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

I get the sense that there are more physicists on this forum than mathematicians, so perhaps not everyone's seen basics of mathematical infinity. Infinity has a very specific meaning in mathematics, and the word "unlimited" doesn't come into play.

Arfa brane and phyti, are you both saying that

a) You need to review the basics of mathematical infinity, which does not correspond to the concept of "unlimited." (I showed that by giving two examples of infinite but bounded sets); or

b) You know about set theory but you have philosophical objections to it, so you are trying to make your own definition. Perhaps you feel that set-theoretic notions of infinity don't match with your personal intuition.

I can't tell if you are needing a review of set theoretical infinity, or explicitly rejecting it. If people are curious about set-theoretical infinity, there are lots of Wiki pages I could point you to. And if your objections philosophical, I might even agree with you! But I'd like to understand your objections better.

This is mathematically incorrect. We do know that there are more points in the unit interval than there are integers; that there are the same number of points in an interval as there are in the entire real number line; and that there are fewer points in the real numbers than there are subsets of the real numbers. You can't put as many points as you like in an interval. There are constraints.

These things are proven beyond dispute. The only question is whether you need to review the math, or you're explicitly rejecting the math. I would like to understand where you're coming from.

Here too, mathematics disagrees. We have many transfinite cardinals and ordinals. In fact contemporary set theory is all about large cardinals, which are infinite quantities so big that their existence can't even be proven from standard set theory. Large cardinals are mathematical objects of intense study in the past couple of decades. http://en.wikipedia.org/wiki/Large_cardinal

In fact "all numbers are finite in value" is similar to "all numbers are positive," or "all numbers are rational," or "Negative numbers don't have square roots." Those are historical beliefs about numbers that were overthrown by mathematical discoveries. And since 1870 we have a comprehensive theory of infinity, and we know about lots of specific infinite numbers.

So again, it's helpful for me to understand why so much standard math is being ignored here in favor of vague, imprecise talk about things that are "unlimited." Lots of things in the real world are unlimited but finite. Grains of sand on a beach, ideas in the mind, images on a computer display. Lots of things are bounded but infinite, like the two examples I gave. "Unlimited" is vague. Mathematical infinity is precise.

That's why I think the use of the word "unlimited" is detracting from the clarity of the discussion. Mathematics provides a precise conceptual framework for discussing infinite quantities. We should use it unless there are specific philosophical objections being proposed.

 

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Let's assume numbers are all objects with a value, such that adding 1 to this value increases that value. Basic arithmetic.

So if n is a number, n + 1 is a larger number. Nothing controversial so far.
Suppose infinity is a number, then infinity + 1 is also a number which is larger than infinity.

But, there is no number larger than infinity (!) (a reasonable assumption). Hence infinity cannot be a number and cannot therefore have a value q.e.d.

Further, I see no problem with associating the two concepts "infinite" and "unlimited in value", although the latter phrase appears to abuse the concept of value, it still works.

 

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Basic finite arithmetic. Adding 1 to a transfinite cardinal does not increase its value.

Not controversial, but restricted to finite numbers. If k is a transfinite cardinal, then k + 1 = k. Your statement can be made correct as follows: "If n is a finite number, n + 1 is larger than n."

No, that's just false. It's only true for finite numbers. It's false for transfinite cardinals. This has been well known since Cantor's first paper in 1874.

Reasonable in 1870. Proven false in 1874. We know lots of transfinite numbers, all distinct and having various properties and behaviors that distinguish them from each others.

The power set of any cardinal is strictly larger than the cardinal itself. The number of subsets of the integers is a strictly larger infinity than the number of integers. The set of real numbers has strictly larger cardinality than the set of integers. As I said earlier, these are 150 year old mathematical truths. If your training is in physics, consult Penrose's The Road to Reality, where he discusses the basics of transfinite cardinals.

Since all your premises are wrong, you haven't got an argument.

I can only repeat my question, and ask for a direct answer.

Are you ignorant of modern math, or rejecting it? That's a straightforward question.

If you're ignorant, I can point you to the relevant Wiki pages. If you're making a philosophical point, what is the point you're trying to make?

In other words if you say, "I'm a finitist, I don't believe in infinite sets," that's a defensible position in the philosophy of math. Or, "I'm an ultra-finitist, I don't even believe in sufficiently large finite sets." [For example if there are only 10^80 atoms in the universe, how can a number like 10^10^10^10 have any meaningful existence?] Or, "I'm a constructivist, I believe in infinite sets but only when their elements can be generated by an algorithm." Constructivists believe in pi and sqrt(2) but they don't believe in the full set of real numbers. Constructivism is having a resurgence these days due to the influence of computer science.

Finitism, ultra-finitism, and constructivism are all valid positions in the philosophy of math. Perhaps you're implicitly taking one of those stances, in which case it would be helpful for you to be explicit.

But ignorance is not a valid position in any scientific discipline. Let alone the deliberate sophistry of your "proof" built on assumptions that have your conclusion baked in. Having seen some of your other posts, your line of argument in this thread seems beneath you. How did you manage to learn topology without being exposed to basic set theory?

 

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I think this discussion is about the concept of value, what it means for a number to have or "be" a value.

So we distinguish between those mathematical objects which are or have a value, and those objects to which no value can be assigned. There are the included concepts of larger or smaller--numbers are larger or smaller than other numbers, this is how we distinguish numbers from each other. We know that numbers belong in infinite sets, but what is this "infinite"? Is it, or can it, be a number if it can't have a value so we can distinguish it from larger and smaller numbers?

I can't see how any of this follows from anything I've posted. You have no evidence that I've been declaiming infinite sets. Or that I'm ignorant of set theory.

So how do you define numbers, is there such a thing as an infinite number, or is that just a convenient turn of phrase that really is meaningless? Are you really saying you can write down the value of some transfinite cardinal, not just say if it's larger than some other "number"?

And I have to say, your dialog style comes across as imprecise at best, for instance here:

Where you say "no that's just false", what's just false, the supposition that infinity is a number? Are you then agreeing with the argument that it isn't a number after all?

 

It's nice to see a response allowing alternative opinions, from someone with an open mind. The norm seems to be "this is the way it is", by someone with an attitude, and thus the reason why I visit forums less or not at all.
You will probably put me in the constructivist category. I'm not sure about Arfa.

 

primary textbook: dictionary
synonyms for infinite:
immeasurable, endless, countless, unlimited, inestimable, vast, unbounded, boundless, interminable, limitless, never-ending

Quotes from wiki on subject:
Infinity (symbol: ∞) is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics.

Early Greek
He used the word apeiron which means infinite or limitless.

Real analysis
...the symbol , called "infinity", is used to denote an unbounded limit. means that x grows without bound, and means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then
...means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.

The article differs from your opinion.

Theoretical applications of physical infinity

This part is a good reality check for those who accept abstraction as real outside the mind.

someguy #37

The set {1/n} is infinite only because n is an element of N. The limit 0 applies to the final value of the function 1/n, not to the sequence of values or the set. Though the function cannot equal 0 by definition, it is used as a practical solution, since there is no significant error in real world applications.

The real number world is considered a continuum. In the manner of Dedekind, the unit interval can be divided into smaller intervals, which contain as many points as the previous interval, ad infinitum. Out of frustration, you might say "what's the point?".

The key word "uncountable" supports the idea that "infinity" is not a number in the
sense of a measure with an assignable value.

To express my opinion of only one infinite set, in the style of the late George Carlin,
"If you never run out of integers, what's the problem?".

 

No, it is not "meaningless"

First let's agree that the ordinals can be defined recursively as follows.......

The successor set \(\emptyset^+ = \emptyset \cup \{\emptyset\} = \{\emptyset\}\)

Likewise \(\{\emptyset\}^+ = \{\emptyset\} \cup \{\{\emptyset\}\}\)

and so on. By relabeling \(\emptyset = 0,\,\,\,\{\emptyset\}=1\) etc one arrives at the conclusion that for any ordinal \(x\) then there is a successor \(x^+= x \cup \{x\}\). And moreover, again recursively, that every ordinal contains all its predecessors as subsets.

One says that the smallest transfinite ordinal that contains all possible predecessor sets is \(\omega\).

Question: Given the above, is the invented symbol \(\omega\) any less "meaningful" than the invented symbol \(5\)?

 

I'll respond to this and to arfa's post in detail later on.

For the moment I just want to point out that it's a category error to use an imprecise dictionary definition when a precise technical definition is available.

It's like doing physics by looking up the dictionary definitions of mass, energy, force, and momentum; instead of studying how physicists define them.

I'm arguing that "limitless" is not a technical term in math; and that we do have an explicit and logically sensible definition of "infinite set" in math; and that you need the precise mathematical definition to know for sure whether a given collection is infinite or not.

The word "limitless" is vague and has no technical meaning. Here are four things in the real world that are unlimited yet mathematically finite.

* All the grains of sand in the world;

* The human imagination;

* The breadsticks at Olive Garden; and

* The collection of possible images on a modern computer display.

More later.

 

All; Thanks for your help, but the problem has been thoroughly solved in the offline conversation in which it started. The conclusions we arrived at are as follows:


1) Discussing the resolution or acuity of an n pixel display in terms of discrete states does not demonstrate that the number of images that can be displayed on it is finite. Period. This is because it is not possible for all of the pixels to reach their final state (thicknesses of LCD pixels) instantaneously. This is true whether you are displaying a colored pixel or turning the display off. In fact, this type of delay makes slower displays problematic for gamers, and everyone here knows this. The possibility that within a given time interval, an LCD pixel has not settled to the thickness electronically intended means that the display has artifacts and states, however transient, that are not taken into account with discrete mathematics calculations of how many different images the display may be able to show, intentionally or otherwise.

2) Discussing the resolution or acuity of a high resolution CCD camera or the human eye is likewise a misleading approach to a determination of whether or not there is a finite number of images possible for humans not gifted with Tetrachromatic vision by similar discrete math arguments. The human eye itself is an evolutionary compromise, and for the most part it doesn't function or deal with an image that is focused upside down on the retina in a manner that does not lose much of the information transmitted, for example, the sense of polarization of the light received, to name only one. It is therefore ludicrous to make a calculation based on something like 500,000,000 colors we are able to distinguish, or 70,000,000 shades of grey, to try to make the same discrete mathematics calculation with an eye as one might try to do with a high resolution HDMI display. CCD cameras up to 400 megapixels have been produced. So have femtosecond cameras capable of capturing images of a femtosecond pulse of light traversing a coke bottle, or are even fast enough to capture images of objects hidden behind opaque screens the way we would expect a light version of radar to operate. But these devices too have discrete limitations and may transfer images that may not capture all of the available visual information that is present.

3) Folks like Brian Greene have attempted to show that inasmuch as the known universe contains something like 10^80 nucleons (baryons, atoms) within a sphere of radius 13.8 billion light years, then there is only a finite arrangement of them possible since the Big Bang, and therefore we are living in a very limited, deterministic, finite and discrete universe.

This idea is unrealistic, and it is garbage. This universe is a composite of matter and energy, plus an infinite amount of time. Matter absorbs and re-emits energy continuously, and it may do so in any of an infinite (not discrete) number of directions, and matter or energy in this universe may likewise move in an infinite number of directions with respect to each other, while the energy that is absorbed or emitted may travel in any of an infinite number of other directions. Unless one is talking about arrangements of energy and matter in the universe in which either it is ALL matter, in the same place (in which case time dilation is infinite), or else it is ALL energy and is dispersed evenly throughout the universe (in which case, we have entropy death), the universe will be capable of producing an infinite variety of images independent of ideas about how best to display them, at whatever frame rate, or what sensory organ(s) to best observe them with.

The only thing finite in this discussion is Brian Greene's limited imagination or understanding of the reality of the universe. Discrete math doesn't begin to describe even the smallest part of it, other than possibly to a gamer calculating his or her odds of winning. Probability is actually a poor substitute for infinite time, except perhaps in QFT. It only works there because the only other choice would be a whole lot of unmanageable infinities and also because a nanosecond is an eternity on that scale.

 

My understanding of "meaningful" in mathematics is "well-defined", certainly both symbols are well-defined. However, \( \omega \) seems to be missing something, which is a value, which is something all finite numbers have (or are), which was my point.

 

Well that settles the matter once and for all! I'm glad you privately got the answer that makes you happy.

Still a man hears what he wants to hear
and disregards the rest

-- Paul Simon, The Boxer.

[

 

My point being that a transfinite "number" isn't a number, but an infinity.

And yes, I do understand the difference between aleph zero and aleph one; the current wisdom dictates that the second is a larger infinity than the first. How much larger is it though? Is it infinitely larger hence there is nothing to be gained from the concept of value (unless someone can suggest something)?

So that the inclusion in the quoted statement of yours, of the phrase: "does not increase its value", I claim is not meaningful. Infinite numbers are either not numbers, or completely different kinds of numbers from finite ones which do increase in value if you add another number to them.

I have seen several online tutorial articles that say infinity isn't a number, and I assume they mean any kind of infinity.

 

In your earlier post you referenced the idea of value also. I admit I'm confused. By value I assume you mean the numeric value. For example the value of "5" is 5 and the value of "47" is 47, using quotes to carefully distinguish between a string of symbols and the abstract thing it represents. "47" is a character string with no meaning. 47 is a number -- an abstract, Platonic number if you will -- with a certain value. We haven't defined value, but I think we agree that the value of "47" is 47.

But when you say that a transfinite number has no value, that's incorrect. To a mathematician, "\(\aleph_0\)" is a string representing a particular transfinite cardinal whose value is \(\aleph_0\). And what number is that? It's the number of natural numbers 1, 2, 3, 4, 5, 6, ...

What is the value of the string "\(2^{\aleph_0}\)"? Well, it's the transfinite cardinal \(2^{\aleph_0}\). What number is that? It's the cardinality of the set of real numbers. It expresses, as best as humans are able to express, the "number of" real numbers. I'm putting "number of" in quotes to make happy those who dislike semantically conflating "number of" with cardinality. But no matter how you slice it, mathematicians have an object they call \(2^{\aleph_0}\), and they can put it into one-to-one correspondence with the real numbers.

These are achievements of the human mind. And yes, perhaps they're strange and counterintuitive. But no more so than, say, quantum physics. The twentieth century was when humans discovered the profound limitations of "what's obvious," and were forced to make the leap to "What seems to be actually true."

Once one is familiar with basic set theory, it's just as sensible and familiar to say, "The value of \(2^{\aleph_0}\) is \(2^{\aleph_0}\)" as it is to say "The value of 47 is 47."

To be sure, when we talk about the "value" of \(\omega_1\), the first uncountable ordinal; I agree with you that this concept strains the imagination. (I mention this example because \(\omega_1\) is a number whose existence can be proved without the Axiom of Choice; but that is nevertheless an extremely strange and challenging mathematical object. http://en.wikipedia.org/wiki/First_uncountable_ordinal

Nevertheless, just as the quantum theory is counterintuitive and difficult to comprehend, yet is regarded as true in the practice of physicists; likewise, modern mathematics has an elaborate taxonomy of infinite numbers; and each one has a very specific value.

Ok so you now answered my question. You DO know some set theory, you are just making some sort of philosophical objection to it. Now I understand. Before, I did not understand.

So you say there is "nothing to be gained from the concept of value." I can't say, because the "concept of value" is something YOU are introducing into the conversation.

There is no such thing in math as the "concept of value." I suppose it may relate to discussions in philosophy regarding the "use" and "mention" of a thing. Or as I indicated above, between strings like "47" and the numbers (whatever numbers are!) they point to.

I don't think you have defined "value" and I don't think your concept is clear, at least to me.

But the value of 47 is 47 and the value of \(\aleph_0\) is \(\aleph_0\). And now I can leave off the quotes, because essentially everyone implicitly understands this. If you wish to reject the last 141 years of progress in mathematics (dating from Cantor's 1874 paper) you are free to do so; but the burden is then on you to be extremely clear in your ideas.

So: What is value? If 47 is the value of 47, then why isn't \(\aleph_0\) the value of \(\aleph_0\)?

 

ps ...

I'm sure those are tutorials for beginners and the general public that say, "Infinity is not a number" when they don't want people using infinity in the wrong way, for example asking what is \(1/\infty\) as people often do.

Once you get past calculus and into the realm of actual math, there are many strange objects that are "numbers," such as p-adics, for example. Transfinite numbers would be in that category. Things that are regarded as numbers by mathematicians, but not without a lot of context, so it's just better to tell people "infinity is not a number." You're reading nursery rhymes as if they were advanced literature.

There is no actual definition of "number." A number is whatever we need it to be in a given context. The integers mod 5 are numbers, even though they're circular: 0, 1, 2, 3, 4, 0, 2, 3, 4, ...

Wiki gives transfinite cardinals, transfinite ordinals, infinitesimals, hyperreals, and the Surreal numbers as examples of "nonstandard" numbers in math. http://en.wikipedia.org/wiki/List_of_types_of_numbers

And I notice that they left off the p-adics, which are very strange beasts but perfectly good numbers. http://en.wikipedia.org/wiki/P-adic_number

 

Sure there is the concept of value in mathematics. This I suppose can be defined as something finite, although the phrase "infinite in value" might still be useful. A finite number like 5 is something we might regard as a "quantity" that covers all sets of 5 things, say. But then quantity looks like another word for value. I have to ask though, you've never encountered the phrase "numerical value"??

If a number is something with a finite value and something to which the rules of arithmetic apply, then infinities are not numbers but something else. Then it's just an argument about semantics and whether to interpret "an infinite number" as an abuse of concept.

 

How disingenuous are you trying to be? Sure, if I'm teaching a child that 3 + 4 = 7, I'll explain that 7 is the value of 3 + 4. How do you get from that to ... well, to anything? 5 is a quantity, fine. In the context of set theory, \(\aleph_0\) is a quantity. It's the quantity of natural numbers. If you dislike modern math, that's your right. But we do need to start from a base of accepting what is already known and what is already standard math.

I don't care whether you regard transfinite ordinals and cardinals as numbers or not. I agree it's a matter of semantics. You are the one who seems to think there's something magical about the word "number," when I have already explained that in math, there is no general definition of the word "number." A number is whatever we agree to call a number in a particular context.

Understand that you are in error when you say, for example, that "Adding 1 to a number makes it larger." For example in the integers mod 5, adding 1 to the number 4 gives 0. The integers mod 5 are a field in which you can add, subtract, multiply, divide, even talk about polyomial equations for example; but they have no transitive order on them.

If you say, "Adding one to a number gives you a greater number," that's TRUE about the natural numbers, TRUE about the real numbers, FALSE about the complex numbers, FALSE about the transfinite numbers. No statement about "numbers" has any truth value at all until you say which numbers you're talking about.

So the word "number," in the abstract, has no meaning at all. So if we're talking about the natural numbers, they have certain properties. And if we're talking about the transfinite numbers, they have different properties.

You keep thinking the word "number" has some meaning in math that it doesn't have.

The question is this thread is whether certain collections are infinite. Mathematics has a perfectly sensible definition of when a collection is infinite. The word "unlimited" does not appear in this definition, nor is the word "unlimited" related to the mathematical definition of infinity.

It's true that poets, the mathematically unsophisticated, many ancient philosophers, and sadly even Wikipedia all think infinite collections can be described as "unlimited." However this is no longer how infinite sets are defined in mathematics. So if you wish to know whether a given collection of things is infinite or not, you need to use the most precise definition we have, which is the mathematical definition.

I have already given several examples of collections of things in the physical world that can fairly be characterized as "unlimited," even though as mathematical collections they are finite.

The reason the natural numbers are an infinite set is NOT because they're "unlimited," which has no technical meaning in math whatsoever; but rather because we can biject the set of natural numbers to a proper subset of itself. That's how we know the natural numbers are infinite.

The unit interval is bounded, which is certainly a reasonable interpretation of "limited," yet the unit interval is an infinite set.

I'm afraid at this point I'm just repeating myself, so I will have to leave it at that.

But let me ask you something. You seem to know about topology. Now a topology is defined as a set, along with a privileged collection of subsets (called the open sets) such that the open sets are stable under arbitrary unions and finite intersections.

How can you make sense of this definition if you reject or do not completely understand the technical definition of a finite set? How would you know if a collection of sets is stable under finite intersections if you reject the mathematical definition of the word finite?

 

It's not because we know we can add 1 to itself an unlimited number of times?

So what, though?
The unit interval can be open or closed, or rather you can define either set.

If this is directed at me, you will have to demonstrate what it is you think I don't understand about a collection of abstract elements which has finite size.

 

As the op noted in #49, the "display" scenario is settled in his opinion. I think the problem was misleading/unclear in referring to the pixel as an integral thing instead of a display. A painting is then a work in progress. I don't disagree with the conclusion since SR shows we see the far end of a stick at an earlier state than the near end. I also don't think it's significant in everyday activities.
Infinity will be transferred to a new post for my purposes.

 

DanShawen:From your Post #1

The 2,000,000 pixels allow for N = 2E2,000,000 black & white displays. Considering a time factor only results in the duplication of one of the possible displays.

In case you do not understand the above, consider the following.

If you generate one display per second for N seconds, you have shown every possible display. If you make more displays after N seconds, you must repeat one of the N displays. ​

 

Already knew that, from my first post. But pixels on such a display have many levels between on and off (LCD thick or thin). Therefore, it isn't as discrete a problem as you think.

This idea was extended in other posts to the idea that the number of possible arrangements of matter in the universe is discrete. It isn't either, because the universe has both matter AND energy, and also time. Arguments about the resolution of display devices or the human eye notwithstanding, the number of images possible in the real universe is infinite. You could not possibly view them all from an infinite number of angles, or in infinitely subdivided time intervals, if you lived to be many times the age of the known universe.

Thanks for the late comment, however.