What is 'Infinite'?

[NichG]

Here's a more careful statement then.

If x = sum_i=0^n a_i k^-i and y = sum_i=0^n b_i k^-i for a,b non-negative integers less than k, a_0 and b_0 >0, and k being a positive integer>1, then for n finite x=y iff a_i = b_i for all i

For n -> infinity, this is no longer true, because the residual sum can add up to a factor of the base k when a_i = k-1.

[Rockphed]

Here's a more careful statement then.

If x = sum_i=0^n a_i k^-i and y = sum_i=0^n b_i k^-i for a,b non-negative integers less than k, a_0 and b_0 >0, and k being a positive integer>1, then for n finite x=y iff a_i = b_i for all i

For n -> infinity, this is no longer true, because the residual sum can add up to a factor of the base k when a_i = k-1.

Gah! I hate trying to discuss math on these boards because there isn't a good way to properly express everything. Someone should make a board software that properly integrates tex support.

That said, I think I agree with you. I was just mumbling out loud about a related issue.

[keybounce]

Consider these ideas as an answer to "what is infinite?"

For normal practical purposes, "infinite" is about 1.8e308.

For real big purposes, "infinite" is somewhere around 2^(10^120)

It is very, very hard to come up with something that is bigger than that last one and still finite. :-)

(There's about 10^80 things in the *visible* universe. There's about 10^120 quantum events in history of the *visible* universe. There's about 2^10^120 possible ways the *visible* universe could have played out. While it's technically possible to have a bigger number, it would not have any meaning).

EDIT: As pointed out, these numbers represent the visible universe, not the entire universe.

[danzibr]

It is very, very hard to come up with something that is bigger than that last one and still finite. :-)

While I like the rest of your post, I can’t bring myself to agree with this bit.

[georgie_leech]

While I like the rest of your post, I can’t bring myself to agree with this bit.

Mm. It's easy. Here, behold as I make a bigger number than both of them:

(2^(10^120)) x 3

What, still not big enough?

((2^(10^120)) x 3)^2

Still not big enough? Fine, here's a way to keep getting bigger numbers, because I'm lazy and want an easy way to get them:

((2^(10^120)) x 3)^2 +1. If that isn't big enough, add another +1. You can do that literally forever and never run out of bigger numbers.

[Lvl 2 Expert]

Mm. It's easy. Here, behold as I make a bigger number than both of them:

(2^(10^120)) x 3

What, still not big enough?

((2^(10^120)) x 3)^2

Still not big enough? Fine, here's a way to keep getting bigger numbers, because I'm lazy and want an easy way to get them:

((2^(10^120)) x 3)^2 +1. If that isn't big enough, add another +1. You can do that literally forever and never run out of bigger numbers.

Keybounce is not arguing you can't mathematically make larger numbers, they're arguing you'll never have a practical purpose to use any of those numbers, except maybe if you go out of your way to find those purposes for the purpose of this discussion.

[BeerMug Paladin]

Keybounce is not arguing you can't mathematically make larger numbers, they're arguing you'll never have a practical purpose to use any of those numbers, except maybe if you go out of your way to find those purposes for the purpose of this discussion.

I'm not sure it counts as practical, but proving the validity of calculus requires conceptual infinity. Or infinitesimals, but I'd rather not go there.

I suppose there's a lot of folks okay with the application of ideas without any regard to their validity, but they are neither mathematicians nor scientists. Granted, neither field is particularly geared towards practicality....

Although his argument about never having an engineering/descriptive purpose for larger numbers does seem to be valid by current scientific theory.

[Lord Torath]

Consider these ideas as an answer to "what is infinite?"

For normal practical purposes, "infinite" is about 1.8e308.

For real big purposes, "infinite" is somewhere around 2^(10^120)

It is very, very hard to come up with something that is bigger than that last one and still finite. :-)

(There's about 10^80 things in the universe. There's about 10^120 quantum events in history of the universe. There's about 2^10^120 possible ways the universe could have played out. While it's technically possible to have a bigger number, it would not have any meaning).

Okay, at first I thought you were saying 10¹²⁰ was "infinity", and I figure that's way too small. Heck there are estimated to be 10¹⁷¹ outcomes for Go games! And 10¹⁷¹ is 50 orders of magnitude larger than 10¹²⁰. Then I re-read and noticed that it was 2^10¹²⁰. I'm not quite sure how 2^10¹²⁰ compares to 10¹⁷¹, or even 10⁷⁰⁰ (from wikipedia: 10⁷⁰⁰ is thus an overestimate of the number of possible games that can be played in 200 moves and an underestimate of the number of games that can be played in 361 moves.), but I'm pretty certain it (2^10¹²⁰) is a good bit bigger.

[eggynack]

Okay, at first I thought you were saying 10¹²⁰ was "infinity", and I figure that's way too small. Heck there are estimated to be 10¹⁷¹ outcomes for Go games! And 10¹⁷¹ is 50 orders of magnitude larger than 10¹²⁰. Then I re-read and noticed that it was 2^10¹²⁰. I'm not quite sure how 2^10¹²⁰ compares to 10¹⁷¹, or even 10⁷⁰⁰ (from wikipedia: 10⁷⁰⁰ is thus an overestimate of the number of possible games that can be played in 200 moves and an underestimate of the number of games that can be played in 361 moves.), but I'm pretty certain it (2^10¹²⁰) is a good bit bigger.

I mean, worst case scenario, we could just make a bigger version of Go. Such a game would probably crush these tiny numbers pretty fast.

I don't really like this "Measure of objects in reality as infinity" thing. For one thing, as is noted here, most of the estimates are inevitably going to be too small. Like, there's n objects out there, right? But what if I want two of them? Or three? In other words, what if I want the power set of objects in reality? So now I have a much much bigger number. Except, hey, what if I want to consider two of those sets at once? Or, hell, even consider a given object twice? I look at a rock and ask, "Hey, it'd be neat were there two of that rock."

Beyond there being technically no limit to infinity, I am thoroughly unconvinced that there is a practical limit to reality. There are always questions we can ask, sometimes really straightforward and even practical questions, that can invoke values that extend beyond any bound you'd name. There's always n+1, and there's always need for n+1.

[Lord Torath]

I mean, worst case scenario, we could just make a bigger version of Go. Such a game would probably crush these tiny numbers pretty fast.

I don't really like this "Measure of objects in reality as infinity" thing. For one thing, as is noted here, most of the estimates are inevitably going to be too small. Like, there's n objects out there, right? But what if I want two of them? Or three? In other words, what if I want the power set of objects in reality? So now I have a much much bigger number. Except, hey, what if I want to consider two of those sets at once? Or, hell, even consider a given object twice? I look at a rock and ask, "Hey, it'd be neat were there two of that rock."

Beyond there being technically no limit to infinity, I am thoroughly unconvinced that there is a practical limit to reality. There are always questions we can ask, sometimes really straightforward and even practical questions, that can invoke values that extend beyond any bound you'd name. There's always n+1, and there's always need for n+1.

Keep in mind that 10¹²⁰ is a 1 with 120 zeroes behind it. While not nearly as big as 10^10¹²⁰, 2^10¹²⁰ is still mind-breakingly huge. It leaves a dinky little number like 10⁷⁰⁰ looking almost infinitesimally small by comparison. If you want to compare the total number of possible outcomes in two (or more) universes, you might need a bigger number. But even measuring the volume of the visible universe in units of the Plank scale, you only need about 300 exponents.

If you want to play Go on a quantum foam-scale board the size of the known universe, you might need a bigger number. But how practical is that, really?

It's important to realize that keybounce isn't saying that 2^10¹²⁰ is infinity, just that you are exceedingly unlikely to ever need a number bigger than that for a real-universe calculation.

[Grey_Wolf_c]

"Hey, it'd be neat were there two of that rock."

But there isn't. There is just the one universe, and keybounce's point is that his numbers encompasses everything in the universe - not just what did happen, but could possibly ever happen. That's the point of the ludicrously large "every possible quantum event combination" number. Which would include the scenario in which you did have two rocks, by the way.

If keybounce's numbers have an issue, it is that it is considering, as far as I know, merely the visible universe. One of my eternal pet peeves with astronomy is their tendency to call the visible universe just "the universe", despite knowing full well there are stars beyond the visible universe that are not being accounted for in these "total number of atoms in existance" calculations. Now, I know why they do it - there is literally no way to know how much more universe we cannot perceive and indeed will never be able to perceive, since the distance between us is increasing faster than the speed of light - but still, these kind of estimation are therefore incorrect, probably by orders of magnitude that might even dwarf 10¹²⁰.

Grey Wolf

[eggynack]

Keep in mind that 10¹²⁰ is a 1 with 120 zeroes behind it. While not nearly as big as 10^10¹²⁰, 2^10¹²⁰ is still mind-breakingly huge. It leaves a dinky little number like 10⁷⁰⁰ looking almost infinitesimally small by comparison. If you want to compare the total number of possible outcomes in two (or more) universes, you might need a bigger number. But even measuring the volume of the visible universe in units of the Plank scale, you only need about 300 exponents.

If you want to play Go on a quantum foam-scale board the size of the known universe, you might need a bigger number. But how practical is that, really?

It's important to realize that keybounce isn't saying that 2^10¹²⁰ is infinity, just that you are exceedingly unlikely to ever need a number bigger than that for a real-universe calculation.

I'm aware that big numbers are big. I'm aware secondarily that he's not claiming this is "true" infinity. He's just claiming this is practical infinity. And I'm disagreeing. Infinity is practical infinity. Honestly, these massive numbers are way less practical. That's why we use infinity in the first place.

But there isn't. There is just the one universe, and keybounce's point is that his numbers encompasses everything in the universe - not just what did happen, but could possibly ever happen. That's the point of the ludicrously large "every possible quantum event combination" number. Which would include the scenario in which you did have two rocks, by the way.

It only contains the scenario in which there are two rocks if that material is coming from somewhere else in the universe. The idea that a "practical" study of the universe can encompass only the broad situation we're in now (including alternate probabilistic outcomes), but not the situation where we imagine one rock more or less, strikes me as kinda ridiculous. We can see a room with one rock in it, imagine a second rock, and still consider everything that's happening practical and in accordance with standard human imagination. There's no meaningful difference between this situation and the universe bending one I posited.

I guess that what I'm expressing here, on the most basic level, is that it's really trivial to get to "+1", regardless of starting circumstance, in a way that reflects what you might want to do. The idea that any number larger than this one is pointless is rooted somewhat in the idea that the massive number we're already using is pointless. And I don't think it is.

[keybounce]

Yes, I was thinking visible universe. Previous post fixed.

And, as for "+1" , "*3", 2^" (power set), etc -- that's not really a change in scale. Ok, Power set is a change in scale -- that's the difference between 10^120 quantum events, and 2^10^120 possible ways those quantum events could have happened. But another power set of that? What would that represent?

Calculus depends on a meaningful limit theory. (Warning: I last studied limit theory in BC calculus in 12th grade, which was approximately 1982. This might not be correct). Limit theory does not require a perfect infinity, it requires a sufficient concept of an infinitesimal, and it requires that a function's behavior does not change as your infinitesimals gets smaller. It does not require that the infinitesimal be small enough that it cannot be resolved. There is an actual formal number system (conway numbers, do not ask me to explain it, I barely understand it, maybe, probably not) that has an actual number for infinitesimals that you can do math with.

Integration is, essentially, the sum of products -- you are calculating the value of a region by estimating the height of the graph times the width of the area being looked at -- and you are shrinking the width down to that infinitesimal width. 30 years ago I could have actually walked you through the process of solving for the integration process; today, I remember the results but not how to derive them.

Differentiation is essentially "what is the slope" -- f(x-epsilon) compared to f(x+epsilon) over 2*epsilon -- and you need to look at both sides because you cannot take the derivative at a cut-point.

In both cases, you have "Here's a number", which just needs a sufficiently fine step value, versus "here's the formula", which requires high-end symbol manipulation and rules of math.

In anything having to do with physics or the real world, "epsilon" is the plank length or the plank time -- nothing smaller makes any sense. Or a quantum coin toss. Etc. You can work with that as your finite infinitesimal without worrying about something smaller as your theoretical concept.

I'm not concerned with being able to duplicate a ball with a perfect concept of infinity. You cannot duplicate a ball in reality no matter what tech level you have.

Big numbers? Grahm's number -- as big as it was, it certainly was much bigger than 2^10^120 -- turns out to actually be less than 65536, bigger than 12, and last I read the mathematicians working on it had the intuition that it was about 20. Again, it was a number that had a physical concept, that was a simple operation on graphs.

TREE()? Sure, it's huge. Basically, as I understand it, you could not generate the full set in the universe -- any representation of the set won't fit in the visible universe. [It's easier to fill a zfs file system -- apparently, the energy needed to generate enough information at it's minimum would only be enough to boil the oceans on earth. (Disclaimer: This came from a paper I read back when ZFS was new, and I cannot find any reference to this tonight.)]

Go? 10^171 possible outcomes? Ok, so you cannot represent every single outcome of go in the possible universe. Heck, a simple 2x2 board has a ridiculous possible number out outcomes because you can wind up with what looks like repetition but just with different numbers of removed stones. Yes, the vast majority of those will never happen in real life (one person passing, the other placing stones until all but one space is filled, then the passing player plays one stone and the board is otherwise empty with a lot of removed stones. Now repeat this until every possible square is the singleton for each player twice. Now repeat with two spots for each player instead of 1 spot for each player. Etc. Go's trivial cases are crazy numerous.). But OK, you've got more Go outcomes than the universe can describe, and it's a reasonable but excessive finite number. But that's not the same as the number of go board positions (a given spot can be white, black, or empty, so less than 3^181 as not all combinations are legal).

> In other words, what if I want the power set of objects in reality?
That's the 2^10^120 case. (Well, maybe less. The figures I've heard are 10^80 elementary particles now, and 10^120 quantum events over the 13.7by history.)

[eggynack]

And, as for "+1" , "*3", 2^" (power set), etc -- that's not really a change in scale. Ok, Power set is a change in scale -- that's the difference between 10^120 quantum events, and 2^10^120 possible ways those quantum events could have happened. But another power set of that? What would that represent?

Why do I care, precisely, about changes in scale? As long as a given number isn't big enough to capture the entirety of a situation, the next biggest number has utility. What if there's one more quantum event than that number you just said? Then that's actually the true amount of events that could have happened. I don't necessarily agree that numbers need to have a direct reflection in reality. Sometimes there is utility beyond the limitations of the universe. But, if we are stipulating that, then why accept less than full accuracy?

Calculus depends on a meaningful limit theory. (Warning: I last studied limit theory in BC calculus in 12th grade, which was approximately 1982. This might not be correct). Limit theory does not require a perfect infinity, it requires a sufficient concept of an infinitesimal, and it requires that a function's behavior does not change as your infinitesimals gets smaller. It does not require that the infinitesimal be small enough that it cannot be resolved. There is an actual formal number system (conway numbers, do not ask me to explain it, I barely understand it, maybe, probably not) that has an actual number for infinitesimals that you can do math with.

I don't really agree. If you use the actual definitions that underlie calculus, "perfect" infinity shows up all over the place. For a sequence to converge to a value, for example, then there must be no real number such that you can't find almost all of the sequence between that number and the value (I'm paraphrasing a bit). There is no "sufficiently close".

More importantly, calculus is fundamentally dependent on the real numbers, and similarly uncountable infinities. It's just where the study lives. Without that infinite density and full measure, it just wouldn't work. It doesn't work in the naturals, and it doesn't even work in the rationals.

In anything having to do with physics or the real world, "epsilon" is the plank length or the plank time -- nothing smaller makes any sense. Or a quantum coin toss. Etc. You can work with that as your finite infinitesimal without worrying about something smaller as your theoretical concept.

That strikes me as the opposite of true. Measuring things with actual infinity is easy. Measuring things with a ridiculous number of tiny Planck lengths is insanely hard. Calculus, which uses actual infinities, simplifies things. You're introducing worries, not taking them away.

I'm not concerned with being able to duplicate a ball with a perfect concept of infinity. You cannot duplicate a ball in reality no matter what tech level you have.

Why is that the metric of utility? Why must things have the most obvious possible expression in reality for them to be worthy of concern? The fabric of the reals is essential to the functioning of calculus, and the way a mathematical sphere operates is essential to the fabric of the reals.

Big numbers? Grahm's number -- as big as it was, it certainly was much bigger than 2^10^120 -- turns out to actually be less than 65536, bigger than 12, and last I read the mathematicians working on it had the intuition that it was about 20. Again, it was a number that had a physical concept, that was a simple operation on graphs.

When was the upper bound reduced that low? Wikipedia said the upper bound was 2↑↑↑6. I think that's somewhat bigger than 65536, which, from what I'm reading, is 2↑↑↑3.

[gomipile]

Big numbers? Grahm's number -- as big as it was, it certainly was much bigger than 2^10^120 -- turns out to actually be less than 65536, bigger than 12, and last I read the mathematicians working on it had the intuition that it was about 20. Again, it was a number that had a physical concept, that was a simple operation on graphs.

I think you're confusing Graham's number for the smallest value for n that satisfies the Ramsey theory problem Graham's number was applied to. Graham's number is Graham's number, it doesn't get smaller if we find a smaller upper bound on that Ramsey theory problem.

[Lord Torath]

Okay, I had no idea what Graham's number actually was. Now that I know what it is, I still don't know what it is! Small "eeks" aren't going to cut it. This requires big ones!

[Grey_Wolf_c]

Okay, I had no idea what Graham's number actually was.

Grey Wolf

[Peelee]

I refuse to accept Graham's pile of pumpkins.

[JCarter426]

Okay, I had no idea what Graham's number actually was. Now that I know what it is, I still don't know what it is! Small "eeks" aren't going to cut it. This requires big ones!

Allow me to introduce you to Graham's number's bigger sibling TREE(3).

[Lord Torath]

Allow me to introduce you to Graham's number's bigger sibling TREE(3).

See, now TREE(3) just makes me go ? All I have to go on is his word, and needing "2 tower 1000" symbols to prove it's finite using finite math, which is a good bit smaller than G1 (1 sun-tower of 3s). And some comments saying it's very, very hard and technical to explain just how big it is.

[JCarter426]

See, now TREE(3) just makes me go ? All I have to go on is his word, and needing "2 tower 1000" symbols to prove it's finite using finite math, which is a good bit smaller than G1 (1 sun-tower of 3s). And some comments saying it's very, very hard and technical to explain just how big it is.

You need numbers on the order of Graham's number just to prove TREE(3) isn't infinitely big. If you had a Graham's number amount of people, they couldn't begin to imagine how big TREE(3) is.

That's an interesting point about perception, though. As big as Graham's number is, it's conceivably big. You can understand exponential operations and how iterating that process can produce seriously big numbers like Graham's number. Even if you can't really know how big a number it is, you can understand how you got there. In the same way you can think that's a pretty big sack of sand but you don't think about how much sand there is on a beach. Even if you don't know how much sand there is in a big sack of sand, you can still grasp it as an amount that can be measured, unlike a beach... never mind numbers bigger than the amount of all the grains of sand on all the world's beaches.

It's the same with infinity. It's not the concept of a number bigger than all the other numbers that people have a problem with. We can think in terms like "I can't count to a hundred without losing count" or "I'm never going to make enough money to pay off this loan" and conclude there must be some number out there that's just too big and be ok with the concept of something unimaginatively big. But that's not what infinity is. It's not some arbitrarily large number like Graham's number or TREE(3). It's even bigger than those, absurdly bigger, and just the idea of "bigger" is not enough to describe it. It's weird properties and behaviors of infinity like the cardinality of different infinities and those hotel paradoxes where people lose it. When you add those into the mix, it stops being relatable to our reality. And that's something that TREE(3) is guilty of even though it's still finite. There is no physical phenomenon in the universe even close to its scale. It's too ridiculously big to fit in this universe.

[loasmkers]

Then it should start with 0/1.

Yes,you are right ,that is the end and the beginning .

[Rockphed]

See, now TREE(3) just makes me go ? All I have to go on is his word, and needing "2 tower 1000" symbols to prove it's finite using finite math, which is a good bit smaller than G1 (1 sun-tower of 3s). And some comments saying it's very, very hard and technical to explain just how big it is.

Yeah, needing 5 minutes to explain what the problem even is is a healthy reason for me to assume that it is akin to crates of balloon juice.

[georgie_leech]

Yeah, needing 5 minutes to explain what the problem even is is a healthy reason for me to assume that it is akin to crates of balloon juice.

...is akin to what now?

[Rockphed]

...is akin to what now?

Sending someone around for a crate of balloon juice. Similar to sending someone on a snipe hunt.

[Grey_Wolf_c]

...is akin to what now?

Balloon juice is a reference to a type of hazing - usually in the military - where a newbie is sent in pursuit of an article that doesn't exist. In this case, canisters of balloon juice to fill up the balloon, but also quite famously turn blinker fluid, three-pronged compasses for drawing ovals, left-handed wrenches, etc.

Grey Wolf

[Peelee]

Balloon juice is a reference to a type of hazing - usually in the military - where a newbie is sent in pursuit of an article that doesn't exist. In this case, canisters of balloon juice to fill up the balloon, but also quite famously turn blinker fluid, three-pronged compasses for drawing ovals, left-handed wrenches, etc.

Grey Wolf

Even though I got the joke, it still seems like a really crappy joke; it's really easy to get balloon juice. Headlight fluid is clearly the superior hazing method.

[Lord Torath]

Balloon juice is a reference to a type of hazing - usually in the military - where a newbie is sent in pursuit of an article that doesn't exist. In this case, canisters of balloon juice to fill up the balloon, but also quite famously turn blinker fluid, three-pronged compasses for drawing ovals, left-handed wrenches, etc.

"He was sent for the key to the bowsprit to the captain of Battery B. And the captain sent him back for some red lamp-black and a camouflage coat for the sea."
-- The Redheaded Rookie

My response for the "blinker fluid" question was to turn the engine on. "If there's enough juice to start the engine, there's plenty of juice for the blinkers!"

As for "balloon juice", I respectfully submit panel 10 and Panel 7.

[Peelee]

My response for the "blinker fluid" question was to turn the engine on. "If there's enough juice to start the engine, there's plenty of juice for the blinkers!"

"Yeah, there's enough to start it, but it needs to be refilled! You can start a car with almost no gas, but that don't mean it's gonna run a mile down the road! Now run until you puke for givin' me lip!"

As for "balloon juice", I respectfully submit panel 10 and Panel 7.

[georgie_leech]

Sending someone around for a crate of balloon juice. Similar to sending someone on a snipe hunt.

Balloon juice is a reference to a type of hazing - usually in the military - where a newbie is sent in pursuit of an article that doesn't exist. In this case, canisters of balloon juice to fill up the balloon, but also quite famously turn blinker fluid, three-pronged compasses for drawing ovals, left-handed wrenches, etc.

Grey Wolf

I've heard of all of those... except, apparently, for balloon juice My favourite is Plaid Paint