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wtf last won the day on May 28

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  1. I found a picture of the mapping from (0,1] to (0,1). Instead of using the inverse powers of 2, it uses the sequence 1/2, 1/3, 1/4, 1/5, ... See how this works? The 1 at the right end of (0,1] gets mapped to 1/2. Then 1/2 gets mapped to 1/3; 1/3 gets mapped to 1/4, and so forth. The mapping is reversible so this is a bijection.
  2. Walk through it step by step. Convince yourself that the mapping is: * Injective: Different inputs go to different outputs; and * Surjective: Everything in the target set gets hit. By definition, the mapping is a bijection. Stepping through the details will make the idea clear. To see exactly how it works, I suggest using the method I showed at the end (instead of my solution). Map 0 to 1/2, map 1/2 to 1/4, 1/4 to 1/8, etc. What happens is that 0 gets absorbed and all the powers of 2 get slid down one space, mapping [0,1) to (0,1).
  3. 1/x is a bijection from (0,1) to (1, infinity). There's a bijection from [0,1) to (0,1). How do we do this? The rationals in (0,1) are countable so they can be enumerated as [math]r_1, r_2, r_3, \dots[/math]. Now you define a bijection [math]f: [0,1) \to (0,1)[/math] as follows: If [math]x[/math] is irrational, then [math]f(x) = x[/math]. That is, [math]f[/math] leaves all the irrationals unchanged. [math]f(0) = r_1[/math]. And [math]f(r_n) = r_{n+1}[/math]. In other words [math]f(r_1) = r_2, f(r_2) = r_3[/math], and so forth. Now you can see that every rationa
  4. I've wondered the same thing, but about the scientific rather than philosophical effects. We looked at the stars and drew pictures in the sand and worked out trigonometry and early theories of astronomy. What if everything else had been the same but the earth were covered by clouds? We'd still have gravity, but we couldn't look at the heavens. How would science have progressed? Hope this isn't too much of a thread hijack but it's a question that's long been on my mind.
  5. wtf

    prime dilemma

    Yes exactly. I suppose you'd have to subtract the two integers and see if the result is integer-divisible by the modulus. That seems like the most sensible way. Probably not the only way.
  6. wtf

    prime dilemma

    Great question. Yes they are "the same but a little different." In math, mod is an equivalence relation on the integers. We say that [math]a \equiv b \pmod n [/math] for two integers [math]a, b[/math] if it happens to be the case that [math]n | a - b[/math], where the vertical bar means "divides." So for example [math]5 \equiv 3 \pmod 2[/math] and [math]17 \equiv 5 \mod 3[/math]. You can verify that this is an equivalence relation: It's reflexive, symmetric, and transitive. It partitions the integers into [math]n[/math] pairwise-disjoint equivalence classes. It's a fundamental conce
  7. wtf

    prime dilemma

    I don't think they have names, but there's a famous theorem of Fermat that says an odd prime is the sum of two squares if and only if p = 1 (mod 4). https://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squares
  8. I think it's interesting that this question always gets asked in terms of the Internet, because that's the only complex computer system people have a daily experience of. But it's far from the most complex and mysterious computer system. If any computer system were to become self-aware, my bet would be the global supply chain. The system that moves raw materials from here to component factories there to integration sites somewhere else to distribution points somewhere else and ultimately puts a finished consumer good on the shelf at your local big box store, at a price point attractive to buye
  9. Yes, the Leibniz series is known to converge extremely slowly. Fun beginning programming project. I think he was continuing the series, not any previous conversation.
  10. Aha. I did help you. I am pointing out your specific error. You are not asserting the question of whether one set is an element of another. You are interpreting the left hand side as a string that specifies a set. That's your error. You're equivocating "n in N such that n is prime" as a specification, on the one hand, and as a set, on the other. That is YOUR error, not an error in logic books. The logic books DON'T make that error. If it's a string, then you have to take its Godel number to ask if it's prime. If it's a set, you don't. But you can't have it both ways at the same time.
  11. You replied before I edited that out. I actually don't think I can help. I agree with you that [math]\{n \in \mathbb N : n \text{ is prime}\} has a Gödel number, which in fact is not prime (since it will be the product of a bunch of prime powers). Aha I have a bit of insight. if [math]S[/math] is a statement in the language of set theory, let [math]G(S)[/math] be its Gödel number. Let [math]P[/math] be the set of primes. Then [math]G(\{n \in \mathbb N : n \text{ is prime}\}) \in P[/math] is a valid statement with a definite truth value, which in fact we can determine to be False.
  12. Strings talking about sets are not sets. Well the set of primes is a set. But the string "n in N such that n is prime" is not a set! I agree that the string has a Gödel number, and you can assert that collections of Gödel numbers are sets, but I don't think you can ask if "n in N such that n is prime" is an element of the set of primes. Well yes all the statements have Gödel numbers but I don't think it works this way. I'm out of my depth on Gödel numbering here so maybe you can explain why you think it works this way. I hope you get the mi
  13. Looking at your paper, I see some issues. 1. What is a "set definition?" Do you mean a predicate? But predicates don't define sets, that's Russell's paradox. So what do you mean by a set definition? 2. You have the expression [math]X \notin S(X)[/math]. But [math]X[/math] is a "set definition," whatever that is, and NOT a set; and [math]S(X)[/math] is a set. So your expression makes no sense, it's neither true nor false. 3. Likewise [math]M \in S(M)[/math], has the same problem. [math]M[/math] is a set definition and not a set; and [math]S(M)[/math] is a set. Perhaps you c
  14. This popped up on my radar this morning. "Finding primes using a parabola." Looks like it might be of interest to you. https://www.cantorsparadise.com/finding-primes-using-a-parabola-a939c6e8be53 First time I clicked on it the story appeared, then after that it said I needed a paid app. Not sure how this works but perhaps you'll be able to view it.
  15. You have not demonstrated this relationship for 5, 17, and 85. The best you can do is compute the sector area for 5 and 17 and then adjust the spiral constants after the fact. You'd then have to pick different constants for every pair of numbers. And you could do exactly the same for 4, 6, 24; or 8,10,80. First, the primality of the first two numbers isn't needed; and secondly, for each pair of radial lengths you need to adjust your spiral constant after the fact. Don't you see this? What are the spiral constants for 5 and 17?
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