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ETpro's avatar

Is the set of all information infinite or finite?

Asked by ETpro (34466points) March 26th, 2010

If it is finite, then can it be used to completely define anything infinite? At some point as we pursue finite information ever further, wouldn’t we fall of the edge of the world—so to speak—unable to speak any longer about the infinity we are endeavoring to describe.

If it is infinite, like the set of all true, or well-formed formulas in Principia Mathematica (PM) is, then can’t information be coded in such a way as to answer that question? If so, what do you think is the likelihood that a self-referential statement could be correctly coded in information/code similar to the set of PM Gödel Numbers proving that information/code can generate a proof that itself is false?

In other words, does not Gödel’s Incompleteness Theorem apply to any set of infinite information/code in such a way that information/code cannot prove itself in any but trivial applications.

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43 Answers

janbb's avatar

Must be a guy thing.

CaptainHarley's avatar

Most likely a very, VERY large, albiet finite number. Even if you consider the speed, location and direction of every single quantum as “information,” there are nevertheless a finite number of quanta in our universe, making the set of all information also finite, although many orders of magnitude larger.

ETpro's avatar

@janbb Ha! Thanks for a good laugh.

NOTE: Too late to edit, but in the first paragraph of the explanation of this i should have written: At some point as we pursue describing the infinite ever further…

janbb's avatar

@ETpro I’ll bet you dollars to donuts that guys are the only ones who answer this, except for my comments.

jackm's avatar

It is infinite by necessity.

nisse's avatar

I think Gödel’s theorem is about provability. It shows that there are by necessity:

Truths that we can prove
Truths that we cannot prove
Falisities that we cannot prove
Falsities that we can prove

This is the frustrating thing, within formal systems such as PM, there are things that are true, and just because of their “truthiness”, we cannot prove them.

I’ve still to grasp it completely tbh, but Gödel Escher Bach is a good and fun way to examine and try to understand the ramifications of the theorem.

To answer your question (if i understand it correctly), yes, I personally think that the Gödel theorem applies to our set of knowledge in the “non mathematical world” as well. There are truths which we simply cannot prove from inside “our system”, or from our vantage point as human beings.

This is of course only philosophical speculation. :)

MrItty's avatar

The mere existence of the concept of infinity means the set of all information must be infinite.

“Name all the numbers”. The knowledge of which number comes “next” is infinite.

“Describe every cubic centimeter of the universe, and how far it is from the center of the core of the Earth”. The fact that space itself is infinite means that this knowledge is infinite.

lilikoi's avatar

@MrItty made my point.

ETpro's avatar

@janbb You’re on. Meet you at Duncan Donuts. Since I don’t generally concern myself with the gender of my fellow Flutherites, will any female providing a non-tricvial answer to this please announce that I have won the bet?

@CaptainHarley Why do you believe that?

@nisse That was exactly where I felt this led. Thanks.

@MrItty So it would seem to me.

zophu's avatar

Sometimes, the greatest problems to be solved in life, are solved simply by being a healthy human being. That’s not to say that people should live simple lives and not worry about the greater things, like this. It’s that things like this are intuitive to the healthy mind, and not something that needs to be proven (or proven that it can’t be proven which is the inevitable end of your pursuit here.) It’s better that effort go to allowing more people to be healthy than go to “proving” some grand absolute that will only be proven wrong eventually. The mathematician’s folly, I think.

But then, my brain doesn’t work like a mathematician’s, so maybe there’s things that I’m missing.

infact, maybe i misunderstand the question almost entirely

nisse's avatar

@zophu: For me being a healthy human being means being interested in understanding the mysterious world we live in, hence the interest in these sort of questions. :)

zophu's avatar

@nisse heh, you’re right

Mariah's avatar

Hmm… I might argue with the people who say that since there are infinite numbers, information in the universe is infinite. Because numbers, unless they truly represent something in the universe, are essentially manmade. We can imagine a number like 10^1000000, but is that number a real thing if it doesn’t apply to anything physical?

Then there’s the idea that perhaps we can get infinitely small, that we can just keep splitting distances in half forever. But probably the existence of fundamental particles disproves that (I wonder, though, if empty space has a fundamentally small unit).

@MrItty I would also argue that we do not know for sure that space is infinite. The universe may just be a big “loop,” if there exist more dimensions than the three we’re used to.

cockswain's avatar

Wouldn’t it be infinite, since you could always subdivide or expand any value into infinitely larger or smaller units? Like if I wanted to define, say, the speed of a fastball. Couldn’t I define it at 99.01mph, 99.013, 99.0127, etc, infinitely? Further, couldn’t I devise infinite units to measure the speed?

It seems that in essence if you can find one infinite value anywhere in the universe, all possible information must be infinite. maybe my reasoning is too simplistic for this discussion

MrItty's avatar

@Mariah it doesn’t matter if numbers represent something real or man-imagined. The question is about “information”. Not “information pertaining to tangible items”. Whether the question is “What is the next number?” or “What does mankind’s consensus imagine the next number to be?” it’s still knowledge. Still information.

zophu's avatar

Well, whether or not everything is actually infinite, we may always have to see it that way. Perhaps the condition of observance, or even existence, demands that all perspectives had require there to be an infinity—if it is only the infinity of cycling unknowns. For, even if we learn everything, there may always be things that are forgotten.

A comfortable guess is that information is not infinite, but it might as well be, for it will always seem infinite in comparison to our finite capacity for information.

i’m trying out putting “for” before statements. i don’t really like it that much. sounds too old-timey

lilikoi's avatar

@Mariah We are used to four dimensions. Space doesn’t need to be infinite in order for information to be infinite, as there are other examples of things that are infinite as @MrItty pointed out prev.

@zophu Information is knowledge gained. To say that information is finite is to say that there is a point where we can be all knowing. Surely on the individual level, this is not possible. Collectively as a species, I don’t think it is possible either: We can’t ever know the last number because there is no last number.

There is also what I’ll call a psychological question of objectivity. We are constrained to a human perspective. Everything we perceive is through a “human lens”. Can we learn everything about ourselves from within, or would something else looking at us as we look at fish in a bowl be able to know things that we are physically incapable of realizing due to our constrained perspective?

Will we ever be able to know with certainty what it is like to be another species?

janbb's avatar

@ETpro Dang – I lost. Boston creme or glazed?

ETpro's avatar

@MrItty Whether the universe is infinite or not, a thought experiment can construct a vehickle that travels not in the loop of space-time, but in such a way that it departs the space-time universe and thus defines infinitely more space ebtween itself and the universe as it infinitely travels outward. Ergo there must be infinite room for space-time to expand into.

@cockswain Not only can I define real numbers that define real things in infinitely finer detail, I can define them separately in all the languages that have ever existed (well—I actually can’t right now but given infinite time to learn all those languages I could) and then I can go on to invent infinitely more code systems in which to define those things.

@lilikoi That, my friend, is the crux of the biscuit I’m trying to bake here. :-)

@janbb On OPM? Boston cream. :-)

Mariah's avatar

@lilikoi I meant four spatial dimensions. And I only said the thing about space not being infinite because of @MrItty‘s statement, “Describe every cubic centimeter of the universe, and how far it is from the center of the core of the Earth”. The fact that space itself is infinite means that this knowledge is infinite. Also, does information necessarily have to be knowable? By the Heisenberg Uncertainty Principle, we can never simultaneously know the exact position and velocity of any particle, yet certainly this information exists.

@ETpro it departs the space-time universe and thus defines infinitely more space ebtween itself and the universe as it infinitely travels outward. I just don’t know that we know enough about the universe to say that this thought experiment is valid. We don’t know that the universe is infinitely large, or that it doesn’t loop so that the vehicle would end up back where it started (I can explain further what I mean by this later, but I am in a hurry right now).

lilikoi's avatar

@Mariah By definition, information is something that is known. Perhaps we cannot sense the exact position and velocity of a particle in real-time, but we know and can prove it exists and in the case of a projectile for example we can track position and velocity over time and generate a database to collect this information.

Just to clarify, there are 3 spatial dimensions and time, the fourth dimension which is not spatial. Physicists think there are more, but I have enough trouble with 4 dimensions and don’t like to think about any more than that.

LostInParadise's avatar

If space is finite then information is finite also. Information would consist of all possible arrangements of particles and energy.

Mariah's avatar

@lilikoi It’s not considered information until someone knows it? Well, I’ll be darned. I was under the impression that information about an object was somewhat synonomous to its properties; that it exists whether anyone knows about it or not. I guess that changes this debate quite a bit.

My point exactly. There are three spatial dimensions that we can observe, but if physicists are correct and there really are more than three, it might be possible to travel away from a point in a “straight” line and end up right back where you started. Thus you couldn’t get infinitely far away from anything.

CaptainHarley's avatar

@ETpro

Most current cosmological theories about the universe conclude that it is finite, but unbounded.

ETpro's avatar

@Mariah I get your point about multiple undiscovered dimensions. Mine was that if a straight line ultimately brings me back to where I started, then my wonderful, flying though-machine will fly in a unique crooked line such that it leaves the boundaries of space-time. How do you conceive that space-time is bounded? What do you run into at its outer limit? Can you tell as you approach that limit that you are nearing a wall, or do you just vanish when you penetrate it?

And then, can I not divide a single inch into infinitely smaller increments, or is there some limit where division simply fails to work? If I can divide a single inch into infinitely small increments and define what i do or do not find there, I can produce and infinite amount of information about a single cubic inch—I don’t need an infinite universe to get to infinity—even though as @CaptainHarley notes, our best evidence today is that my machine could travel on infinitely far filing reports every on trillionth of a millimeter about what it found at a given point before it.

Your point about the definition of information is well taken. I tripped over that one just a day ago. But to answer the question, can infinity exist or dies it, we do not have to actually produce it. We can easily demonstrate that the set of whole all integers, or all primes, or all Fibonacci numbers are all infinite sets. We don’t have to actually calculate the infinite set to see that it goes on forever. The same goes for demonstrating that the amount of information that is potentially available is infinite.

Anyway, excellent challenges. If I have missed some part of it, or there is more to add, please chime back in.

@LostInParadise Space is perhaps a poor place to look for infinity. Math contains lots of know infinities. Wouldn’t fully describing any one of them produce an infinity? Perhaps we ask, “What is the complete set of squares/” We could sit in one place in space-time and given an infinite amount of time to calculate, build an infinite store of information answering that one number property—squarness.

Mariah's avatar

@ETpro
Disclaimer: Obviously, I am very very much a layman so somebody correct me if I say anything ridiculous.

How do you conceive that space-time is bounded? What do you run into at its outer limit? Can you tell as you approach that limit that you are nearing a wall, or do you just vanish when you penetrate it?
My train of thought is (and obviously this is just speculation, and but one of many many possibilities) that there could be a fourth spatial dimension that we don’t perceive. I can’t think of any way to explain the looping effect this could have without using this metaphor: imagine that there’s a creature who thinks he’s living in two dimensions that exists on the surface of the Earth. He wouldn’t be exactly two dimensional because of the Earth’s curvature. However, since the Earth’s radius is very very large in comparison to him, the curvature is very gradual and he therefore doesn’t realize it exists. Now, he plants a flag in the ground and starts walking in what he thinks is a straight line leading directly away from the flag. In due time, he circles around the Earth and ends up right back where he started at his flag. It’s very hard to think about four dimensions, but I’d suppose by extrapolation that it’d be possible that space itself could have a slight curvature in the fourth dimension that would bring us right back to Earth if we sent a spacecraft far enough away. No boundary, no “wall of the universe.” Just walking in circles. It’s a kind of neat concept to think about.

And then, can I not divide a single inch into infinitely smaller increments, or is there some limit where division simply fails to work?
Up higher I wrote this: “Then there’s the idea that perhaps we can get infinitely small, that we can just keep splitting distances in half forever. But probably the existence of fundamental particles disproves that (I wonder, though, if empty space has a fundamentally small unit).” I think because matter is quantized, you can’t legally divide something as many times as you want. But who knows, really.

Part of the reason I’ve taken such interest in this question is that I once asked a very similar question at the debate site I used to frequent before it shut down, except that I think I phrased it, “Does infinity exist in the physical world?” By this I meant that numbers could only be included if they were shown to actually exist somewhere other than our minds. Somebody mentioned irrational numbers (infinite precision) and I shot them down with the claim that irrational numbers probably don’t show up in the real world. I said that there probably doesn’t exist a circle whose circumference is equal to its diameter times exactly pi. Simply due to the fact that the real world isn’t like the Cartisian plane: perfect curves cannot exist because matter is quantized, and so two particles sitting next to each other would technically make a straight line – rather than a circle, it would be a shape with an extremely large number of sides that is almost a circle. However, now that I think about it further, aren’t black holes supposed to be perfect spheres? That would be a physical manifestation of pi.

Goddamn, sorry this is so frickin’ long.

ETpro's avatar

I am delighted to what you added. Thanks. I dispute that numbers are not information, thought. You cannot say that data or states in the real universe are not information until someone knows them, and they say that numbers are not information because they exist only in our heads. Know things, knowledge, is fundamental to the definition of information

The thought of the exact shape of a black hole is quite profound. Thanks for that.

Mariah's avatar

@ETpro I wasn’t the one saying that data isn’t information until it is known. That apparent definition of the word was news to me. But yes, of course if we include numbers as information, then information would be infinite. I simply think it makes for a more interesting discussion if we rule out numbers. Everyone knows there’s an infinite number of numbers. :]

ETpro's avatar

@Mariah OK, challenge accepted. Let’s return to my fantasmagorical thought spaceship, because I do not believe that you have fully grasped its wondrous properties. Let’s allow that the universe, while unbounded, needs no bounds because a 4th spatial dimension (or 5 or 12 or whatever sting theorist are up to now) exists that makes travel in a straight line eventually bring me right back where I started. Let’s also allow, to make this challenge interesting, that you cannot divide an inch forever. Let’s say that the smallest meaningful measurement you can make is equal to the size of the smallest existing sub-atomic particle. That seems a reasonable concession since we have set numbers themselves aside. What good would the number for half that particle size be as information?

So traveling in some corkscrew pattern would take me through vast quantities of a finite universe and I could measure our smallest meaningful unit each time I traveled a distance equal to it, producing a really huge number, but still a finite one.

But my ship knows all that. So it has been programmed with a clever algorithm that prevents it ever traveling in a straight line or a path that retreads old ground. Instead, it travels a path that will eventually take it directly through every speck of the universe—every part that is as big as our smallest particle. Now my ship will not meet its starting point but will eventually stumble upon the intersecting point with another dimension and fly right out of the unbounded universe. I am now outside space-time, except that as I move on I am creating it.

But how do I radio home about my discovery?

LostInParadise's avatar

@ETpro By information, I was assuming that we are talking about actual objects. In mathematics, you can make things up ad infinitum. Any set of consistent axioms defines a mathematical system, which may or may not be of much interest. This mathematics tells us nothing about the real world until it is applied.

@Mariah To understand how the universe may be both bounded and finite, imagine a two dimensional universe on the surface of a sphere. Our 3 spacial dimensions may be the analog of the surface of a 4 dimensional object. A good introduction to these ideas is The Shape of Space

mattbrowne's avatar

As @MrItty said the mere existence of the concept of infinity means the set of all information must be infinite. I think what is finite is the means to actually encode it, for example when we add up all possible quantum states of our universe (multiverse).

malevolentbutticklish's avatar

@MrItty: “Name all the numbers. The knowledge of which number comes next is infinite.” <== I have to argue with you on this..

As @mattbrowne says there may be a limited number of encodings. If you start at one and name the next integer this means that there is a largest integer which can be represented and eventually you won’t be able to name the “next” integer. You could then call the largest integer Z and say it is Z+1 but now you have multiple encoding schemes and some states must be used to specify the encoding scheme. Since no encoding scheme can record more integers than Z and at most there can be Z encoding schemes the total number of integers that can be represented by any method is less than Z^2.

LostInParadise's avatar

There must be a largest number that could possibly have any practical application. If you look at information in terms of the real world, it is finite.

ETpro's avatar

@LostInParadise I really prefer working with dictionary definitions. Making up our own renders discussion ever more difficult.

@malevolentbutticklish Per the definition of information, English or Words in any language aren’t the only meaningful things qualifying as information. Numbering systems including base 10, binary and others are all included. And it is very easy to name a bigger integer than any supplied in either base 10 or binary.

@LostInParadise I would say that any integer than answers a question like, “Is it a square?”, “Is it a cube?”, “Is it a prime?”, “Is it a Fibonacci Number?” or “Is it a Godel Number?” all have been proved to supply infinite sets of meaningful answers.

malevolentbutticklish's avatar

@ETpro:
a) You do acknowledge that If there are a limited number of encodings then there must be a limited number of integers which can be expressed individually and exactly?
b) “And it is very easy to name a bigger integer than any supplied in either base 10 or binary.” <== I am not sure what you intend by this?

ETpro's avatar

@malevolentbutticklish To your first point, no. The encodings, 0 and 1 are a very finite set and are quite sufficient to the task. As to what I mean about there always being additional integers available, we need look no further than Euclid’s proof of the Infinitude of Primes which has withstood peer review for over 2,300 years and still stands.

malevolentbutticklish's avatar

@ETpro: This peer review does not take into account the possibility of a finite set of multiverse states (nor do they wish to). It is no small wonder that when you ignore the finite set of multiverse states some answers change. The concept of integers going on forever exists just like the concept of time travel but the reality may very well be otherwise.

ETpro's avatar

@malevolentbutticklish It doesn’t matter whether there is only one universe of there is some finite set of multiverses. What we are discussing here is the set of all possible information’s size, not how many universes there are. The more, the merrier.

malevolentbutticklish's avatar

@ETpro: “the set of all possible information’s size” must fit inside a finite sized universe. Therefore, while the “conceptual size” of that set is unlimited the real size of the set is in fact limited. The actual limit may be really rather large, but it is there.

ETpro's avatar

@malevolentbutticklish The conceptual size is what’s at issue in Gödel’s Incompleteness Theorem.

LostInParadise's avatar

I looked at the definition of information in the online Merriam-Webster and it is pretty vague about what the term covers. At any rate, in the world of matter and energy there is a finite amount of information. In the world of mathematics there is an infinite amount. We can open up a whole other discussion as to the reality of mathematics. One curious aspect of numbers is that the number line is covered mostly by numbers that are undefinable.

malevolentbutticklish's avatar

@ETpro: The “conceptual size” is unlimited but any “facts” which cannot be made to function on a finite size set of information aren’t facts at all. They are fiction. You can write a book about such “facts” but you could never apply them to the universe with physics.

ETpro's avatar

@malevolentbutticklish What do you conclude from that?

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