Infinite Martingale
Quote: WizardLikewise, I would say the "measure" of the chances of ever getting a red in an infinite roulette game is 1. However, the probability is not 1. If the probability is not a true 1 then the infinite Martingale player might lose.
Actually the probability is 1 but it happens "almost surely" and not "surely". Similarly like in dart throwing scenario where you can have an outcome with probability 0.
I refer to the article:
http://en.wikipedia.org/wiki/Almost_surely
which says: "The difference between an event being almost sure and sure is the same as the subtle difference between something happening with probability 1 and happening always."
Quote: Jufo81Actually the probability is 1 but it happens "almost surely" and not "surely". Similarly like in dart throwing scenario where you can have an outcome with probability 0.
I refer to the article:
http://en.wikipedia.org/wiki/Almost_surely
which says: "The difference between an event being almost sure and sure is the same as the subtle difference between something happening with probability 1 and happening always."
From the article you quote: "If an event is sure, then it will always happen, and no outcome not in this event can possibly occur." Conversely, if an event is "not sure", there must be a possibility of another outcome.
I only see one possible outcome (completion of the game) in the situation we discuss - the player wins a bet. If you claim this outcome is "unsure", there must exist some other way to end the game. What is the other possible outcome you are considering?
The article, while being illustratively close to the reality is actually mistaken about the the coin-toss example. The notion of "almost surely" ("almost always" etc.) is not usually applied to discrete measures, because in a countable set, the only null subset is the empty one, and therefore "almost always" and "always" are equivalent. This is exactly the kind of stuff I did not want us to get into earlier, because it only muddies the waters, and has really nothing to do with the question being discussed.
Quote: Asswhoopermcdaddy
What is the impact of infinity on the following types of games:
1.) Positive Expectations - win
If you look at my previous post about the game where you win with 51% chance and lose with 49% chance then even by starting with a bankroll of 1 unit there is larger than zero probability that you will never bust, in other words if you do a random walk on integer axis, starting at point +1, you might not reach zero even in infinity.
Quote: Asswhoopermcdaddy
2.) Neutral -0- Expectations - ??????
If you start with bankroll A and your target bankroll is B then the probability to reach B without losing A is p=A/B. If B ->Infinite then p->0, so we can say that you will not become infinitely rich in neutral expectation game starting with a bankroll of any finite size "almost surely", in other words Wizard's soft 1. If both A and B are infinite then the expression becomes Infinite/Infinite and is undetermined.
Quote: Asswhoopermcdaddy
3.) Negative Expectations - lose
You will always lose any finite size bankroll with probability 1 although I am not sure if it's "almost surely" or "surely". As for infinite bankroll I am not sure, I guess this was the question this thread is all about.
Quote: weaselmanFrom the article you quote: "If an event is sure, then it will always happen, and no outcome not in this event can possibly occur." Conversely, if an event is "not sure", there must be a possibility of another outcome.
I only see one possible outcome (completion of the game) in the situation we discuss - the player wins a bet. If you claim this outcome is "unsure", there must exist some other way to end the game. What is the other possible outcome you are considering?
I'd guess the martingaler with infinite bankroll has to accept that there is a chance that his doubling will never end and the bet will never be resolved. This is probably why martingaling can't beat negative or zero edge game even in infinity and therefore infinite martingaling would have zero or negative expectation.
Quote: Jufo81I'd guess the martingaler with infinite bankroll has to accept that there is a chance that his doubling will never end and the bet will never be resolved.
But that's not an outcome. If the bet is not resolved, the game is not finished, and must continue in order to produce the outcome.
Quote:Maybe this is why his martingaling can't beat negative edge or zero edge game in infinity and thus even infinite martingaling would have zero or negative expectation.
Actually, it does not have a negative expectation. The expectation (as the some of probabilities times outcome over all possible outcomes) is 1*1 = 1 + ???
Once again, there is no other outcome, because there are no (non-trivial) null subsets in a discrete set.
Quote: weaselman
Actually, it does not have a negative expectation. The expectation (as the some of probabilities times outcome over all possible outcomes) is 1*1 = 1 + ???
Once again, there is no other outcome, because there are no (non-trivial) null subsets in a discrete set.
I think there is the outcome that the bet is never resolved and the +1 unit win is never reached. You may not accept it as valid outcome so I guess we disagree on that.
Quote: Jufo81I think there is the outcome that the bet is never resolved and the +1 unit win is never reached. You may not accept it as valid outcome so I guess we disagree on that.
If you insist on counting infinite sequences as "outcomes", what in your view is the difference with a +EV game then?
Let's say, red pays 1.5 to 1, not 1 to 1. Will you say that "infinite martingale" would still "not work" in such a game, since there is still that additional "outcome" of never seeing the red at all?
Quote: weaselmanYou do realize, that the game with this hypothetical infinitely long streak would never end, right? If the game is still in progress, how can you conclude that the player lost?
I am assuming a timeless quality to the experiment. As if every spin could be done instantly, although still sequenced for purposes of the Martingale.
Quote: boymimboActually, according to Heisenburg's Uncertainty principle, the dart is much bigger than the size pi and takes up a finite space between 0 and ten, and therefore, the probability of hitting pi could be measured. If the dart was extremely small, then measuring the position of the dart affects the actual measurement itself and therefore, you could only say with a percentage of certainty that the dart hit pi.
For purposes of my example, we have to throw away particle physics, and assume the tip of the dart is infinitely small. To put it another way, if you pick two numbers anywhere between 0 and 10, including the irrationals, what is the probability they will be equal? I know it is humanly impossible to pick such a number, so we have to suspend reality a bit here.
Quote: boymimboIn any case, infinity really means nothing.
I didn't want to say this, for fear of it being misinterpreted, but I claim that for practical purposes there is no such thing as infinity. It is an artificial construct to help us understand principles of mathematics.
Quote: WizardI am assuming a timeless quality to the experiment. As if every spin could be done instantly, although still sequenced for purposes of the Martingale.
Still. What criteria do you use to declare that the player has lost?
Quote:
I didn't want to say this, for fear of it being misinterpreted, but I claim that for practical purposes there is no such thing as infinity. It is an artificial construct to help us understand principles of mathematics.
I am not sure what you mean by "practical purposes", but it sounds like mathematics (along with physics and statistics among others) isn't one of them in your view?
How about irrational numbers? Would you say, they don't exist either in this sense?
Define N= number of times you can double your bet on a given game before you break the Maximum at a given table. The most common setting for N in real world casinos is N=6, but you can on occasion find a table that will permit N=10. For example if a table has a $5 minimum and a $500 maximum the most you can bet is 100 times the minimum bet. Find N such that N is the largets number possible where 2^N<=100. The answer is N=6 because 7 doesn't work (2^7=128).
Define Minimum Bankroll as the minimum bankroll you require to support a losing doubling sequence N long (MINbankroll = 1+2+4+..2^N). For N=6
MinBankroll = 1+2+4+8+16+32+64= 127). Closed form solution for MINbankroll = 2^(N+1)-1.
You need a well defined rule for saying that the Martingale "worked". Several possible definitions are available, but probably the simplest is that you reach 2*MINbankroll before suffering N losses in a row.
It's not a well publicized result, but in the case of a zero house edge game, the probability that the Martingale system will "work" is roughly 50% of the probability that you will lose N times in a row. If you think about it logically, that is not surprising since you have a 50% chance of losing each game.
What also is true is that the above result is fundamentally independent of N. If you find a table that permits N=10 (there have been such tables in downtown Las Vegas that I have seen), the above result is the same as a table that only permits N=6.
As a limiting case, if you permit N=100, 1000, a million, the probability does not change. So the limiting case as N approaches infinity does not change the probability of the Martingale "working" (using the above definition).
================================
There also seems to be some confusion about a game "not resolving". You can work with a severely limited game of N=1, where you only have a bankroll of 3 units. The game ends when the player loses two in a row, or the player reaches 6 units. There is a logical outcome where this simple game does not resolve despite an infinite number of plays. However it the likelihood of this outcome approaches zero in the limit the longer you play.
================================
Incidentally these types of questions are as old as speculative thought has been recorded. Greek philosophers such as Parmenides and Zeno posed these kind of questions and developed some elaborate theories about the nature of reality based on the seeming paradoxes that resulted.
One simple paradox was to imagine a stairway with right angles that extended a cumulative of 4 units in length and 3 in height. The length of the stairway is always 7 no matter how many steps you have. If you theoretically have hundreds of steps, the length is always 7. But if you let the number of steps go to infinity, then the stairway gradually approaches a straight line. The length of the straight line is length 5 by Pythagorean Theorem.
As a result of these paradoxes Wierstrauss developed a formal definition for the limit in the 17th century. It is a bit complex to explain here.
Quote: pacomartin
As a limiting case, if you permit N=100, 1000, a million, the probability does not change. So the limiting case as N approaches infinity does not change the probability of the Martingale "working" (using the above definition).
You have demonstrated it yourself with your own example about the staircase, that you cannot this easily make a leap from a finite sequence to an infinite one - there may be surprises along the way. The important difference between the finite (however large) bankroll, and the infinite one is that the latter is ... well ... infinite :) In the first case, there are two possible outcomes of the game, while in the second case there is only one (you must continue playing until you win), and therefore it is impossible to ever see any other one.
If you amend the problem with an additional criteria as you suggested (stop after X bets, or whatever), then yes, it won't work, but if you look closely at your criteria, you'll see that it is equivalent to limiting the size of the bankroll, which makes it a completely different problem.
Quote: weaselmanStill. What criteria do you use to declare that the player has lost?
I don't know how to put it into words, which to prove my other point that there is no such thing as infinity.
Quote:I am not sure what you mean by "practical purposes", but it sounds like mathematics (along with physics and statistics among others) isn't one of them in your view?
How about irrational numbers? Would you say, they don't exist either in this sense?
I mean that you can't point to anything physical in which there is an infinite quantity. Irrational numbers exist as much as any number exists.
Also, I think paco's staircase is a great example. I think of it as starting with a single step, going from (0,0) to (0,1) to (1,1). The distance is 2. Then you decide the step is too big and make a half way point, going from (0,0) to (0,.5) to (.5,.5) to (.5,1) to (1,1). The distance is still 2. Then you divide again, and the distance is still 2. No matter how many times you divide the steps by 2, the total distance remains 2. I claim that if you could do infinite divisions the distance would still be 2.
By the same logic, the Martingale player will always lose 2.70% no matter how large his bankroll. You have to approach a different outcome to get there.
Quote: Wizard
I mean that you can't point to anything physical in which there is an infinite quantity. Irrational numbers exist as much as any number exists.
They do have an infinite number of decimal places though. If they exist, then here it is, I just met your challenge :)
It is hard to claim that they do not - dividing a circumference of a circle by its radius is hardly a metaphysical procedure.
If that's not physical enough, consider an arrow of time, which is infinite in at least one direction by most commonly accepted current models.
Quote:I claim that if you could do infinite divisions the distance would still be 2.
You could just as well claim that it would be zero - after an infinite number of divisions the length of each step would become zero :)
It just illustrates that, like I said before, you cannot apply induction to infinity as if it was just another number. "An arbitrary large number" is fundamentally different from "larger than any arbitrarily large number".
BTW, note that the final result in pacomartin's example does not have to be 5 - it is only one of many possible answers, depending on how exactly you perform the divisions (the resulting line does not have to be straight).
Quote:By the same logic, the Martingale player will always lose 2.70% no matter how large his bankroll.
Yes. As long as the bankroll is finite, he will eventually lose it (on average), no matter how large it was. But if there is no limit at all, the notion of "lose" ceases to exist.
Quote: pacomartin... One simple paradox was to imagine a stairway with right angles that extended a cumulative of 4 units in length and 3 in height. The length of the stairway is always 7 no matter how many steps you have. If you theoretically have hundreds of steps, the length is always 7. But if you let the number of steps go to infinity, then the stairway gradually approaches a straight line. The length of the straight line is length 5 by Pythagorean Theorem. ...
I think I disagree. More specifically, I disagree with the part where you seem to make the jump from "approaches a straight line" to where you act as if you actually have a straight line. With the infinite number of infinitesimal, right-angle steps, each segment of the broken line is either vertical or horizontal, with a total length of 7. You never have a diagonal straight line of length 5, since you never ever travel in a diagonal direction.
Quote: Wizard
I mean that you can't point to anything physical in which there is an infinite quantity. Irrational numbers exist as much as any number exists.
In Physics infinite quantities are met every day. For example: a particle with electrical charge creates an electric field around it and the the strength of the electric field goes towards infinity as you approach the particle. There are probably many more examples. So saying that in physics you don't encounter the concept of infinity is not correct.
Quote: Jufo81In Physics infinite quantities are met every day. For example: a particle with electrical charge creates an electric field around it and the the strength of the electric field goes towards infinity as you approach the particle. There are probably many more examples. So saying that in physics you don't encounter the concept of infinity is not correct.
Approaching infinity and infinity are not the same thing. This sounds like the concept of infinity, which I have no problem with.
Quote: WizardApproaching infinity and infinity are not the same thing. This sounds like the concept of infinity, which I have no problem with.
Well how about black holes which are a result of of a star collapsing to singularity, and singularity implies that some quantity (in this case mass density) goes to infinity. It doesn't just approach infinity but reaches it and a black hole results from that.
Quote: Jufo81Well how about black holes which are a result of of a star collapsing to singularity, and singularity implies that some quantity (in this case mass density) goes to infinity. It doesn't just approach infinity but reaches it and a black hole results from that.
Before I address that, how much space does a black hole occupy?
"Infinity" really becomes a meaningless term when actually discussing infinity, like "ground" is in a geological context. In this case it's a mathematical infinity as a result of division of a finite value by zero.
What's more, in most theories regarding singularities, they are not true geometrical singularities, but only project on three-dimensional space as points. In all theories, universe is considered to be at least four-dimensional, ranging from spacetime where time acts as a full dimension, not a separate concept, to a full boat of dimensions as in String Theory, up to in some theories even an in[de]finite number.
Quote: WizardBefore I address that, how much space does a black hole occupy?
From what I understand of theory on the subject, a black hole singularity occupies zero space. In physical theory, though, it is held to have, at least for some purposes, a finite size, specifically one Planck length. Since any geometry smaller than Planck length is impossible, or rather does not make physical sense, a size of zero Planck lengths is impossible as well. In many ways it's like a pixel of the universe, but in this way it's not (as you could have 0 pixels). So in physical sense 0 p.l.=1 p.l., and for different purposes either value is used. Although I can't get my head around it well either.
Quote: WizardBefore I address that, how much space does a black hole occupy?
To be honest I am not an expert on black holes so I would have to consult Stephen Hawking to get an answer. The point I was trying to make is that infinity is not an alien concept in physics. Then again one would need to specify what counts as a "proper" infinity. Does singularity count as one or would the infinity need to be something more special.
Quote: Jufo81Well how about black holes which are a result of of a star collapsing to singularity, and singularity implies that some quantity (in this case mass density) goes to infinity. It doesn't just approach infinity but reaches it and a black hole results from that.
I thought about suggesting a black hole example yesterday, but decided against it. The singularity in the blackhole is really just the artifact of the model, describing it (general relativity). All it means is that the GR is not equipped with adequate tools to properly describe the physics of this extreme high mass/low spatial extent situation.
The string theory (the current most promising candidate for the "Theory Of Everything") on the other hand is capable of describing the physics of the black holes (and of everything else, including Big Bang) without singularities. According to string theory, the space, in which the black hole's mass is concentrated, is not zero sized, but has an extent of about Plank length, thus, the density is not really infinite, just extremely high.
The problem in GR is not with infinity per se though, but rather with singularity. If Wizard said that singularities do not exist in the physical sense, I would agree with that statement - anything, that has physical meaning has to be infinitely differentiable. But the infinity without a singularity is a perfectly valid and sensible physical object.
Quote: DocI think I disagree. More specifically, I disagree with the part where you seem to make the jump from "approaches a straight line" to where you act as if you actually have a straight line. With the infinite number of infinitesimal, right-angle steps, each segment of the broken line is either vertical or horizontal, with a total length of 7. You never have a diagonal straight line of length 5, since you never ever travel in a diagonal direction.
Well, you would be correct. I was showing it as an example of how constructing limits, particularly where infinity is involved can be confusing if you try to use your intuition. The father of mathematical analysis , Karl Weierstrass improved Cauchy's definition of the concept of limits so that they meet a formal rule to help avoid similar incorrect intuitive leaps.
With regards to the Martingale analysis no matter how large of an N you choose, the minimum bankroll gets larger and larger to cover the bets. The probability that you will double your bankroll is almost independent of N, but as N gets larger and larger small differences in N matter less and less.
In extreme cases, like N=1 or N=2 you will see some difference.
A surprising number of people think that an infinite bankrolled Martingale system works. I have had several people with advanced degrees in mathematics and physics say that believe it makes sense. There are a lot of people who believe that table maximums were invented to stop people from using Martingale. Table maximums were invented simply to control the standard deviation of the casino business. A small casino in downtown Vegas makes an average of less than $500 per day profit. They simply can't afford the risk of having a gambler making $10K bets. He could clean out the cashier, and destroy the profit for the quarter or give the business a record month. I refer you to "Double or Nothing" about the two young gamblers who bought the Golden Nugget from MGM Mirage in 2004 and lifted the table limits. A gambler beat them for $8.5 million on a craps streak.
Quote: pacomartin
The probability that you will double your bankroll is almost independent of N, but as N gets larger and larger small differences in N matter less and less.
In extreme cases, like N=1 or N=2 you will see some difference.
...
A surprising number of people think that an infinite bankrolled Martingale system works.
I think, this is a communication issue more than anything else. Like you said yourself earlier - you need a clear definition of what "works" means. If by "works" you mean "lets you double the bankroll", that is obviously not true when the bankroll is infinite, so, under your definition it does not work, and I would also be surprised if anyone claimed that it did.
Another possible definition of "works" is that you cannot lose your bankroll, which obviously is true if the bankroll is infinite. So, in this definition it does work.
Yet another possibility (the one, being discussed in this thread) is "you definitely will be ahead at some point". This event has the probability of 1 if you r bankroll is infinite.
I thought I heard Carl Sagan say once that you could put a black hole on the head of a pin. That would seem to imply they have some size greater than zero. They can't have zero space because then they, well, just wouldn't exist. It doesn't seem reasonable that they could occupy infinitely small space, like how much space Π takes up on the number line. Even if a black hole is the size of a neutrino it still has non-zero size, and thus the density would not equal infinity. It may also spill over into the 4th dimension or more.
Martingale
I'm starting to think that the original question is absurd. Not only would you need an infinite bankroll for the Martingale to possibly work, but infinite time as well. To say that the Martingale would win with an infinite bankroll seems to imply that wins would grow up. However, at any defined point in the future, there would be an expected loss, because the player might be in the midst of a long losing streak.
To combine the two topics, suppose the casino moves into space to avoid the sun collapsing and frying the earth. Eventually, the matter of the casino would decay or it would fall into a black hole. If this happens when the player is in the midst of a bad losing streak then he may very well finish negative. Of course, the player's bankroll would still be infinite, so he wouldn't care.
I'm starting to feel another story by DarkOz in the making.
Quote: WizardEventually, the matter of the casino would decay or it would fall into a black hole. If this happens when the player is in the midst of a bad losing streak then he may very well finish negative.
Actually, from the point of view of an outside observer, the fall into a black hole will take infinite time :)
Quote: weaselmanActually, from the point of view of an outside observer, the fall into a black hole will take infinite time :)
Not sure I agree. I thought the eventual fate of the universe will be one of the following:
1. The big crunch.
2. Everything becomes a single black hole.
3. All matter decays.
If we rule out 1 and 3 then eventually everything should fall into a black hole. Once matters enters one it never leaves, and new black holes holes keep popping up and the existing ones draw more stuff in. Eventually the black holes will combine with each other. It would just be a matter of time.
Quote: WizardI thought I heard Carl Sagan say once that you could put a black hole on the head of a pin. That would seem to imply they have some size greater than zero. They can't have zero space because then they, well, just wouldn't exist. It doesn't seem reasonable that they could occupy infinitely small space, like how much space Π takes up on the number line.
As far as I understand the basic theory, black hole singularities indeed have "zero" size, but there is no true zero size in physics, rather there is a minimum size, Planck length, any length below which is meaningless. So it's zero and non-zero at the same time. But the non-zero size isn't even infinitely small, it's rather pretty finite.
The physical probability of hitting Pi on a finite sized number line is actually also finite and measurable, equal to 1.6e-35/L (a small coefficient could participate though in some cases), where L is the length of the number line. If you hit within Planck length of Pi, you hit Pi, because lengths smaller than that do not physically exist and everything within one Planck length is the same point, thus your hit was exact.
Quote: P90The physical probability of hitting Pi on a finite sized number line is actually also finite and measurable, equal to 1.6e-35/L (a small coefficient could participate though in some cases), where L is the length of the number line. If you hit within Planck length of Pi, you hit Pi, because lengths smaller than that do not physically exist and everything within one Planck length is the same point, thus your hit was exact.
Interesting. What would happen in football if one team kept causing "half the distance" to the goal penalties until the distance to the goal was between 1 and 2 Planck lengths. Since they couldn't get any closer to the end zone, would it just be declared a touchdown due to the constraints of physics?
Quote: WizardInteresting. What would happen in football if one team kept causing "half the distance" to the goal penalties until the distance to the goal was between 1 and 2 Planck lengths. Since they couldn't get any closer to the end zone, would it just be declared a touchdown due to the constraints of physics?
Just sounds like Zeno's paradox all over again. Give 'em the TD.
Aren't those the same thing?Quote: Wizard1. The big crunch.
2. Everything becomes a single black hole.
Sigh.... I assume the team making the errors is the Jets.Quote: WizardInteresting. What would happen in football if one team kept causing "half the distance" to the goal penalties until the distance to the goal was between 1 and 2 Planck lengths. Since they couldn't get any closer to the end zone, would it just be declared a touchdown due to the constraints of physics?
Quote: WizardTo combine the two topics, suppose the casino moves into space to avoid the sun collapsing and frying the earth. Eventually, the matter of the casino would decay or it would fall into a black hole. If this happens when the player is in the midst of a bad losing streak then he may very well finish negative. Of course, the player's bankroll would still be infinite, so he wouldn't care.
The casino already is a black hole. Money gets sucked into it and never comes out, and the amount of money it can absorb is infinite.
Quote: Wizard[W]ould it just be declared a touchdown due to the constraints of physics?
Yes. That's why they call it physical education.
Note also that to be that certain about the ball's position you'd have to be highly uncertain about its momentum (Heisenberg), so the quarterback sneak ought to work just fine.
Quote: WizardNot sure I agree. I thought the eventual fate of the universe will be one of the following:
1. The big crunch.
2. Everything becomes a single black hole.
3. All matter decays.
If we rule out 1 and 3 then eventually everything should fall into a black hole. Once matters enters one it never leaves, and new black holes holes keep popping up and the existing ones draw more stuff in. Eventually the black holes will combine with each other. It would just be a matter of time.
Well, first, 1, and 2 are the same thing.
Second, #3 isn't quite right - the matter won't decay (decay into what?) - it will just continue expanding spatially, making the universe very very empty place (but note, that galaxies do not expand themselves, so locally there won't be very much difference in densities).
There are also other scenarios as well, most notably, the expansion slows down asymptotically, so that the size of the universe keeps increasing forever, but never overcoming a certain limit. This is considered one of the most likely scenarios in the "classical model".
But that's beyond the point. The fall into a black hole takes infinite time from the point of view of the outside (stationary) observer - the one, that's not falling. From the guy's that's falling down perspective, the time is very finite.
Since there are no "outside observers" to the universe (all observers are inside), there is no contradiction - if the universe were to collapse into a black hole, it would happen after some finite time interval.
Quote: P90
The physical probability of hitting Pi on a finite sized number line is actually also finite and measurable, equal to 1.6e-35/L (a small coefficient could participate though in some cases), where L is the length of the number line. If you hit within Planck length of Pi, you hit Pi, because lengths smaller than that do not physically exist and everything within one Planck length is the same point, thus your hit was exact.
First, the number line being infinitely long, this is still zero :)
And second, if what you were saying was correct, it would be entirely impossible to hit the Pi - if it was, then it would also be possible to express Pi as a point in a number line, located within a given number of Plank lengths from zero, and that would make it rational, which it is not.
Quote: weaselmanFirst, the number line being infinitely long, this is still zero :)
Ah, but it doesn't have to be complete. Even in a complete one, it's perfectly reasonable for it to have a finite scale, e.g. 1=1cm.
Quote: weaselmanAnd second, if what you were saying was correct, it would be entirely impossible to hit the Pi - if it was, then it would also be possible to express Pi as a point in a number line, located within a given number of Plank lengths from zero, and that would make it rational, which it is not.
This would be the case if Planck lengths were pixels. But while for some purposes they are, there are differences. Specifically, while a length of 0.3 Lp can't exist, a length of 82.4 Lp can, even though there is no way of telling it from 82.1 Lp, there will be uncertainty instead, so measurements down to fractions are meaningless. But as far as just existing, it can do that much, the universe isn't pixelated.
So the number Pi, even being a mathematical abstraction, still has a physical position exactly Pi scale units from zero. It's not a range of units and not an approximation, but an actual point in space exactly there. At the same time, it's physically (but not mathematically) equal to any point that is not at least a Planck length away from it, though more precisely they are the same point. Note the language, not "is less than 1 Lp away", because there is no less, but "is not at least 1 Lp away". It can be a tricky concept, but the short version is that the concept of space has no meaning within Planck length.
Quote: DJTeddyBearAren't those the same thing?
This is getting out of my area, but I didn't think so. I thought the "big crunch" was where the universe collapses on itself, regardless of the state of matter. Think of the universe as a balloon first expanding and then contracting into a single point.
The black hole theory suggests that eventually everything gets sucked into a black hole. From there the black hole might eventually cease to exists due to the big crunch or particle decay.
I think I may have to do another street shout out to my dad if this gets much heavier.
Quote: P90
This would be the case if Planck lengths were pixels. But while for some purposes they are, there are differences. Specifically, while a length of 0.3 Lp can't exist, a length of 82.4 Lp can, even though there is no way of telling it from 82.1 Lp, there will be uncertainty instead, so measurements down to fractions are meaningless. But as far as just existing, it can do that much, the universe isn't pixelated.
If "measurements down to fractions are meaningless", then there are no fractions, whether you want to call them pixels doesn't change the fact, that there is only a finite number of them in any closed line segment, making it impossible for irrational (or even rational) numbers to exist.
Quote:So the number Pi, even being a mathematical abstraction, still has a physical position exactly Pi scale units from zero.
It's not a range of units and not an approximation, but an actual point in space exactly there.
Well, if it is, then where would you place Pi+1e-38 and Pi-1e-38? And an infinite number of other numbers between those two? They can't all be in the same "pixel", because the "dart" has to be able to independently hit each one of them without hitting any of the others (meaning that each of them is it's own independent separate object).
A black hole is in fact matter. It has dimensions and it has mass. However, the gravitational pull is such that light cannot escape from it. At the center of the black hole is the singularity. According to Hawking, it is possible for a black hole to lose mass because it can radiate energy. He has proposed a model where black holes can evaporate. No one knows what happens to matter at the point of singularity. It's one of those great mysteries. However, it is maintained that the mass inside the black hole doesn't decrease so I would think that the mass is retained and that the density at the singularity approaches infinity (but is not truly infinity).
Once you hit the event horizon, your time stops because you are moving at the speed of light. You are energy, and energy does not know time.
It is unclear whether there is enough dark matter in the Universe to create a big crunch where the universe collapses upon itself. The black hole theory is not valid because it would emit radiation faster than it could absorb it by sucking out things around it. My belief (and that's all it is) is that the Universe is still expanding and will expand forever, though we won't know. The most current value of the Hubble constant is 70km/s which means that the Universe, from our point of view, is still expanding rapidly and is 13.7 billion years old.
My first year astronomy professor was none other than Tom Bolton, who discovered the first Black Hole at Cygnus X-1.
If you put a stack of dollar bills, infinity miles high, on Red, and another stack, infinity feet high on Black, which stack weighs more?
Then again, it's a damn interesting tangent!
Quote: boymimboHowever, it is maintained that the mass inside the black hole doesn't decrease so I would think that the mass is retained and that the density at the singularity approaches infinity (but is not truly infinity).
Yeah, it does actually decrease. That's what Hawking means by "evaporate" (it doesn't actually become vapor :)). Every black hole radiates energy at a rate inversely proportional to it's mass (by the Hawking model you mentioned), and constantly decreases in size accordingly, albeit very slowly if we are talking about stellar (massive) black holes.
Eventually, given enough time, it becomes smaller and smaller (and the smaller it becomes, the more intensively it radiates, and the faster its losing its mass), and eventually disappears completely, without a trace.
Quote:
Once you hit the event horizon, your time stops because you are moving at the speed of light. You are energy, and energy does not know time.
Actually, you do not notice anything special at all about crossing the event horizon. It is only the observer, looking at you from distance, who will see some odd stuff - from his standpoint you will never cross the event horizon at all, you'll be moving slowing down asymptotically, never quite reaching the point of no return.
By your own clock, the jorney will only take finite time. No, you will not reach speed of light (technically, the event-horizon itself is a light-like surface, so if you were orbiting the black hole exactly at the EG level, you could be said traveling with the speed of light, but that would be a very misleading thing to say for a lot of technical reasons).
Quote:
It is unclear whether there is enough dark matter in the Universe to create a big crunch where the universe collapses upon itself.
It is almost universally accepted now that there isn't.
Quote:The black hole theory is not valid because it would emit radiation faster than it could absorb it by sucking out things around it.
The whole theory is not valid?
Where did you hear this? What exactly is "black hole theory"?
And what exactly is wrong with emitting faster than absorbing? Pretty much every object you see around (especially hot ones) emits radiation faster than "sucking out things around it", and yet they are all pretty "valid" :)
Quote:The most current value of the Hubble constant is 70km/s which means that the Universe, from our point of view, is still expanding rapidly and is 13.7 billion years old.
You got the units wrong. It's actually about 72 km/s/Mps, where Mps stands for Megaparsec. It doesn't have very much to do with the age of the universe - only it's rate of expansion.
My point is that the mass within the black hole doesn't decrease as a result of the singularity, meaning that the mass isn't disappearing out of the singularity.
Megaparsec is Mpc, and estimates are between 70-73. A rough estimate of the age of the universe comes from the inverse of the Hubble constant.
Quote: boymimbo
My point is that the mass within the black hole doesn't decrease as a result of the singularity, meaning that the mass isn't disappearing out of the singularity.
Not as a result of singularity , but it does get carried away by the radiation, making black hole lighter over time. In this sense, it does disappear.
Quote:Megaparsec is Mpc, and estimates are between 70-73. A rough estimate of the age of the universe comes from the inverse of the Hubble constant.
Yes, Mpc, of course.
Re. age of universe, the chain of reasoning is the other way around. The 1/H thing is only true when the deceleration parameter (that tells whether the expansion speeds up or slows down overtime) is zero. We don't know what the actual value is of this parameter is, so, we have to estimate the age of universe by other means, and then, knowing that, we compare it to 1/H to determine the value of the deceleration parameter. (This was the common approach till about 15 years ago, and, using we had presumed the parameter to be zero, because the observed age was so close to the 1/H ... But then it was proven that the expansion is actually accelerating, but at a rate, which just coincidentally makes the age of universe match 1/H at this particular point in time ... go figure!)
Quote: weaselmanIf "measurements down to fractions are meaningless", then there are no fractions, whether you want to call them pixels doesn't change the fact, that there is only a finite number of them in any closed line segment, making it impossible for irrational (or even rational) numbers to exist.
Lengths of less than 1 Lp are not meaningful; all and any interactions within 1 Lp are exactly the same, distance does not exist there. Lengths of 255.5 or 255.1 Lp are potentially meaningful; the force of gravitational interaction, for instance, might be theoretically lower at 255.5 Lp than at 255.1 Lp. Any attempt to measure the difference to a precision of less than 1 Lp will hit uncertainty, but it can be used theoretically with success.
As I said, Lp is not a pixel, even if it often works like one. You could better think of it as dots on a monochrome CRT display (assume perfect phosphor). The resolution of such a display is theoretically unlimited, and a dot can be placed to any precision at any point on the screen.
However, if you actually try to get unlimited resolution, you will find that at somewhere around 3,000x3,000 (for a 9" screen) dots will begin overlapping, so a 18,000x18,000 picture will only have a discernible resolution of something like 4,500x4,500. You can still place a dot at 0.785431x0.267883 and it will be at a different distance from the edges than if you placed it 0.785432x0.267884, but if you place both, they will not be distinguishable.
A similar thing happens in physics at distances and time intervals smaller than Planck units, except it's considerably more fundamental, so instead of merely overlapping they perfectly coincide.
Quote: weaselmanWell, if it is, then where would you place Pi+1e-38 and Pi-1e-38? And an infinite number of other numbers between those two? They can't all be in the same "pixel", because the "dart" has to be able to independently hit each one of them without hitting any of the others (meaning that each of them is it's own independent separate object).
Any direct measurements will stumble into uncertainty. But Planck length is the distance light travels in Planck time. If you know a turtle travels at 1/1,000,000,000 light speed, you can define its location down to fractions, even though you'll never be able to verify it directly. But the turtle will behave as if it travels at a speed of 1/1,000,000,000 Lp per Tp.
So Pi+1e-38 is located Pi+1e-38 scale units from zero, and Pi-1e-38 is located Pi-1e-38 scale units from zero, but the distance between Pi+1e-38 and Pi-1e-38 is 0, for all physical intents and purposes, and for all physical intents and purposes they coincide. Any physical object (in our case a dart) that hit Pi+1e-38 scale units from zero did hit Pi, as did it hit Pi+1e-38 and Pi-1e-38.
Quote: P90Lengths of less than 1 Lp are not meaningful; all and any interactions within 1 Lp are exactly the same.
You don't have to repeat yourself, I heard you the first time.
I get really confused when people do this, don't quite understand what I am supposed to do now - repeat all my previous points too? Or just shut up and go do something else?
Your argument begs the question. It is based entirely on a particular (very technical and highly conditional) result of a particular model - the string theory. Note though, that your discretization of the continuum makes calculus, and everything based on it invalid. And without calculus and differential geometry, the string theory itself cannot be formulated.
Quote: weaselmanYour argument begs the question. It is based entirely on a particular (very technical and highly conditional) result of a particular model - the string theory. Note though, that your discretization of the continuum makes calculus, and everything based on it invalid. And without calculus and differential geometry, the string theory itself cannot be formulated.
This behavior is actually not exclusive to string theory. What string theory does is explain why known space behaves that way, one of the explanations. String theory operates with elements that are not definable in the terms of currently known space.
But, for instance, the most prominent meaning of Lp is in describing a black hole's entropy, which is A/(4*Lp^2). This is a specific number, not a near-infinity. At the same time, black hole's density still can be said to be infinity, but if any object was smaller than 1 Lp in size, it would behave as if it is was 1 Lp in size (so for all known space purposes it is), and so this figure factors into all of its properties.
We're going way off the rails now, this discussion would be more in place on physicsforums.com, though going into the detail of string theory would be hard even there.
Quote: P90This behavior is actually not exclusive to string theory.
No? Where else do you find it? (black hole entropy example you are quoting is a direct result of string theory).
AFAIK, all the predecessors of string theory - notably, the Standard Model of classic QFT and GUTs - are based on dot-particles having zero spatial extent. That property has posed irreconcilable differences in trying to integrate the quantum physics with GR, which in turn gave rise to the string model.
Quote:but if any object was smaller than 1 Lp in size, it would behave as if it is was 1 Lp in size (so for all known space purposes it is), and so this figure factors into all of its properties.
This is not true (not within the realm of string theory anyway). Paradoxically, an object of size R*Lp with R<1, making it smaller than Lp is predicted to exhibit the same properties as an object of size Lp/R, which would be larger than Lp. This opens some mind-blowing possibilities of entire universes inside black holes, forever separated from our own by their event horizons.
