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Everything posted by uncool

  1. Bit more than that; don't forget the inner product. I'd say that that's the central aspect of Hilbert spaces. Depends on what you mean by "aspect" or "attribute"; I'd also be careful about what you mean when you say "something like its own dimension". If you mean that each particle (and simplifying to quantum mechanics, where there is no particle creation or annihilation), then kind-of, but not really. To explain further, and in some formality: the state space for a combination of particles comes from the (completion of the) tensor product of the state spaces of the individual particles, not a direct sum. Sorry, but I don't really have a less technical explanation for what that means. I'd say that each eigenstate a QM "entity" can be in can be thought of somewhat separately, yes (until you get some time evolution that mixes between eigenstates). To my recollection (which is admittedly rusty), the use of "state" can be somewhat ambiguous; it can either be used to mean a single eigenstate (of an unspecified operator), or can mean an arbitrary mixture. In the latter case, I'd say that the answer to your question is "no"; for example, the the states |a>, (3/5) |a> + (4/5) |b>, and |b> cannot really be thought of as in "different dimensions".
  2. On what level do you want an answer? Because on one level, it was a bunch of wars and abolition movements. But that doesn't answer how all those wars and abolition movements came together. On another, it was a bunch of philosophical movements that led to ideas of liberty. But that doesn't answer why those movements came to the forefront. On another, it was a result of economic forces giving rise to systems that outcompeted slavery. But that doesn't answer why they hadn't happened earlier. And so on and so forth.
  3. Yes, that makes sense; I had read your statement as "So he could determine in principle that Alice has performed a measurement [instantly], but he wouldn't be able to tell which outcome Alice got until he made the measurement", because the second half of the statement was qualified by timing while the first half was unqualified. Sorry about that.
  4. ...if Bob could determine whether Alice has made a measurement, then Alice could send a message faster than light purely by choosing whether to perform measurements, if I'm understanding what you're saying correctly.
  5. ..."protected class" is a designation based on various US laws, most prominently the Civil Rights Act of 1964. And in that law, the classes are not restricted to the "oppressed" classes. Both "men" and "women" are classes that are protected, all religions (including Christianity) are protected, and so on. Explicitly: "All persons shall be entitled to the full and equal enjoyment of the goods, services, facilities, privileges, advantages, and accommodations of any place of public accommodation, as defined in this section, without discrimination or segregation on the ground of race, color, religion, or national origin.[...] All persons shall be entitled to be free, at any establishment or place, from discriniination or segregation of any kind on the ground of race, color, religion, or national origin, if such discrimination or segregation is or purports to be required by any law, statute, ordinance, regulation, rule, or order of a State or any agency or political subdivision thereof."
  6. ...or until it failed under internal stress, or until it was shattered by random space debris, or ... it would be an absurd undertaking on logistical terms alone, even before considering actual use. the largest artificial satellite that we have ever launched is the international space station, which is 73 meters by 109 meters. you're talking about a mile wide object. that's 1609 meters. then you have to consider the feasibility of using a mile-wide mirror. that is, you need a properly curved, smooth, reflective surface that achieves a specified position in orbit above earth. and I'm pretty sure swansont meant 6000 degrees as an absurd upper limit, not a reasonable estimate, although i'm not sure that all of the necessary assumptions for that theorem are met (i haven't checked thoroughly, though, so i'd default to trusting him!)
  7. https://www.law.cornell.edu/uscode/text/18/1512#c_2 "Corruptly" isn't statutorily defined, as far as I can tell; there is some case law on the definition. https://www.ce9.uscourts.gov/jury-instructions/node/732 If he urged people to the Capitol in an attempt to impede the counting of the electoral votes (which multiple courts have ruled is an official proceeding), and if he did so with some consciousness of wrongdoing (which is something that might be inferred), then yes, that is enough, if I've read it right.
  8. I read the question as a general question of whether Congress has any power to bring charges and punish under its own authority. But even with the interpretation of charges being specifically about the Jan 6 insurrection, I think a case could be made for Congress having the power to punish the insurrectionists itself. Not that that is what is happening, or even that it is likely to happen, but that that is within its power.
  9. While they are not the same, part of my point with the link is that Congress's power is not limited to finding someone in contempt, but may include the ability to prosecute those charges itself. The link I provided gives examples of both houses of Congress themself trying cases where someone was accused of attempting to bribe a member. From the link: "Ultimately, Mr. Anderson was found in contempt of Congress and was ordered to be reprimanded by the Speaker for the “outrage he committed” and discharged into the custody of the Sergeant-at-Arms. Mr. Anderson subsequently filed suit against Mr. Thomas Dunn, the Sergeant-at-Arms of the House, alleging assault, battery, and false imprisonment. [...] The Anderson decision [by the Supreme Court] indicates that Congress’s contempt power is centered on those actions committed in its presence that obstruct its deliberative proceedings. " Unless you are making a distinction between "bringing criminal charges" and imprisoning someone for contempt, I don't see how the context changes my point - not that this is likely to happen, but that Congress may have the power to do the latter.
  10. As an explanation of what is likely to happen, I think this is correct; as an explanation of what can happen, I think this is incorrect. Congres has the inherent power to hold people in contempt, and can do so without involving the executive branch whatsoever, and this can include fining or imprisoning the accused contemnor for the length of that session of Congress. And this can be for either a coercive purpose - trying to force the accused contemnor to do something - or a punitive one - punishing them for failing to do so. https://sgp.fas.org/crs/misc/RL34097.pdf With that said, it apparently hasn't happened since 1935, and was rare before then as well.
  11. I don't think that anyone really answered the question, because the argument the question is about uses assumptions that aren't that obvious. The statement "Force is proportional to mass" is somewhat imprecise; the fully precise statement is "Force is proportional to mass when acceleration is held constant." That is, if you have two masses which are accelerating in the same way, then the force on one divided by its mass will equal the force on the other divided by its mass. Alternatively, for any situation with forces and masses, F/m is a function of the acceleration. Similarly, the statement "Force is proportional to acceleration" is imprecise, with a more precise version being "Force is proportional to acceleration when mass is held constant." If you have two objects with the same mass, the force on one divided by its acceleration will equal the force on the other divided by its acceleration. F/a is a function of the mass. So we have that F/m = f(a) for some function f, and F/a = g(m) for some function g. Then F/ma = f(a)/a = g(m)/m. f(a)/a is also a function of acceleration independent of mass, and g(m)/m is also a function of mass independent of acceleration. And the only way that can happen is if f(a)/a = g(m)/m is constant - is independent of both mass and acceleration. Call that constant C. Then F = C*ma - in other words, force is proportional to the product of mass and acceleration.
  12. …no, NTuft, that is not why e^(i pi) = -1. Your blind substitution is not correct. You are claiming that e^(x*i*pi) = x cos(pi) + i x sin(pi) = -x, if I’ve remade your equations correctly. This simply isn’t true. It has nothing to do with the actual justification of the original equation (which has to do with McLaurin series), and is self-contradictory with a bit of thought.
  13. …none of this is accurate, and this can be checked directly by e.g. wolfram alpha. e^(i’ pi) = e^ (i sqrt(2) pi) = cos(sqrt(2) pi) + i sin(sqrt(2) pi) ~ -0.266 - 0.963 i
  14. …I have to admit, I’m…uneasy with this reasoning. It suggests that there is policy interest in who gets to reproduce based on speculation on future criminality, which has some connections with eugenics. Those connections strengthen when you add in the conservative framing of abortion being connected to the right of the fetus to life - under that framing, this argument suggests that some people don’t have a right to live because of speculative criminal activity. I’m not suggesting that this is what you meant to suggest. Just that it may have unintended implications, especially among people with other framings.
  15. Worth noting it was used on both “sides”: by slavery supporters to say that their “rights” to own slaves shouldn’t end when they visited free states (or when slaves successfully ran away to free states) and by abolitionists to say that African Americans should have the same right to be free in all states
  16. Additionally, the Senate could refuse to confirm any extra Supreme Court justices if it changed its mind after allowing more. It would be very unlikely, but the Senate has both direct power over the number of justices (through legislation) and indirect control (through confirmation). What are you defining as “rights” here? Does e.g. the right to buy alcohol count as a “right”? The right to buy guns? The existence of a patchwork of rights sounds like the same thing as different states having different laws. What you are looking for is probably covered by the “Full Faith and Credit” clause of the Constitution.
  17. 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.
  18. 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.
  19. 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).
  20. 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.
  21. 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.
  22. 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.
  23. 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.
  24. 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.
  25. 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.
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