that you have to travel is infinite, but the sum of the segments is finite AND and the sum of the times to travel each segment is also finite at constant speed. So it's exactly equivalent to a non-supertask. Technically it's not a supertask at all. Not so with the game.
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  47. The hyperreal numbers are useful in physics, economics, and game theory.
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  48. Jorge C. M. That's definitely not what the rules of the checker game are.
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  49. L4Vo5 Not quite. They did not consider limitations to what happens when pieces move. They did not need to, after all. The point is that having the sums be equal is necessary but not sufficient. Much like in the pebbling the chessboard game.
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  50. Perplaxus 10^122 is the binary size of the biggest number that can be stored in the universe, not the number itself. But what is the biggest number we can represent? There is no such biggest number, since you can make arbitrary notation to represent arbitrarily large finite quantities, and that does not even account for transfinite quantities, of which there is no largest representable member. So it is not sensible question. Instead, the question that makes sense is the largest number that can be stored in the universe, which is what was addressed in the video.
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