OK, so now I will ask you to consider something (not accept, just consider). People have been studying set theory formally for more than a century. In the mean time, many other fields, like theoretical computer science, rely on the diagonalization argument extensively, and we have some foundational results, like the undecidability of the Halting problem, are proven in a very similar manner. It's possible to prove the uncountability of the reals without it, as Cantor did, before he came up with the diagonalization proof.
Now, consider something before you put large quotes around the proof word: what is more probable? That the collective minds of mathematicians of various generations spanning hundreds of years is wrong? Or that you are the one who's not understanding something?
You are essentially saying that I should accept it without understanding it, and without understanding where the mistakes in my thinking are?
Asking questions about something does not mean "I don't believe it to be true". It simply means "I don't understand this particular thing about it".
Shouldn't it be the exact reverse? I mean in the cases of runs that have been console-verified? Shouldn't it be the console version that's considered the main primary version, with perhaps an ancillary version of the run that has been "fixed" to run on the emulator, if necessary?
TASes on the site do not aim to only work on console.
That's why I said "in the cases of runs that have been console-verified". If a run has not been console-verified, then it would need to have only the emulator version.
The other problem is people who want to play back these tases will not know that the main file is actually the console verified one, where it might not sync. It’s just a step that is completely unnecessary and would cause confusion, especially to encoders looking to re-encode older movies.
Have two links, one for the version that works on the console, and one that works on the emulator. The official time should be that of the console.
Instead, what you're doing is, you are reassigning every real number to a natural number, which unlike the reals, must have a terminating digit expansion, and apply something analogous to the diagonal argument there. Of course it does not work, because if it did, you would prove that any infinite set is uncountable, you have not used any property of the reals that distinguish them from an arbitrary infinite set.
But the thing is, the argument (ie. the proof by contradiction) starts with the assumption that the set of reals (or, more precisely, the set of all infinite strings) is countable, ie. enumerable, ie. listable. It has to make this assumption, or else it cannot say "take the first digit of the first string, the second digit of the second string, etc."
So my thinking is: If we assume that the list is countable, then it shouldn't make a difference if we replace every element with a natural number (eg. its position in the list). It shouldn't make any difference what we replace every digit of every element in the list with, as long as the resulting remapping is unique. From the point of view of the proof, it shouldn't make a difference.
So if we do that, we are replacing the original list of infinite strings with a list of finite strings, and the only thing that the diagonalization does is to tell us "there are no infinite strings in the list", which is self-evident to the point of being tautological. Of course there aren't, because we replaced them with finite lists in the first place. The only thing that the "proof" is now saying is, in essence, "if you replace all infinite strings with finite strings, there will be no infinite strings."
(In addition, if we do that remapping, and we use eg. base 10 strings, the diagonalization will produce a rational number, because it will have an infinitely repeating digit. Ostensibly rational numbers were in our original list, so it didn't really create a new number.)
If the counter-argument is "you can't replace all the infinite strings with finite ones in a unique way" you'll have to explain why. The answer cannot be "because the set of all infinite strings is uncountable" because that's precisely what we are trying to prove, and we cannot assume what we are trying to prove.
What bothers me (not a whole lot, just a bit, but anyway) is that it appears to me (and please correct me if I have misunderstood) that the version of the run that works on the emulator seems to be considered the "main" "primary" "official" version, and this version will only get the "console-verified" badge if it has been demonstrated to run on the original console... even if some minor modifications to the run are required (such as adding some lag frames here and there). In other words, the "fixed" version of the run seems to be considered some kind of secondary optional extra, perhaps provided alongside the actual primary emulator version.
In other words, it's as if the emulator is considered the main authority of how the TAS should work, with an ancillary side variant of it that's fixed to work on the actual console.
Shouldn't it be the exact reverse? I mean in the cases of runs that have been console-verified? Shouldn't it be the console version that's considered the main primary version, with perhaps an ancillary version of the run that has been "fixed" to run on the emulator, if necessary?
One thing where this would make quite a visible difference is if the two versions of the run are of different lengths, in frames. Shouldn't the official time be that of the console, not that of the emulator, if they differ?
First, for some reason you stick with this way of ordering. But it doesn't list all real numbers, because as I said initial statement allows to list any real number. And many real numbers has inifinite binary digits. For example 1/5 is 0.110011001100 (period 1100) but it's not in your list. None of your numbers has infinity representation. For each of them there is position from which all digits are zero.
Since I don't have an advanced maths background, I'm not very good at explaining my thinking, but I will try.
If I understand correctly, Cantor's diagonal argument is a proof by contradiction (of sorts). It says, essentially: Let's assume that the set of real numbers is countable. This means we can list all the numbers. We proceed to show that using this list we can construct a number that does not exist in the list. Thus the original assumption that the list contains all the real numbers was incorrect.
The thing is, if we make the assumption that the set is countable, and thus we can list all of them in a particular order, it shouldn't matter if we "remap" the numbers to the natural numbers. If we are proving that there are numbers that cannot exist in this countable set, then it shouldn't matter in which order we list the numbers in question, or what the values of those numbers are (as long as every one of them has a unique value).
1/5 is in the list. It's just that it has been "remapped" to eg. the value 16 (using the Calkin-Wilf sequence). If we are just trying to demonstrate that there are numbers that do not exist in the set of rational numbers, this shouldn't make any difference.
If the counter-argument is "but you can't do that with irrational numbers like sqrt(2) or pi", then you are already assuming that irrational numbers are not countable and thus cannot be listed, and thus cannot be "remapped" to the natural numbers. So you are already assuming what you are trying to prove (ie. that the set is uncountable and thus unindexable), which makes the argument circular and contradictory.
Its basic usage is poof that real numbers from 0 to 1 are uncountable.
I think there's a contradiction in the argument, then.
That's because the argument starts with the assumption that you can list all the numbers between 0 and 1 in some order (because if you couldn't, then it would be impossible to "choose the first digit of the first number, the second digit of the second number, and so on".)
If you can list all the numbers in the set as a list, that means that you can assign a natural number to it: Its position on the list.
I don't see any reason why you couldn't list the numbers in their base-2 increasing order... which then results in the "diagonal" number being 1111111... (or 0.111111... which would just be 1), which doesn't really tell us anything.
(Most particularly, if you were to list the numbers in the order 0.000..., 0.100... 0.010..., 0.110... and so on, and you included 1.000... in the list, the "diagonal number" would give you 0.111111... which is just 1.0. A number that's already in the list. Thus the diagonal argument actually fails to give a new number in this case.)
Thinking about Cantor's diagonal argument, I realized that I don't actually understand how it proves anything about there existing uncountable sets.
Any countable set can be "indexed", ie. every element in the set can be assigned a unique natural number. Thus we can represent any countable set as simply the list of natural numbers, in order. So we list them eg. in binary, with the least-significant digit first:
0 = 0000000...
1 = 1000000...
2 = 0100000...
3 = 1100000...
4 = 0010000...
5 = 1010000...
etc.
Cantor's diagonal argument takes the first digit of the first number, the second digit of the second number and so on and so forth, and inverts them, and constructs a new number that "doesn't exist in the set":
X = 1111111...
But this is just a number with infinitely many 1-digits in it. Which technically speaking is infinity. But one could argue whether this is an actual number at all. Sure this number with an infinite number of 1-digits doesn't exist in the set of natural numbers... but that isn't very surprising nor telling, because this number is just "infinity". Infinity isn't actually a number (not a natural number, not a real number).
So I don't really understand how this proves anything. It's just essentially saying that infinity is not part of the set of natural numbers. Which is true by definition. It doesn't say anything about uncountable sets.
I suppose it makes intuitive sense that if there's an infinite set of numbers that's "smaller" than the natural numbers, it must be a countable set (because if it were uncountable, then it would pretty much by definition be larger than the natural numbers, not smaller, I think.)
And if it's countable, it means that every element can be assigned an index number. Which, all in itself, provides a 1-to-1 mapping, ie. bijection, between this set and the natural numbers.
But I was thinking that, perhaps, it could be possible to construct a set of size aleph-0 or "smaller" (if that's even possible) where it's not possible to index the elements because there's no "least element" that can be chosen or pointed out.
(But, then, perhaps it's not possible to construct such a set. Maybe it's a bit like the rational numbers: At first it looks like the rational numbers would be this kind of set: You can't eg. decide that 0 is the "first" rational number and then choose the next one to be "the smallest rational number that's larger than 0" because such a number doesn't exist. However, there are other ways to unambiguously assign unique indices to each distinct rational number.)
Thinking about infinite sets, a question rose to mind.
The continuum hypothesis famously posits that there is no set whose cardinality is strictly between that of the integers and the real numbers.
This got me thinking: It seems to be kind of taken for granted that the set of integers is the "smallest" possible infinite set. Why?
Would the question "is there an infinite set whose cardinality is strictly smaller than that of the integers?" be something completely ridiculous to ask? Is the answer self-evidently "no"? Would it be as ridiculous as asking if there's a smaller non-negative number than zero?
I tried reading about cardinality and aleph numbers, but I didn't really find something that would definitely state that there cannot be an infinite set which cardinality is strictly smaller than that of the integers. Maybe because it's such a self-evident thing that it isn't even worth mentioning? But is it?
I think that if a particular TAS file does not sync in the actual console without modifications, then it should not be marked as "console-verified" because, well, it isn't. If it doesn't run on the console, then it doesn't, period.
If it can be made to work on the real console by modifying it, eg. by adding or removing some input, then it should be this new TAS file that ought to be published alongside the original, and only this one should get the "console-verified" mark, because this is the TAS file that actually works on the console.
If this modified version which works on the console does not work on the emulator because of deficient emulation, then I think the one that works on the console should be considered the primary version of the TAS, and the one that works on the emulator should be considered an ancillary version of the TAS that's "fixed" to work on the emulator (at least until the emulator is fixed to emulate the console better). I think it would be a bit backwards to consider the file that works on the emulator to be the primary main version, and the one that works on the real console the secondary version, as if it were the emulator that's like the highest authority on how the original game binary should be run, rather than the original console hardware.
So I just learned about this mouse, and it's on sale: https://amzn.to/2ZAHLHj
I placed my order, so I'll get it whenever Amazon figures out how to ship it.
It appears to be at this moment cheaper at Microsoft's own store ($39.23), and I think they offer free shipping (even internationally). (I once bought an Xbox One controller wireless USB adaptor for the PC, and they shipped it for free to Finland.)
On that note, the mouse I recommended, the Logitech G402, seems to also be within the price range at Amazon ($42).
Fits large hands nicely, should have a nice size to it.
Has some weight to it, none of this lightweight junk that you can barely tell you're moving it.
Has a body, so you can actually wrap your hand around it.
Has a decent wheel, scrolls nice and smoothly and can click.
Wired, USB.
At least two buttons besides the wheel.
1000 DPI or higher.
No craziness around the sides, dozens of buttons.
Primarily for right hand use, but should be able to be used left handed as well.
Price is $50 or less (really shouldn't be more than $15, but junk these days...)
Good quality, will last several years without issues.
I can only speak from experience about the mouse I have, which is the Logitech G402.
I think it fits nicely all your criteria, except perhaps the "no craziness around the sides, dozens of buttons" part. But I don't think that's an issue. The extra buttons don't get in the way, and sometimes there are actually nice uses for them (eg. I like one of the thumb buttons working as a web browser "back" button. Surprisingly handy.)
Just so we're in the clear, I will not attempt to explain why only in terms of arithmetic for the following reason: there seems to be a widespread misconception that everything mathematicians do should be neatly expressible using numbers, addition, and multiplication.
I did not ask because I think that everything should be expressible using arithmetic. I asked because my strongest knowledge and experience of mathematics is, essentially, high-school level math, or what I would call "practical math" (arithmetic, elementary algebra, analytic geometry, basic trigonometry). Once you start throwing integrals, derivatives and infinite summations and products into the mix, it starts going over my head fast. (I know how to derive most stuff, and I remember something from high-school and university integration lessons, but that's about it.)
In other words, a very esoteric answer using advanced postgrad math wouldn't be very useful of an answer for me, personally.
The discussion that we had, which was totally reasonable, was whether it's possible to arrive at the result 1+2+3+...=-1/12 without complex analysis, which I think is totally possible. There is this derivation, common in physics, where one introduces an exponential regulator. It's possible to define a metric where functions that differ by a pole of the form 1/epsilon^2 are very close together, so that the summation does, in fact, converge to -1/12.
If the sum of the natural numbers is -1/12 according to these summation methods, why is the sum of the reciprocals of the natural numbers infinity?
(If you could keep the answer as close to arithmetic as possible, I would be grateful.)
Wow...I was literally just going to post this video in here. Really darn mental!
I have to confess I fell for the conspiracy theory that Disney plagiarized Kimba the White Lion. I somewhat pride myself for being knowledgeable about conspiracy theories and detecting them when I see them and thus being quite immune to them when I see the signs. In this case, however, they got me. I swallowed it hook, line and sinker. I didn't see any warning signs, and I just believed it. The videos made about the subject were incredibly convincing and I just allowed myself to believe them without doing any research for myself.
That debunking video is worth every minute of its 2+ hour length.
Wouldn't it be a contradictory concept to state that a non-convergent sum converges to a finite value? It's either convergent or non-convergent. It cannot possibly be both at the same time.
A submitter is allowed to write anything they wish in their submission text, as long as it does not break the site rules regarding adult content, hate speech content, other uncivil material, etc. This includes any sort of political content, regardless of whether it's left-wing or right-wing or if it tackles any sort of specific political issue, as long as, again, it does not violate the site rules. TASVideos is not a political arbiter.
That being said, the submission topic is for discussing the submitted movie itself, not politics. Therefore, discussion of the political statement itself should be done in another topic.
I find those two paragraphs to be contradictory. Why is a TAS submission message allowed to contain off-topic political messages that are unrelated to the TAS itself, while the discussion topic is restricted to only discussing the TAS itself and nothing else? Why doesn't the same rule apply to both, given that both are related to the exact same thing (ie. a TAS that has just been submitted)?
The major problem in allowing political messages to be expressed in such text is that political messages, especially of the modern nature, tend to be very provocative, divisive and controversial, and they tend to rile people up, and derail the conversation. Regardless of what opinion you may have or which side (or no side) you may be, I don't think it's constructive to allow people to be annoyed and riled up like that, inducing them to derail the conversation. It's also not very constructive to allow such provocative statements to be made but then, inconsistently (and slightly hypocritically) ban people from responding to it, giving the original statement special status exempt from criticism or commentary in the same context where it was made.
There's nothing wrong in the site being a political arbiter, in the sense of "political topics are not allowed at all in submissions, neither the submission text itself nor in the submission discussion thread". All political discussion being forbidden is a common practice at many forums, regardless of what side the political discussion may be. (And no, there is no need to start nitpicking about what is and isn't "political". The same kind of common sense and judgment can be used as when determining whether something is offensive, discriminatory, etc.)
Off-topic remarks in general might not be something to ban from submission texts, but I'm all for categorically banning political statements of any kind from them. The reason is clear: They are provocative and divisive, and they cause animosity, disagreement and bickering, they only make people angry, and they derail the conversation about the subject in question.
There are literally thousands and thousands of political online forums out there. There is no need to make this into one. We don't need yet another one of them. If you want to discuss politics, choose one of those.
Yes. It was perfectly clear that you would prefer that we avoid discussing certain categories of facts that are uncomfortable to think about for a certain segment of our userbase.
So uncomfortable, in fact, that a sub-group of those who find them uncomfortable will refuse to accept that these are facts, and instead loudly decry their validity.
This is exactly the kind of rhetoric I didn't want to see. I know for a fact that this particular political topic will eventually devolve into exactly this. Soon direct namecalling and accusations will start to be thrown around.
I suggest this topic be locked before it devolves even further, because it will, with 100% certainty.
Saying "let's not discuss politics" implicitly declares certain topics as "political" and certain topics as "not political."
Actually it doesn't do it implicitly. It does it quite explicitly, because it's true. Some topics are political and some aren't. And everybody understood perfectly what I was referring to in this case.
Therefore, "let's not discuss politics" is, itself, a political statement
That doesn't make much sense. That's like saying that "let's not discuss math" is a mathematical statement.
The irony is that your post in the bundle topic is what triggered this entire fork of discussion, so, you really only have yourself to blame at this point.
Is it like a reverse psychology thing? Asking for no politics causes political discussion. The person who asks for no politics is to be blamed for the politics.
Out of curiosity I wrote a small program that simulates this. I used the ISAAC rng to generate random bits (ISAAC is a cryptographically strong RNG, which ought to mean very good quality of randomness and no bias for any of the bits).
Since this is very fast to simulate I ran it for 10 million iterations.
From all those 10 million tries, the minimum number of tosses was, rather unsurprisingly, 10. The maximum was 16929. The average number of tosses was 1023.2
(Conspicuously this differs from your calculated result by almost exactly 9. I wonder if you counted only up to the first toss that's followed by a string of 9 of the same tosses. Ie. you didn't include those 9 subsequent tosses into the count.)
I think something somewhat similar might have been asked before, but anyway.
There are videos out there where people toss a coin and get 10 of the same side consecutively (ie. either 10 heads, or 10 tails, consecutively). These videos are unedited and untampered. There is no trickery going on.
Of course the method is pretty simple: Just keep tossing a coin until you get 10 of the same side consecutively, and show that part only.
This got me thinking: What's the expected average number of times you'll need to toss the coin in order to get 10 of the same side consecutively?
If you repeated this process over and over (ie. keep tossing the coin until you get 10 consecutives, then start over; for the sake of simplicity we juts forget the previous tosses, ie. we don't count it as "the same" is you throw the same side an 11th time; we just count it as everything having been reset and we are starting over, so it's so far just 1 of that side having been tossed) a million times, what would be the average number of tosses?
