Finite or Infinite Number of Possible Images on an LCD Display?

Brian Greene would have been correct then, but it wouldn't make any difference to a mere mortal if it was infinite or not. Acts of terrorism won't change that... Maybe you should try to explain to him that the number would be so large anyways that, even if he died of old age, he wouldn't even be able to put a dent in the number of possible images that could be made if it wasn't infinite even with a super computer! Mathematicians probably wouldn't even be able to come up with an exact number for it.

 

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Some Posts to this Thread are mind boggling. From Post #37 by SomeGuy1

The set (1/n) has an infinite number of members & the sum of the members is infinite. How can you claim that it has a limit of zero? I suppose you actually mean that the value of the last term approaches zero as n grows without bound. If this is your intended meaning, your semantics are not those used by knowledgeable mathematicians.

From Layman Post #63

The correct value is 2E2,000,000. Consider smaller numbers of pixels.

Code:

3 Pixels[b]:[/b] 2E3 = 8 possibilites: 000, 001, 010 [b] . . . :[/b]111, not 3! = 6

4 Pixels[b]:[/b]2E4 = 16 possibilities: 0000, 0001  [b] . . . :[/b] 1111, not 4! = 24

The factorial values are many orders of magnitude greater than the true values.

FraggleRocker, Phyti, & Sarkus seem knowledgeable. Most other Posters need to read Cantor’s essay on trans finite numbers.

Cantor has a valid definition of such numbers & follows it with very lucid analysis. He provides a definition of sets with an infinite number of members & his essay defines larger sets.

 

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Of course it would be more correct to say that the sequence (1/n) has that limit. Or if you prefer, the set {1/n} has a limit point. My conclusion is unaffected by your comment. I gave an example of bounded infinite set that has a limit point. The concept of infinity is therefore not necessarily associated with the words "unbounded" or "limitless."

Truly my post boggled your mind? Did you need me to further explain this example? Perhaps you're not familiar with limit points. I'm not sure how to take your comment. It's a very clear example that shows that the words "unbounded" and "limitless" do not exhaustively characterize the class of infinite sets.

 

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I think you forgot to convert the binary number to the base 10 decimal system. 111 in binary = 7; 1111 in binary would equal 25, so the numbers are actually larger than the factorial of the number of pixels. Plus, you took 2,000,000! out of context. I only meant to say it was about that much or an approximate answer. I didn't think it would be necessary to really add a number after that, because it would be completely insignificant to the first term. Then the number is so large an exact value would most likely be rounded to where that wouldn't matter anyways.

 

Another way to solve the problem could be to figuring out what 0.111... x 10^2,000,000 is in binary rounding down to the nearest whole number and converting that to the base 10 digit system.

 

The answer would be 01010^0111101000010010000001 - 1 in binary, but I couldn't find a binary calculator online that used scientific notion. I added one to 0.111... x 10^2,000,000, so the number would only be 1 x 10^2,000,001, that would be easier to work with. Then those two numbers are just 10 to the 2,000,001 power minus 1 in binary. It would be nice to know what the number would be in base 10, but I don't know of a binary to base 10 converter that uses scientific notation. It would have to be less than 2,000,000 digits long, since it is about that long in binary and converting to base 10 gives less digits or decimal places.

 

On second thought, the answer could just be 1 x 10^2,000,001 - 1 in base 10 notion. That would make it about 0.999... x 10^2,000,001, if it is a valid operation to convert binary and base 10 that way using scientific notation. Then it would just be one away from having a one with 2,000,001 decimal places.

 

Someguy1, your point was a good one. Any mathematician reading your example would assume a mapping \(f:N \to Q\,\,\,\,n \mapsto \frac{1}{n}\) which has a limit \(\displaystyle\lim_{n \to \infty}f(n)=0\).

Moreover she would see immediately that for all \(n \in N\) that \(\frac{1}{n} \in [0,1]\) which is obviously bounded above and below.

Obviously (she would say) that the mapping \(s:N \to Q\) that defines a sequence of the rationals in \([0,1]\) is a special case of this, but quite superfluous here.

She might also say that as a topological space \(R^1\) contains the subset \([0,1]\) as a closed set in the so-called standard topology. And since any closed set is equal to its closure, and since the boundary of any set is its closure less its interior (in this case \((0,1)\)) we have \(\{0,1\}\) as the boundary (or limit set)

All this in spite of the fact that the cardinality of \([0,1]\) is infinite

 

Their would definitely not be zero number of images that could be displayed on an LCD display...

 

From my Post #59

Fraggle Rocker & some other Posters seem to understand the above.

I find it hard to believe that there are Posters who do not understand that a finite number of pixels can only provide a finite number of displays. As indicated by the above, the time factor does not increase the possible number of displays.

BTW: A Poster who has not read & understood Cantor’s essay probably does not understand the concepts of trans-finite numbers. This seems to apply to some Posters here.

I am not sure about the limitations of analogue displays. TV & computer monitors fire particles (electrons, I think) at the screen. I wonder if such screens have an infinite number of spots which can be activated. I suspect not, but a designer of such screens can provide an answer here.

The Heisenberg Uncertainty Principle might support the notion of an infinite number of possible displays for an analogue screen, but I have no idea of its applicability here.

 

There are some Posters with limited computing skills. From my Post # 82

From Layman Post #84

You need some further education. It is true that binary 111 = decimal 7. However, 3 binary bits provide 8 values (You forgot about 000 = 0).

Binary 1111 equals 8 + 4 + 2 +1 or 15, not 25. Ergo: 4! (or 24) is greater than binary 1111 (or decimal 15)

2,000,000! is many orders of magnitude larger than 2E2,000,000.

 

From my Post #59

Fraggle Rocker & some other Posters seem to understand the above.

I find it hard to believe that there are Posters who do not understand that a finite number of pixels can only provide a finite number of displays. As indicated by the above, the time factor does not increase the possible number of displays.

BTW: A Poster who has not read & understood Cantor’s essay probably does not understand the concepts of trans-finite numbers. This seems to apply to some Posters here.

I am not sure about the limitations of analogue displays. TV & computer monitors fire particles (electrons, I think) at the screen. I wonder if such screens have an infinite number of spots which can be activated. I suspect not, but a designer of such screens can provide an answer here.

The Heisenberg Uncertainty Principle might support the notion of an infinite number of possible displays for an analogue screen, but I have no idea of its applicability here.

 

SomeGuy1 From your Post # 83

Your example is the set 1/n [Id Est: 1 + 1 /2 + 1/3 + 1/4 . . . . ]

The sum of the above set grows without bound as n increases. Id Est: In ordinary language, the sum is infinite.

Is your term ( bounded infinite) intended to mean that the sum of the terms is bounded (Id Est: Not infinite, using ordinary language)?

Perhaps it would help if you explained what you mean by bounded infinite set and limit point.

 

Proof:

Assume the number of possible images of the universe is finite. Collect them all either in hard copy or on a computer.

Sort them by name, description, type, etc.

Take a selfie of yourself doing this.

You have added one more picture never taken before. You can only do so because infinite time exists. By mathematical induction, the assertion is proven.

 

If your selfie was already in the collection, then you didn't add one more picture, you added a duplicate of an existing one. The selfie in the collection could have been generated by a program designed to produce every possible combination of pixel outputs. I am assuming we are still talking about a fixed number of pixels, and a finite number of pixel states.

 

The point is, time doesn't stop just because you took what you believe to be the last picture, and as long as time continues, more images are possible.

But you can't take that next selfie at all until or unless you believe you have them all.

It doesn't matter how fast or slow you capture the images. It doesn't matter (after a certain resolution) how detailed the images are. You can always take more smaller images and block them together to render sufficient detail of your image subject, and display them as a scrolling banner.

Even if we believed that space was actually discrete like that, time isn't, and therefore by means of extension alone, neither is space.

 

The possible pictures is infinite in extent just like this universe expansion is infinite in extent

 

I had to look up "Id Est." It's Latin for "that is." In other words this is the familiar "ie." In which case ... your usage is laughably wrong.

The set {1/n} is the set {1/2, 1/3, 1/4, 1/5, ...}. That set has a limit point of 0.

We could also consider the sequence (1/n), which is the sequence 1/2, 1/3, 1/4, ... That sequence has a limit of 0.

You don't seem to know the difference between a set, a sequence, and a series. That's perfectly forgivable if you're here to learn something about math. But it's very revealing (and not in a good way) when your purpose in being here is to accuse other people of not knowing any math. It makes you look silly.

I never referenced a sum. I referenced a set; and a sequence.

You are correct that the infinite series 1/2 + 1/3 + 1/4 ... is unbounded. But that's no problem for my argument. I could just as easily use the set {1/2, 1/4, 1/8, ...} whose elements can be shown to sum to 1. But as I already noted, I'm not talking about an infinite series here. And the fact that you don't understand that frames your earlier snark in a negative way.

At no time have I ever been discussing any infinite series. I have been discussing both the set {1/n} (where it's implied that n goes from 1 to infinity) and the sequence (1/n). The former has a limit point, namely 0; the latter has a limit.

Ah. Thank you for simply asking for an explanation of some technical terms. Glad to oblige.

My point is simply a semantic one. I think that people often confuse themselves by saying that something is infinite when it is "limitless" or "unbounded." So I'm giving specific examples of things that are infinite, on the one hand; yet limited and/or bounded.

It's true that these are mathematical examples. But why not? It's in the field of mathematics that humans have in the past 140 years achieved a spectacular breakthrough in our multi-millenia-long struggle to understand infinity. You referenced Cantor so I know we're in agreement here. So you'll forgive me my mathematical examples, but these are most definitely on point.

* Consider the set {1/n}. This notation means that we are considering a set of numbers {1/2, 1/3, 1/4, ...}. It's extremely worth noting that there is no order implied. I could have listed them as {1/47, 1/98989, 1/2, 1/34242, ...} and it would make no difference. It's membership in the set that's important, and there is no notion of order.

This set is considered in math to have a limit point of 0. That means that if I'm at the point 0 on the number line; if I draw a little circle around myself; then no matter how small that circle is, it will included members of the set {1/n}. That's the definition of a limit point.

* There is a related notion of a sequence (1/n), which is the ordered lists of numbers 1/2, 1/3, ... which has a mathematical limit of 0. That means that if I'm standing at 0 and I draw a tiny circle around myself; then that circle must contain ALL of the points of the sequence (1/n) from some point onward. For example everything after 1/200 must be inside the circle, or everything after 1/100000001. As long as an entire "tail" of the sequence must live entirely within any circle about 0, then we say 0 is the limit of the sequence.

But the point is that both the set {1/n} and the sequence (1/n) are obviously infinite in anyone's view. Nobody would argue that they're finite sets or sequences. The list 1/2, 1/3, ... never ends. Yet, it has a limit. And as an unordered set of points, it has a limit point.

Therefore I conclude that it's not necessarily the case that a thing that's infinite must also be without limit. Am I stretching the meaning of limit? Maybe a little. But in math, the meaning of limit is unambiguous. And why not hold concepts of infinity to a mathematical standard? As we all agree, it's in the field of math that we finally have a grip on infinity.

* You did correctly note that if we happened to consider the sum of all those terms, it would be unbounded. That turns out to be irrelevant to my argument; but even so, had I anticipated your objection I'd have simply used the sequence 1/2, 1/4, ... which is infinite, has a limit, and also has a finite sum. That would have saved us that bit of confusion.

* Finally, the concept of boundedness. If you take the set of points {1/n} on the number line, it's clear that you could take your pencil and draw a big circle around the whole lot of them. In other words an infinite set of things, in this case mathematical points, are able to fit quite comfortably inside a finite region of space. Mathematical points are the only thing we know with that property. Electrons, photon, quarks, and all the other objects of physical science don't have that property.

So we have shown a set that is infinite, yet has a limit, and is bounded. Therefore, it is semantically inaccurate to say that infinite things must be unlimited or limitless and unbounded.

* As one final example, we're all familiar with the closed unit interval of real numbers, denoted [0,1], defined as the set of real numbers between 0 and 1 inclusive. This is an uncountably (in the sense of Cantor) infinite set that is nevertheless bounded, and in fact even includes its endpoints. It's an infinite set with a beginning and an end. Therefore it also removes the word "endless" from our vocabulary of the infinite.

But please, if you take nothing else from this reply, understand that I was making a semantic remark. I think it's wrong to say that an infinite set is a set that's unbounded or unlimited. In order to make my point, I used examples from mathematics. But I was not primarily making a point about math; I was using math to make a point about the usage of the English language. I really hope that's clear.

 

"I was using math to make a point about the usage of the English language."

Yes, you did. And thanks.

Both math (in all of its manifestations) and the English (or other human) language are accessed by means of symbols, which are tools of finite minds. Alphabets have limited numbers of symbols just as numbers of a particular radix re-use symbols representing quantities less than the radix over and over again to access and manipulate numbers of greater or smaller magnitudes.

Finite minds cannot know, much less understand either their own limits or infinity, whether bounded or not, with or without using symbols as tools to that purpose.

The topic of this thread is a good example of exactly how our finite minds utterly fail to grasp exactly what infinity is, or what it is about our reality, if anything, that is infinite. If time abruptly stopped tonight at the stroke of midnight, and the universe that was expanding one second before it ceased to exist were examined from the uncertainty principle limit to the physical limits of that known universe, any argument over whether it was ever finite or infinite in terms of matter or energy or any arrangement(s) thereof, becomes moot. Static universes are finite precisely because they are static. Dynamic ones are not.

 

It is a figment of our finite minds that anything is static. Our finite memory is what produces this artifact.

I don't blame Euclid for this limitation. I possess the same one. The difference is, I recognize it as not the same as reality.