a consequence of the axiom of choice is that every set can be given a well ordering. and well orderings always have smallest elements, but they may not have largest elements.
so there is someone who is the least gay, but there may not be a single person who is the most gay.
I'm not sure if I understand. There might not be the "fully" gay person but there is person(s) who is more gay than anyone else, thus making them gayest
By performing measure-preserving transformations to non-measurable sets and acting surprised when at the end of the day measure isn't preserved. I don't blame AC for that. AC only implies the existence of a non-measurable set, which is in itself not totally counter-intuitive.
Yeah, I'm really out of my element talking about these abstract mathematical topics... I'm too much of an empirical scientist it seems!
Give me a set and I'll find a way to measure it! ...actually I'd like to know what's the mathematical definition of measurement, there's probably a trick there... 🤔
Measure theory gets pretty abstract, so I recommend using a source thats not me. For the Real Numbers, the standard measure (usually the Lebesgue measure) is a way to generalize notions of lengths of intervals to sets that are not intervals or easy unions of intervals. For any set E, we can find various sequences of open intervals (potentially infinitely many intervals) whose union will contain E. The Lebesgue outer measure of E is defined as the infimum of the sum of the lengths of those sequences of intervals (infimum is similar to minimum except that an infimum may not actually be attained). The set E is considered Lebesgue measurable if it satisfies am additional criterion (Caratheodory criterion) in which case the Lebesgue measure is equal to the Lebesgue outer measure. If E does not satisfy that criterion, then it is non-measurable. A measure has desirable certain properties that an outer measure does not satisfy, which is why we prefer the former.
Apologies to all math people for any inaccuracies in the above description, it's been a while.
"Measure" is meant in the specific sense of measure theory. The prototypical example is the Lebesgue measure, which generalizes the intuitive definition of length, area, volume, etc. to N-dimensional space.
As a pseudo definition, we may assume:
The measure of a rectangle is its length times its width.
The measure of the disjoint union of two sets is the sum of their measures.
In 2), we can relax the assumption that the two sets are disjoint slightly, as long as the overlap is small (e.g. two rectangles overlapping on an edge). This suggests a definition for the measure of any set: cover it with rectangles and sum their areas. For most sets, the cover will not be exact, i.e. some rectangles will lie partially outside the set, but these inexact covers can always be refined by subdividing the overhanging rectangles. The (Lebesgue) measure of a set is then defined as the greatest lower bound of all possible such approximations by rectangles.
There are 2 edge cases that quickly arise with this definition. One is the case of zero measure: naturally, a finite set of points has measure zero, since you can cover each point with a rectangle of arbitrarily small area, hence the greatest lower bound is 0. One can cover any countably infinite set with rectangles of area epsilon/n^(2) so that the sum can be made arbitrarily small, too. Even less intuitively, an uncountably infinite and topologically dense set of points can have measure 0 too, e.g. the Cantor set.
The other edge case is the unmeasurable set. Above, I mentioned a subdivision process and defined the measure as the limit of that process. I took for granted that the limit exists. Indeed, it is hard to imagine otherwise, and that is precisely because under reasonably intuitive axioms (ZF + dependent choice) it is consistent to assume the limit always exists. If you take the full axiom of choice, you may "construct" a counterexample, e.g. the Vitali set. The necessity of the axiom of choice in defining this set ensures that it is difficult to gain any geometric intuition about it. Suffice it to say that the set is both too "substantial" to have measure 0, yet too "fragmented" to have any positive measure, and is therefore not well behaved enough to have a measure at all.
Well that's interesting: in order to define unmeasurable sets, you relied on the axiom of choice... I suppose it might be possible to define unmeasurable sets without AC, but maybe not!
Every time I encounter the axiom of choice implying a bunch of crazy stuff, it always loop back to requiring AC. It's like a bunch of evidence against AC!
I find it interesting that the basic description of AC sounds very plausible, but I'm still convinced mathematicians might have made the wrong choice... (See what i did there? 😄)
It's required, but nontrivially so. It has been proven that ZF + dependent choice is consistent with the assumption that all sets of reals are Lebesgue measurable.
you could think about it this way: one sphere and two spheres have the same “number” of points. (in the same way that there are just as many real numbers as there are real numbers in the interval (0,1).)
so, it becomes “”plausible”” that you could use one sphere to construct two spheres (because in some sense, you aren’t “adding any new points”).
but in the real world, “spheres” only have a finite number of atoms. so if we regard atoms as “points”, then it’s no longer true that one sphere and two spheres have the same number of “points”. and in some sense, this is why the sphere duplication trick doesn’t work in the real world.
it’s also worth mentioning that you have to do some pretty fucked up and unusual things in order to actually duplicate the sphere, and if you don’t allow such weird things to be done to the sphere, then it’s no longer possible to duplicate it, even with the axiom of choice.
The axiom of choice doesn't say one way or another whether the spectrum in "the standard order" (is there a standard definition of more/less gay?) is a well ordering, only that there is some well ordering.
yeah this is true. i should have clarified a bit better that a well ordering wouldn’t give you a “least gay” person in that sense of the word. it would be more correct to say there is a well ordering ⊰, and so there is a “⊰”-least gay person. but of course a “⊰”-least gay person could be in the middle of that spectrum.
but the number of people on earth is finite, so in fact the usual ordering is a well-ordering in this case. so i guess those two mistakes i made cancel each other out, and the axiom of choice isn’t even needed here.
How is a spectrum supposed to not have a total ordering? To me saying sth is a spectrum always invokes an image of being able to map to/represent the property as an interval (unbounded or bounded) which should always give it a total ordering right?
How is a spectrum supposed to not have a total ordering?
I'm pretty sure a spectrum is always totally ordered. You can't say "this point on the spectrum holds no relation to that point", because then it's not a spectrum.
You can impose a partial ordering on it. HSL uses a hue angle. If we assume full saturation and lightness and pick an arbitrary direction to be positive, there's your partial ordering. But not total ordering, there is no most clockwise color.
I feel like this doesn't qualify as an ordering relationship, because of the circular nature of the wheel: for any two elements a and b (a ≠ b) on the wheel it's both true that a is further clockwise than b and b is further clockwise than a (just keep rotating). This violates the antisymmetry property that an ordering relation should have.
You can fix it by establishing some point on the wheel as "least clockwise" (essentially unfolding it into just a straight line) but that immediately establishes a total ordering.
Ig thats where most of my confusion comes from, to me saying sth is a "spectrum" always evokes sth along the lines of gay <--------------------> straight (ie one dimensional) with things mapping into this interval. But ig if you also include more than one axis in your meaning of "spectrum" there wouldn't be as straight forward of an ordering for any given "spectrum". + Like @saigot said technically even the 1 dimensional spectrum can have more than one order and the "obvious" one is just obvious because we are used to it from another context not because its specifically relevant to this situation.
Gynophilia, androphilia, romantic attraction, sexual attraction etc. absolutely makes everything complicated yeah. And then there's cultural stuff and minor personal preferences. There's no real end to how many axis you could legitimately argue for including in a sexuality chart.
It all comes down to definitions. First off, Totally Ordered is a property of the function that compares two elements not the set you are talking about. most sets have total orderings (if the axiom of choice is true then all sets have a total ordering). With Fields and vectorspaces there is the concept of a totally ordered Field which is essentially when the total ordering is compatible with it's field operations (e.g the set of complex numbers has many total orderings, but the field of complex numbers is not an ordered field).
So it really depends on how we define the sexuality spectrum. So long as it's simply a set then it has a total ordering. But if we allow us to add and multiply the gays then depending on how we define those functions it could be impossible to order the gay field.
Also a total ordering doesn't mean that there is exactly 1 maximal element (it would need to be a strict total ordering to have that property), so we can all be the gayest.
Also the definition of 'gay' and 'gayest' is poorly defined. This assumes that gay is some sort of scalar, where in reality it's a projection from a multidimensional 'queerspace' that can change the appearance of the spectrum wildly depending on the methodology the one projecting uses.
I think that same as you can't get an arbitrarily accurate measurement of length, since at sub-atomic scales you can no longer perfectly define where exactly an object ends (and eventually you'll reach planck scales and quantum foam and the task becomes even more impossible), it might not be possible to measure gayness with enough accuracy to decide between the most gay contenders. Let's be real - gayness is probably extremely fuzzy if you look into it closely.
Wouldn't it be like 9.99 into infinity? 🤔 and since the human population (at least currently living) is not infinete, then at some 9.999999 there wouldn't be anyone with a higher value?
(I don't know math)
Even in theoretical math, 0.999 repeating ends up being exactly equal to 1. In fact, any terminating decimal can be rewritten in a similar manner. For example, 0.25 is exactly equal to 0.24999999 repeating
No, but each individual human is assigned identically one gayness value, therefore the number of values we must sort is equal to the number of living humans
But the possible number of outcomes is not limited by the subset of living humans. While we may have a currently highest number that doesn’t mean it IS the highest possible, nor that there is exactly one of them.
While we may have a currently highest number that doesn’t mean it IS the highest possible
I would argue you can only be gay if you're alive, so you only need to compare living people, the theoretical maximum doesn't matter, only the actual maximum of a finite number.
It is true that the gayest person currently alive, the gayest person ever in history, and the gayest person who could possibly exist may well exhibit three different levels of gayness; however, I believe that, were one sufficiently determined, it would be possible to find all three.
Okay, so Earth exists. This means for a set volume of space (say about the size of the solar system) there is some positive probability that it contains a planet that is indistinguishable from earth. Let's assume the universe is infinite. If we can search an arbitrary volume instantly, our probability for finding a duplicate of earth approaches 1 as our volume increases. This means the probability we will never find a duplicate of earth is exactly 0, which means that we will find a duplicate upon searching a finite volume. Since in our hypothetical the search is instant, we can perform this search again, locating a second duplicate of earth. Following this process, we can locate an arbitrary number of perfect earth duplicates in a finite ammount of time. This means that if Earth arose from natural processes in an infinite universe, there are infinitely many exact duplicates of earth with life that includes specimens genetically identical to humans.
This implies that there is no one gayest person in the universe.
Most of the entertainment comes from the non-obvious corollaries of the intermediate results. For example, every person has infinitely many perfect yet distinct copies that are identical right down to personal histories.
There's more, if you care to take the time to think about them. I was just using the conversation as an excuse to expose more people to the implication.
There are an infinite number of numbers between 0 and 1, and yet there is no repetition. Pi and other irrational numbers are infinite yet non-repeating. I wish I knew the name for this kind of thing because I'm sure it's been discussed in philosophy (a kind of opposite, eternal recurrence, has been discussed a lot).
I don't think anyone knows enough about the universe to say whether or not there is infinite variety in macroscopic stuff, so I don't think anything can be ruled out.
I don't think anyone knows enough about the universe to say whether or not there is infinite variety in macroscopic stuff
There are finitely many particles in the observable universe (that is to say, an infinite number will not fit), and a finite granularity to discern the position of those particles. That means there are finitely many configurations of particles within the volume of the observable universe.
Therefore, there are finitely many discernable things, so in a meaningful sense we can say with confidence that there's a finite variety of macroscopic things.
Whether or not an infinite number of particles will fit or not is not important, no ? I'm not sure what you mean by finite granularity. There is no "grid", space is continuous, the planck length and the fact that push on each other doesn't really factor in. By virtue of space being continuous and particles being finite, means you can configure stuff in infinite ways.
Edit: Not quoting you with the reference to a grid. I know that's not what you mean.
Yes, but distance is still continuous, a minimum measurable distance (between stuff) doesn't make space granular. I suppose there might be a minimum measurably meaningful number of configurations, but I'm not super convinced.
My claim accounts for the possibility that the distances of particles may physically differ by amounts more granular than a plank length. My statement was that they are indiscernable. There are infinite copys of every person more closely identical than the two most similar identical twins. So closely identical that no physically possible device could ever distinguish between them. We cannot know if space is continuous. We simply know that if it is not continuous, it is of granularity as fine or finer than the plank length. So there is a meaningful sense in which there are finitely many macroscopic objects.
So far nothing in the universe has shown to be infinite, hence any material representation of Gabriel's Horn can be painted since paint has a thickness.
The Plank length is the shortest possible distance between two material points. Though at that scale even vacuum is tempestuous.
Even a progression from "Closed Source" to "Source Available" is nice progress I think.
If we assume that the License is not restrictive we may be able to fork Winamp into a codebase that might actually be Fully Open Source Software and track changes of the upstream as we need.
Just tried the mobile version of Winamp looking for a better Android music player. The interface looks great tbh, but man, they got no support for wma files and some of the tags are not read correctly (on the songs I own). I know this is not an issue for most of the people, but for me, it's really limiting on the music players I can use on my phone. Sadly, Winamp doesn't fit in this category either. Thank god there is Foobar that supports all these from day 1
try Poweramp, it's paid but it costs like 10 cents and comes with a 3 day trial (no subscriptions or other bs)
definitely the most feature packed player with no competition
Thanks! I'll check it out. Foobar is fine and does a lot of things that I do not see other players doing. What I didn't like was the fact that it has no adaptive icon and there is no way to display the lyrics of a song. Plus that I was looking for something that was following the material design guidelines more.
Edit: just checked it. Seems like it's not properly reading some artist tags, so it cannot find them:
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