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Everything posted by uncool

  1. Again: the set of algebraic numbers is countable. There is a bijection between the set of integers and the set of algebraic numbers. And by making an infinite set of exponents, you are again making a map from a countable set (the set of pairs of bases and exponents). The image of a countable set under any map is countable. This isn’t that hard to prove - it’s not an axiom, it’s a theorem.
  2. Part of the point of “forcing” is that it is not constructing new sets within the same model - which is your approach - but instead constructing a new model of the real numbers.
  3. The set of algebraic numbers is countable; therefore, if the set of powers you use is also countable, then the set you get as a result will again be countable. In fact, the output set will be of the same cardinality as the exponent set, for whatever exponent set you choose (possibly subject to some rare exceptions).
  4. The difference between competitive sports and politics is that in competitive sports there’s a clear, objective measure of “better”. Tiger Woods has won because he did better on the golf course. There is no single objective “better” when it comes to the judiciary. There is no “best judge” that can be objectively determined, merely a pool of judges with better qualifications, among whom the President picks. So by picking a black woman from that pool, the President isn’t deliberately choosing a worse candidate - merely a particular one from among a pool of similarly qualified candidates.
  5. again: if a set can be indexed by pairs of natural numbers, then it is countable. i’ is countable, so it can be indexed by the set of countable numbers, so the set of all powers of the form xy where x and y are in i’ can be indexed by pairs of natural numbers, so that set is also countable.
  6. Still countable. Can be indexed by a triple of integers - numerator, denominator, and exponent - and the set of triples of integers is countable (for largely the same reason that the set of pairs of integers is countable). Again, there is nothing “inherently” uncountable about transcendental numbers. I see no specific reason to assume it would, unless you assume the negation of the continuum hypothesis to begin with. Which is your right, but you would then only have a circular argument.
  7. So it is a set that can be indexed by ordered pairs of integers, namely the base and the prime whose square root is the exponent. It will again be countable, as the set of ordered pairs of natural numbers has a bijection with the set of natural numbers. You keep trying to use the fact that these numbers are transcendental as if that gives you a natural connection to uncountability. It doesn’t. The fact that the set of all transcendentals is large doesn’t mean that any particular subset will be large.
  8. What do you mean by {ℕi'}? If you mean the set of all i’-indexed sequences of natural numbers, that set has cardinality equal to that of the continuum. It clearly has the same cardinality as ℕℕ, and it’s not hard to establish that that has the same cardinality as the reals.
  9. The set of all transcendentals has cardinality equal to that of the reals. The set of all square roots of primes has cardinality equal to that of the natural numbers, by rather obvious bijection. Neither of those have “intermediate” cardinality. If you are claiming to have constructed a set of intermediate cardinality, what is it? And your point is? Cardinality is not about underlying members. Just because the elements of a set are transcendentals and there are lots of transcendentals doesn’t mean that set is large. The cardinality of the set containing just pi is still 1.
  10. It’s trivial to construct a bijection between the set of prime numbers and the set of square roots of prime numbers. The set of prime numbers is obviously countable, so the set of square roots of prime numbers is countable, too. It sounds like you are trying to talk about the set of digits of the square roots of prime numbers. This is still countable, though slightly less obviously, but it is not the same as the set of square roots of prime numbers. It should be, if you can make clear statements. My point is that you have not done so. I have been: I can’t determine what statements you are proposing to be true.
  11. A better article would be able to articulate that axiom choice is much less about what’s true and more about what’s convenient. These mathematicians have come up with some axioms which work together in an apparently surprising way, and therefore may be more convenient. That doesn’t make them more true, merely more likely to be used in the future.
  12. You’ve returned to word salad; I can’t determine even what statements you are proposing to be true beyond the supposed conclusion. The pop-sci article is…poorly informed. This statement: makes no sense - one axiom implying another is not a way to gauge whether they are “more likely”. And the first half of the article keeps implying there’s a “battle” over whether the continuum hypothesis is “true”, before the second half reveals the actual state of affairs: it is independent of our current axiom set, meaning it could freely be true or false, depending on which model you choose.
  13. that’s less salad, at least. it’s clear enough to be simply wrong, rather than incomprehensible. the set of square roots of prime numbers has cardinality equal to that of the natural numbers.
  14. not a criminal filing it's a brief in a civil case where one of Trump's lawyers is trying to keep his documents out of Congress's hands those documents being the subject of a subpoena from the select committee to chapman university, which employed the lawyer
  15. Yell "Fire!" - it's worth noting that at least in the originating country of the relevant phrase ("shout 'Fire!' in a crowded theater"), the original case is outdated and generally considered superseded by a more permissive standard, especially since the original was a case of jailing people for opposing the draft. Slander someone - this is true, but the definition of "slander" is often heavily circumscribed (especially, again, in the US). Praise Nazis - that depends heavily on where in the Western world; again, in the US, you can praise Nazis. And to the extent that people can't elsewhere, I think that is a bad thing. Not because anyone should praise Nazis, but because the government should not be able to assume the power to prevent it. "If there is any fixed star in our constitutional constellation, it is that no official, high or petty, can prescribe what shall be orthodox in politics, nationalism, religion, or other matters of opinion, or force citizens to confess by word or act their faith therein."
  16. To be held criminally accountable. Rittenhouse may still be held civilly accountable to the estates of the deceased and to the injured (although self-defense is sure to play a role there, too). The general public can also freely choose whether it wants to associate with Rittenhouse or not.
  17. For Wisconsin, the statutory requirement with the assumption of provocation is: "A person who engages in unlawful conduct of a type likely to provoke others to attack him or her and thereby does provoke an attack is not entitled to claim the privilege of self-defense against such attack, except when the attack which ensues is of a type causing the person engaging in the unlawful conduct to reasonably believe that he or she is in imminent danger of death or great bodily harm. In such a case, the person engaging in the unlawful conduct is privileged to act in self-defense, but the person is not privileged to resort to the use of force intended or likely to cause death to the person's assailant unless the person reasonably believes he or she has exhausted every other reasonable means to escape from or otherwise avoid death or great bodily harm at the hands of his or her assailant. The privilege lost by provocation may be regained if the actor in good faith withdraws from the fight and gives adequate notice thereof to his or her assailant." with the caveat: "A person who provokes an attack, whether by lawful or unlawful conduct, with intent to use such an attack as an excuse to cause death or great bodily harm to his or her assailant is not entitled to claim the privilege of self-defense." https://docs.legis.wisconsin.gov/statutes/statutes/939/iii/48
  18. Yes, you can actively seek out situations where you are probably going to be threatened. This is a good thing; it is the basis, for example, for the Civil Rights marches, or for counterprotests at e.g. racist rallies. And in doing so, you do not lose the right to self-defense - whether with a firearm or otherwise. In Wisconsin, at least, you only lose your right to self-defense when you actively provoke an attack. Additionally: "stand your ground" is specifically about the duty to retreat. Kyle Rittenhouse did retreat, several times. Whether he did so enough may be up for debate. Further, Wisconsin does not have a stand-your-ground law.
  19. Um. If you mean Jacob Blake, then Rittenhouse didn't shoot or harm him. That was by police, and was the subject of the protest itself. Otherwise, none of the 4 people Rittenhouse shot at were paralyzed. He killed 2 (Rosenbaum and Huber), hit and "vaporized" the bicep of 1 (Grosskruetz), and missed 1 ("Jumpkick man", who was not identified during the trial). Not quite. According to the judge, the prosecution was not allowed to refer to them as victims, as that was the question the trial itself was answering.
  20. Let's start with this: "Since the conjecture is correct with n, we have B ≠ Ø." Your definition of B was: "B is also the set of the odd and prime numbers y which qualify the expression y < 2n". How does the conjecture relate to the nonemptiness of B (which, if anything is obviously true - there is always an odd number below 2*n)? It sounds like you're trying to define B based on summing, but you have not done so.
  21. Because the property (a + b)*c = a*c + b*c requires it. Or, to slightly modify what John Cuthber said: if you cancel someone else's debt, you are giving them money.
  22. So how is that n proven to exist in the Actor system? What you're saying reminds me of the fact that e.g. any strictly decreasing sequence of ordinals is, in fact, finite, even though (if the initial value is infinite) it can be arbitrarily long. However, I still don't see what it is about the Actor system that forces the "stop" message to be eventually acted on.
  23. I guess my question is, if the "stop" message can be postponed for an unbounded time, why it couldn't be postponed forever, analogous to the Turing machine algorithm.
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