What is 'Infinite'?

[Crisis21]

The subject about infinity and infinities came up in the discussion about comic 1138 recently and I thought I'd continue it here.

The concept of infinity is best portrayed using numbers, and infinities there are referred to as 'aleph-x', starting as 'aleph-0'. But apparently there's more than a little controversy over which set corresponds to which aleph and as I'm no mathemetician (this is less than a hobby for me) I'm not going into that.

What I have heard used is 'countably infinite' and 'uncountably infinite' which is a bit simpler, but for personal purposes I'm going to declare a 'semi-countably infinite' class as well. I'll explain all below.


Countably infinite can refer to both Natural numbers (i.e. all whole positive numbers from zero on up, such as 0, 1, 2, etc.) and Integers (positive and negative whole numbers (etc., -2, -1, 0, 1, 2, etc.). What this means is that given an infinite amount of time, I could conceivably count to any given number in the set of either Natural numbers or Integers. Though for the latter, I'd obviously have to pick whether to count positive or negative from zero and given up counting the other side. The point still stands that I could count to any number given enough time.

Semi-countably infinite is a property I give to Rational numbers (such as fractions, like 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, etc.). I can attempt to count these like natural numbers and integers, but there's a problem. You see, there are an infinite number of rational numbers in between each natural number and integer. Meaning that unless I skip an infinite number of them (i.e. cheat), there are values I will never be able to count to even with an infinite amount of time allowed. Using the parenthetical example above for counting fractions of value less than 1, I would never reach the value of '1' (i.e. 2/2, 3/3, etc.) even with infinite time. Or if I started counting halves (1/2, 2/2, 3/2, etc.) I would never get around to counting thirds (1/3, 2/3, 3/3, 4/3, etc.). And if I wanted to start at zero and count the smallest rational number first, I'd never be able to even get started because no matter how small a rational number I tried to start with, there would be an infinite number of smaller numbers I'd have skipped.

Of course even with that problem, all rational numbers can still be represented (i.e. written) in a finite space (granted that space may be inconceivably huge depending on the exact value you're talking about, but it is still finite). That's not the case with the next set, the Real numbers.

Uncountably infinite is how Real numbers are called, and it means that any attempt to count them is doomed to failure before it begins. This is because Real numbers contain values like pi, which while having a very specific numerical value, require an infinite amount of space in which to represent them due to having an infinite number of decimal places. So since we can't calculate pi to the last decimal place, we can't determine which Real numbers come directly before or after it in a counting sequence, ergo we can't even attempt to count Real numbers.

And then there's Imaginary numbers which is a whole other headache given that it basically (to my limited understanding) takes the one dimensional nature of Real numbers and adds a second dimension to the whole thing. Would 'incomprehensibly infinite' be a thing?

[tiornys]

As a mathematician, I have to dispute your example of "semi-countable". Specifically, this claim about the rationals is incorrect: "Meaning that unless I skip an infinite number of them (i.e. cheat), there are values I will never be able to count to even with an infinite amount of time allowed."

This is getting into areas where infinities are extremely unintuitive. However, with infinite time you can indeed count all of the rationals without missing any. Formally, you can create what is known as a 1-1 map from the rationals to the natural numbers. In other words, you can assign each rational number to exactly one natural number such that all rational numbers have an assignment and no natural number is assigned to two different rational numbers. Proving that the real numbers are uncountable involves proving that no such map can exist.

Note that adding imaginary numbers does not automatically increase the size of your infinity. The set of all numbers of the form A + Bi where A and B are natural numbers is a countably infinite set (this can be proven in exactly the same way that you can prove the rationals are countably infinite).

[Cazero]

And then there's Imaginary numbers which is a whole other headache given that it basically (to my limited understanding) takes the one dimensional nature of Real numbers and adds a second dimension to the whole thing. Would 'incomprehensibly infinite' be a thing?
Imaginary numbers is the axis of numbers perpendicular to the axis of real numbers. It's functionaly the same thing. What you're thinking of is called complex numbers. A complex number is what you get when you add an imaginary to a real. It's infinitely broader than just one axis in an uncountable way.
See how you desribed the transition from integers to reals? Try to think of complex numbers as the same transition, except instead of starting from an infinite number of integers you're starting from the set comprised of 0 and nothing else. That's how infinitely more numerous complex numbers are.

I have no idea how aleph-ish any infinite is or how you determine that. Infinity is weird.


edit : apparently, uncountable wasn't the right term?

[Douglas]

Semi-countably infinite is a property I give to Rational numbers (such as fractions, like 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, etc.). I can attempt to count these like natural numbers and integers, but there's a problem. You see, there are an infinite number of rational numbers in between each natural number and integer. Meaning that unless I skip an infinite number of them (i.e. cheat), there are values I will never be able to count to even with an infinite amount of time allowed. Using the parenthetical example above for counting fractions of value less than 1, I would never reach the value of '1' (i.e. 2/2, 3/3, etc.) even with infinite time. Or if I started counting halves (1/2, 2/2, 3/2, etc.) I would never get around to counting thirds (1/3, 2/3, 3/3, 4/3, etc.). And if I wanted to start at zero and count the smallest rational number first, I'd never be able to even get started because no matter how small a rational number I tried to start with, there would be an infinite number of smaller numbers I'd have skipped.
Rational numbers are in fact countable. You can enumerate all of them in an infinite sequence without missing any, you just have to get a little creative. For example: 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1... Every positive rational number will appear somewhere in that sequence if you continue it far enough.

[Crisis21]

Rational numbers are in fact countable. You can enumerate all of them in an infinite sequence without missing any, you just have to get a little creative. For example: 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1... Every positive rational number will appear somewhere in that sequence if you continue it far enough.

My apologies for the error then.


As a mathematician, I have to dispute your example of "semi-countable". Specifically, this claim about the rationals is incorrect: "Meaning that unless I skip an infinite number of them (i.e. cheat), there are values I will never be able to count to even with an infinite amount of time allowed."

This is getting into areas where infinities are extremely unintuitive. However, with infinite time you can indeed count all of the rationals without missing any. Formally, you can create what is known as a 1-1 map from the rationals to the natural numbers. In other words, you can assign each rational number to exactly one natural number such that all rational numbers have an assignment and no natural number is assigned to two different rational numbers. Proving that the real numbers are uncountable involves proving that no such map can exist.

Note that adding imaginary numbers does not automatically increase the size of your infinity. The set of all numbers of the form A + Bi where A and B are natural numbers is a countably infinite set (this can be proven in exactly the same way that you can prove the rationals are countably infinite).

Like I said, I'm no mathemetician, so I was unaware of some of the tricks for counting rationals like what Douglas showed. So, I was in error with 'semi-countable'. It happens and I'm man enough to admit it.

As for Real numbers being uncountable, I would presume that it is hard to create such a map when some of your numbers have values that cannot be represented in finite space. My understanding is that they are called uncountable because no one has been able to create such a map. In this case the absence of proof for existence may be proof enough of nonexistence.

[Douglas]

As for Real numbers being uncountable, I would presume that it is hard to create such a map when some of your numbers have values that cannot be represented in finite space. My understanding is that they are called uncountable because no one has been able to create such a map. In this case the absence of proof for existence may be proof enough of nonexistence.
Actually, there is an explicit proof that no such map is possible. It's called Cantor's diagonal argument, and it goes like this:

Suppose you have a map that uses each natural number to label a different real number. Now let's construct another real number, by way of its digital representation, as follows: for the 1st digit after the decimal place, pick something that is not what the number labeled with 1 has in that spot. For the 2nd digit, pick something that is not what the number labeled with 2 has in that spot. In general, for the nth digit pick something that is not what the number labeled with n has in its nth digit. The resulting infinitely long real number cannot be in your map, because every real number in your map is different from it in at least one digit. This is true no matter what map you started with, so it is not possible for any such map to contain all real numbers.

[tiornys]

Like I said, I'm no mathemetician, so I was unaware of some of the tricks for counting rationals like what Douglas showed. So, I was in error with 'semi-countable'. It happens and I'm man enough to admit it.
As it happens, what you were tripping over is actually an inherent property of infinities, just one that's easier to see in some cases than others. One way to determine that a set is infinite is to observe that it's the same size as a proper subset of itself. In the rationals, that's really easy to observe--your examples show that very nicely. It's a little less obvious on, say, the natural numbers, but as an example you might observe that the set of all even natural numbers is the same size as all natural numbers (the equation X = 2Y is the 1-1 map here).

Worth noting that there are other ways to prove that the real numbers are uncountable besides the diagonalization argument, but diagonalization is a really powerful concept that gets used in a number of places throughout mathematics.

[Rockphed]

Rational numbers are in fact countable. You can enumerate all of them in an infinite sequence without missing any, you just have to get a little creative. For example: 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1... Every positive rational number will appear somewhere in that sequence if you continue it far enough.

Shouldn't it be 1/1, 1/2, 2/2, 2/1, 1/3, 2/3, 3/3, 3/2, 3/1, 1/4, 2/4, 3/4, 4/4, 4/3, 4/2, 4/1, ... Sure, you hit a bunch of things multiple times, (for instance you will have 1 in your list infinitely many times), but it is still a countable set of the rational numbers.

[Douglas]

Shouldn't it be 1/1, 1/2, 2/2, 2/1, 1/3, 2/3, 3/3, 3/2, 3/1, 1/4, 2/4, 3/4, 4/4, 4/3, 4/2, 4/1, ... Sure, you hit a bunch of things multiple times, (for instance you will have 1 in your list infinitely many times), but it is still a countable set of the rational numbers.
There are a lot of ways to arrange the general idea. My version omits all fractions that can be reduced, and groups terms by the sum of numerator and denominator.

[Bucky]

Then it should start with 0/1.

[BeerMug Paladin]

To address the original question, I'd probably argue that "incomprehensibly infinite" is a term that applies to any type of infinity. From aleph-naught to aleph-one and aleph-aleph (if we want to go really nuts here).

Also, to add to the confusion, there's space-filling curves which can take the real numbers and map them to the complex plane. Similarly to how the counting numbers can be mapped to the rational numbers. Those wacky infinities.

Further, there's a curious notion that the infinity that represents the 'amount' (not a math term, hence the marks) of real numbers is the very next smallest infinity after countable infinity. That's an idea that has to be either taken as a fundamental truth or just rejected, as its truth apparently cannot be proven from the common set of beginning, more mundane assumptions.

So, there's essentially a lot of weirdness when you get into talk about infinity. So much so that mathematicians know there can be new rules made just for infinity's sake past a certain point. Not strictly true, I know, but we're talking informally here.

[Devils_Advocate]

There's a series of videos (https://www.youtube.com/watch?v=SLHiq7wZWWM) that explains a lot of this stuff using the idea of Hilbert's Grand Hotel (https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel).

An important point to understand is that infinity is not just an extremely large number. If you think that you could count all the way to infinity given infinite time, and then you'd be done, then you don't really grasp what's being discussed. You wouldn't finally reach infinity "at the end of forever", because there is no end of forever. Counting forever means that you never stop counting.

Anyway, sets -- including infinite sets -- "of the same size" are said to have the same cardinality. The set of natural numbers, the set of all even natural numbers, the set of all integers, and the set of all rational numbers all have the same cardinality. One set plainly having more members than another doesn't means that it doesn't also have exactly as many; to rephrase tiornys's point, it ain't infinite if it's not bigger than itself!

I would expect the set of complex numbers -- i.e. the set of all sums of real numbers and imaginary numbers -- to have the same cardinality as the set of real numbers. After all, each one is equivalent to an ordered pair of real numbers -- just change X +Yi to (X, Y) -- and the ordered pairs of natural numbers are still countably infinite, so I'd expect squaring other infinities not to increase cardinality either.

(Exercise for the reader: Demonstrate that the set of all ordered pairs of natural numbers is countably infinite.)


Countably infinite can refer to both Natural numbers (i.e. all whole positive numbers from zero on up, such as 0, 1, 2, etc.) and Integers (positive and negative whole numbers (etc., -2, -1, 0, 1, 2, etc.). What this means is that given an infinite amount of time, I could conceivably count to any given number in the set of either Natural numbers or Integers. Though for the latter, I'd obviously have to pick whether to count positive or negative from zero and given up counting the other side.
Well, that depends on what you mean by "count", I suppose, but it's quite easy to list them all out by alternating positive and negative: 0, 1, -1, 2, -2, 3, -3...

[NichG]

For ordered pairs of natural numbers, generate all such pairs in sequence as follows:

1. Start at (x=1,y=1), counter k=1
2. Increment x
3. For each y<=x, add that pair. Counter is now at k->k+x
4. Goto 2

This gives you a map from the natural numbers to all ordered pairs of natural numbers. Unordered pairs is just, when (x!=y) add the pair and (y,x). Unordered N-sets of natural numbers is just nesting loops N deep.

How about when N->infinity? Will that change cardinality? Not sure... that should be equivalent to any number that can be written with an infinite number of digits, which means it should map to the reals.

[DavidSh]

How about when N->infinity? Will that change cardinality? Not sure... that should be equivalent to any number that can be written with an infinite number of digits, which means it should map to the reals.

The set of infinite sequences of natural numbers is not countable. Cantor's diagonalization argument applies more directly to this than to the set of real numbers. The set of arbitrarily long finite sequences of natural numbers is countable.

[Avilan the Grey]

A possible sidebar to this...

I don't know if it makes me a simpleton or a genius (or neither), but the concept of infinity is very simple to me. I understand that in certain fields and concepts you have to for formula's sake "specify infinity" but when talking about it otherwise... why? Infinity means infinity. It is it's own definition. It's like asking "What is nothing", which is the opposite of infinity. Nothing means nothing. Infinity means infinity. It is from a non-scientific point of view very basic and easy to grasp, IMHO.

Again I get that when writing a paper on something, or formulating a theorem or something, you might have to put a mathematical definition on it, but from an every day perspective that just makes it complicated where there is no complications to begin with.

[NichG]

A possible sidebar to this...

I don't know if it makes me a simpleton or a genius (or neither), but the concept of infinity is very simple to me. I understand that in certain fields and concepts you have to for formula's sake "specify infinity" but when talking about it otherwise... why? Infinity means infinity. It is it's own definition. It's like asking "What is nothing", which is the opposite of infinity. Nothing means nothing. Infinity means infinity. It is from a non-scientific point of view very basic and easy to grasp, IMHO.

Again I get that when writing a paper on something, or formulating a theorem or something, you might have to put a mathematical definition on it, but from an every day perspective that just makes it complicated where there is no complications to begin with.

The tricky stuff is all in how one can use infinity to do things - even real world things. There's a lot of physical phenomena that are well explained by observing that in some sense the systems are 'closer' to infinity than they are to zero. For example, phase transitions are phenomena that cannot technically exist in finite systems because they violate underlying reversibility and properties of equilibrium distributions (unique ground states, etc). But those properties break down in a particular way for infinite systems, and that breakdown more strongly defines the behavior of e.g. 1kg of water at the freezing point than the finite size effects.

Similarly, the 'almost infinite' (but not really) incompressibility of water causes pressure fields to behave as if they were long-range forces (1/r potential, like gravity) even though they are built out of strictly short-range interactions (1/r^6 potential van der Waals stuff for example).

The countable/uncountable distinction is a bit harder to pin to a phenomenon, but maybe conductance bands might be one. The idea being that the band is dense (like the reals) as a result of something like the infinite ordered set construction (influence from site + 1/10 influence from 1st neighbor + 1/100th from second neighbor + ...) So even though the unit cell eigenstates are countably infinite, the extended material eigenstates are uncountably infinite, giving you contiguous bands of states.

[BeerMug Paladin]

A possible sidebar to this...

I don't know if it makes me a simpleton or a genius (or neither), but the concept of infinity is very simple to me. I understand that in certain fields and concepts you have to for formula's sake "specify infinity" but when talking about it otherwise... why? Infinity means infinity. It is it's own definition. It's like asking "What is nothing", which is the opposite of infinity. Nothing means nothing. Infinity means infinity. It is from a non-scientific point of view very basic and easy to grasp, IMHO.

Again I get that when writing a paper on something, or formulating a theorem or something, you might have to put a mathematical definition on it, but from an every day perspective that just makes it complicated where there is no complications to begin with.
The difficulty about it (like in all areas of speciality, really) comes from when you encounter questions on the topic.
One question people often have about infinity is along the lines of...
"What is infinity - infinity equal to?"
"What is infinity / infinity equal to?"
There's also two different infinities on the real number line and one infinity on the complex plane.

Answering why the answers are what they are is just as important as answering the questions themselves.

[Rockphed]

The difficulty about it (like in all areas of speciality, really) comes from when you encounter questions on the topic.
One question people often have about infinity is along the lines of...
"What is infinity - infinity equal to?"
"What is infinity / infinity equal to?"
There's also two different infinities on the real number line and one infinity on the complex plane.

Answering why the answers are what they are is just as important as answering the questions themselves.

"Infinity minus infinity" needs lots of stuff done to it to be tractable, but "infinity divided by infinity" has been solved (or at least solvable) for 320 years.

And I swear I have seen people write 4 different infinities for complex values (one for each end of a purely real or purely imaginary axis).

[tiornys]

Working with infinity is where a lot of people's intuition trips up. Just look at the contention that tends to arise when people bring up 0.999... = 1.

[Avilan the Grey]

Working with infinity is where a lot of people's intuition trips up. Just look at the contention that tends to arise when people bring up 0.999... = 1.

I think it is all a matter of philosophy vs math. I am definitely on the former side of things, in how my brain works. I have no difference grasping the concept of how long the dinosaurs actually "ruled" the earth (Insects did, and they still do, as they did before that). While some people honestly have a hard time grasping that the civilization(s) of Egypt lasted so long that the construction of the pyramids are closer to our time than to the founding of the Egyptian empire.

The whole 0,999999... thing is funny. The answer to me is "both". It is, from a clear logical standpoint not the same as 1. It's False to say it is. But in any kind of actual mathematical or practical use, it is True.

[tiornys]

From a philosophical standpoint, I'm reminded of that old chestnut: "a rose by any other name would smell as sweet". That is, you can give something many different labels without changing what that thing is.

In the real number system, "0.999..." is just a different (complicated, confusing) label for the mathematical object more commonly labeled as "1". To me, that makes them the same thing in most contexts--i.e. any context where it's being referenced without specifying some number system where it actually represents a different mathematical object. But that might just be me being a mathematician.

[Avilan the Grey]

From a philosophical standpoint, I'm reminded of that old chestnut: "a rose by any other name would smell as sweet". That is, you can give something many different labels without changing what that thing is.

In the real number system, "0.999..." is just a different (complicated, confusing) label for the mathematical object more commonly labeled as "1". To me, that makes them the same thing in most contexts--i.e. any context where it's being referenced without specifying some number system where it actually represents a different mathematical object. But that might just be me being a mathematician.

Of course, but yes, to me the concept of "1" means "one whole thing", when talking about hypothetical things of course (a person is a person even without an arm, for example. Duh). Basically 0.999... is not 1, because no matter how close you get, you never become "whole". But if we instead use the other way of writing it, it suddenly is. 1, 1/3, 2/3 etc. I will TREAT it as 1, but it isn't. Just like 0.333... will never be exactly 1/3. Because decimal math is flawed, it seems. Despite math being the language of creation.

[danzibr]

but it isn't
But it is.

Here are some thoughts on infinity:

At a middle school dance, there's a line of boys on the left side of the gym, and 100 feet away, a line of girls on the right side of the gym. Every 10 seconds they close the distance by half. When will they meet?
A Mathematician would say they never meet.
A Physicist would say they meet at time = infinity.
An Engineer would say after 1.5 minutes they are close enough for all intents and purposes.

[Rockphed]

But it is.

Here are some thoughts on infinity:

At a middle school dance, there's a line of boys on the left side of the gym, and 100 feet away, a line of girls on the right side of the gym. Every 10 seconds they close the distance by half. When will they meet?
A Mathematician would say they never meet.
A Physicist would say they meet at time = infinity.
An Engineer would say after 1.5 minutes they are close enough for all intents and purposes.

d[0] = 100
d[10] = 50
d[20] = 25
d[30] = 12.5
d[40] = 6.25
d[50] = 3.125
d[60] = 1.5625

For the purposes of a middle school dance, I think 50 or 60 seconds is close enough. In fact, I am fairly certain that most people are only about 1.5 feet thick, so at about t = 62 the two lines of people staring bumping in to each other and the chaperones start yelling about inappropriate dancing.

[eggynack]

Of course, but yes, to me the concept of "1" means "one whole thing", when talking about hypothetical things of course (a person is a person even without an arm, for example. Duh). Basically 0.999... is not 1, because no matter how close you get, you never become "whole". But if we instead use the other way of writing it, it suddenly is. 1, 1/3, 2/3 etc. I will TREAT it as 1, but it isn't. Just like 0.333... will never be exactly 1/3. Because decimal math is flawed, it seems. Despite math being the language of creation.
This is a mistake I see a lot as regards this stuff. .999... doesn't become closer and closer to 1, and neither does .333... become closer and closer to 1/3. These are static numbers that are incapable of having any sort of time element. .999... just is 1. It has all the 9's at once, and when you have all the 9's at once what you have is 1. It must be 1, in fact, for roughly a billion different reasons. You're not just treating it as 1. It just is 1. Decimals can be weird sometimes, but they are not flawed. It is when you introduce time elements and treating numbers as other numbers that they seem flawed.

[danzibr]

d[0] = 100
d[10] = 50
d[20] = 25
d[30] = 12.5
d[40] = 6.25
d[50] = 3.125
d[60] = 1.5625

For the purposes of a middle school dance, I think 50 or 60 seconds is close enough. In fact, I am fairly certain that most people are only about 1.5 feet thick, so at about t = 62 the two lines of people staring bumping in to each other and the chaperones start yelling about inappropriate dancing.
When you measure the distance between 2 objects, you don’t do it from their centers.

In this case 1.5 feet seems awfully far.

[Rockphed]

When you measure the distance between 2 objects, you don’t do it from their centers.

In this case 1.5 feet seems awfully far.

... I typically measure either center to center, left to left, or right to right. Who taught you how to measure object locations? I suppose if the 2 groups are facing each-other, front to front is different from back to back. And 1.5 feet is about the length of an adult's arm from elbow to finger-tip. It is a reasonable distance to stand apart waiting for the music to start.

[Lord Torath]

Of course, but yes, to me the concept of "1" means "one whole thing", when talking about hypothetical things of course (a person is a person even without an arm, for example. Duh). Basically 0.999... is not 1, because no matter how close you get, you never become "whole". But if we instead use the other way of writing it, it suddenly is. 1, 1/3, 2/3 etc. I will TREAT it as 1, but it isn't. Just like 0.333... will never be exactly 1/3. Because decimal math is flawed, it seems. Despite math being the language of creation.Let x = 0.999999 and on into infinity.
10x = 9.999999...
10x - x = 9.999999... - 0.999999...
9x = 9
x = 1

Therefore, 0.999999... is exactly equal to 1.

Any real number that ends in a repeating decimal can be written as a rational fraction.

[DavidSh]

"0.99999..." is "1" to the same extent that "1+1" is "2". Sure, as strings of characters they are different, but the conventional meaning of "0.99999..." is the limit of the sequence 0.9, 0.99, 0.999, 0.9999, …. On the usual metric on real numbers, or even rational numbers, this limit is 1. There are other metrics that might give different limits, or might not converge at all.


Just like 1+1 is 2 in about any context in which 2 is defined, but is 0 in the field of integers modulo 2.

[Khloros]

I don't do math, but I do art, maybe this will help you out:


https://video.twimg.com/ext_tw_video/1022874585442775040/pu/vid/640x360/aYWR7JPO1hGdqK4p.mp4?tag=4

[danzibr]

Who taught you how to measure object locations?
A man named Marius Mitrea, my PhD advisor. Here (https://mospace.umsystem.edu/xmlui/handle/10355/45843)'s a link to my dissertation.

I think in this case the most natural ``distance'' would be inf{d(x,y):x\in X,y\in Y}. Not the typical Pompeiu-Hausdorff Metric, but still.

For example, if two cats are having a show down, noses and foreheads pressed against each other, I think it's more natural to say their distance is 0 rather than whatever half a cat + half a cat is.

Likewise, if the center of your body is 100 feet from the center of the building you're within, it'd be awfully strange to me to say you're 100 feet from the building, rather than 0 (given your foot is touching the floor).

[Rockphed]

See, I was taught to measure distance by a guy named Paul who was setting up an assembly line for me to program. And there if 2 stations are touching we still say they are the distance of separation between whatever points we have established as their base points. I suspect that most engineers would approach measurement in a similar fashion. :smalltongue: I think this just goes to show that proper communication often requires making sure both parties are using the same definitions. Hence why every technical paper starts out by explaining its use of terms.

[danzibr]

See, I was taught to measure distance by a guy named Paul who was setting up an assembly line for me to program. And there if 2 stations are touching we still say they are the distance of separation between whatever points we have established as their base points. I suspect that most engineers would approach measurement in a similar fashion. :smalltongue: I think this just goes to show that proper communication often requires making sure both parties are using the same definitions. Hence why every technical paper starts out by explaining its use of terms.
Fair enough.

I retract my "you don’t do it from their centers" statement.

I should've said I don't do it from their centers :P

[BeerMug Paladin]

And I swear I have seen people write 4 different infinities for complex values (one for each end of a purely real or purely imaginary axis).

This... This amuses me.


The whole 0,999999... thing is funny. The answer to me is "both". It is, from a clear logical standpoint not the same as 1. It's False to say it is. But in any kind of actual mathematical or practical use, it is True.

This just makes me think you're using a different definition for … than is in common usage among mathematicians.

[Nifft]

"There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite."

― Jorge Luis Borges, Discusión


Infinity is a cognitive hazard which may or may not have any correlation with reality.

[georgie_leech]

Inasmuch as quantum mechanics may or may not have any correlation with reality, I suppose. (https://en.m.wikipedia.org/wiki/Renormalization)

[Nifft]

Inasmuch as quantum mechanics may or may not have any correlation with reality, I suppose. (https://en.m.wikipedia.org/wiki/Renormalization)



However, even if it were the case that no infinities arise in loop diagrams in quantum field theory, it can be shown that renormalization of mass and fields appearing in the original Lagrangian is necessary.

The thing you're linking seems to be a useful way to REMOVE infinities, and it works even if there is no infinity present. Cool.

Was that supposed to be a refutation, or are you just sharing a cool thing?

[georgie_leech]

The thing you're linking seems to be a useful way to REMOVE infinities, and it works even if there is no infinity present. Cool.

Was that supposed to be a refutation, or are you just sharing a cool thing?

Bit of both. Practical methods to work with infinities are important because they make math hard.

[Nifft]

Bit of both. Practical methods to work with infinities are important because they make math hard.

It's true that practical methods to work around a model's infinities are useful & cool.

However, you're not showing evidence that infinities exist in the universe, just in some models -- and that they can be worked around, and that the theory which works around these infinities also works where there are no infinities. So how is your link supposed to be a refutation?

[georgie_leech]

It's true that practical methods to work around a model's infinities are useful & cool.

However, you're not showing evidence that infinities exist in the universe, just in some models -- and that they can be worked around, and that the theory which works around these infinities also works where there are no infinities. So how is your link supposed to be a refutation?

Just that we keep running into infinities when scientists go and do math with QM, so I'd describe that as at least having some correlation with reality.

[BeerMug Paladin]

Calculus only really exists because of the concept of infinity. So if we avoided conceptual infinity entirely, we would lose access to a rather important mathematical construct.

While one may argue that purely logical constructs within mathematics doesn't necessarily have any direct correlation with reality, I think that's a philosophical assertion that some may accept and others may reject. I see no way to really settle such a dispute in either direction.

[gomipile]

It's true that practical methods to work around a model's infinities are useful & cool.

However, you're not showing evidence that infinities exist in the universe, just in some models -- and that they can be worked around, and that the theory which works around these infinities also works where there are no infinities. So how is your link supposed to be a refutation?

Quantum electrodynamics (the theory of the interaction of photons and charged particles) is a quantum field theory, and it is one of the most precisely verified theories we have.

So, it may not be precisely true with zero error, but QED is at least approximately true with an error of less than ten parts in a billion in the parts of the universe where human life is possible.

[Grey_Wolf_c]

The most immediately obvious presence of infinity in the real world I can think of: energy requirements for a given speed for a massed particle:

https://upload.wikimedia.org/wikipedia/commons/3/37/Rel-Newton-Kinetic.svg
(link (https://en.wikipedia.org/wiki/Tests_of_relativistic_energy_and_momentum))

Grey Wolf

[NichG]

Critical opalescence is a phenomenon where you're seeing a divergent lengthscale emerge during a second order phase transition. Normally, density fluctuations in a fluid sit around the atomic scale, but as you approach the critical point those fluctuations become macroscopic and as a result are visible to the eye.

Similarly, divergence of the specific heat in the same context.

Resonance phenomena also involve an infinity, though they have a more obvious cutoff when dissipation is present.

An easily accessible one is the chirp when a spinning disc settles on a table - the frequency of the sound of the wobble diverges to infinity in finite time. There's a similar divergence in a bouncing ball coming to rest (each bounce loses a fraction of the energy, but the time between bounces decreases as the energy decreases, so the limit is finite only because the frequency becomes infinite)

You can argue that none of these cases is actually infinite and you'd be right - there's always a cutoff. However, in terms of a parsimonious description of reality, approximating these cases as infinite allows a much higher compression ratio of what you can predict versus what information you need to put in. As such, the infinite description is actually a better approximation than the best finite description we can construct in many of those cases. E.g. to describe the bouncing ball without infinity, we need to know what material it and the table are made of to the extent of being able to model microscopic plastic deformations, what the air in the gap is doing (including Mach effects because the vanishing air gap on contact could give rise to supersonic flows).

[Avilan the Grey]

Again, it's philosophy vs math.
I am arguing from the former. 0.999... is not 1. However, mathematically it is. I KNOW that. That's what I meant when I said "I'll treat it as 1".

Again, though, my initial argument was that people seem to have a problem grasping the very concept of infinity (or nothingness, on the other end of the scale). I find that confusing. No, I cannot fathom how incredibly large (and becoming larger all the time) the Universe is. But I can understand it in a philosophical sense. Just imagine something without an end.

There. Done.
That wasn't so hard, was it?

[eggynack]

Again, it's philosophy vs math.
I am arguing from the former. 0.999... is not 1. However, mathematically it is. I KNOW that. That's what I meant when I said "I'll treat it as 1".

But what is your philosophical basis for claiming inequality here?

[NichG]

Again, it's philosophy vs math.
I am arguing from the former. 0.999... is not 1. However, mathematically it is. I KNOW that. That's what I meant when I said "I'll treat it as 1".

Again, though, my initial argument was that people seem to have a problem grasping the very concept of infinity (or nothingness, on the other end of the scale). I find that confusing. No, I cannot fathom how incredibly large (and becoming larger all the time) the Universe is. But I can understand it in a philosophical sense. Just imagine something without an end.

There. Done.
That wasn't so hard, was it?

Now that you're imagining it, how bright is it's sky?

Putting a label on something and understanding it are different things.

[georgie_leech]

https://youtu.be/TINfzxSnnIE

[Lord Torath]

Again, it's philosophy vs math.
I am arguing from the former. 0.999... is not 1. However, mathematically it is. I KNOW that. That's what I meant when I said "I'll treat it as 1".

Again, though, my initial argument was that people seem to have a problem grasping the very concept of infinity (or nothingness, on the other end of the scale). I find that confusing. No, I cannot fathom how incredibly large (and becoming larger all the time) the Universe is. But I can understand it in a philosophical sense. Just imagine something without an end.

There. Done.
That wasn't so hard, was it?Can you explain the difference between 0.999999... and 1?

[Grey_Wolf_c]

Can you explain the difference between 0.999999... and 1?

Alternatively, if .999... is different from 1, there should be infinite decimal numbers between them (just like there are infinite number between, say, 1 an 1.000000000001). Anyone claiming that they are different should therefore be able to provide at least one example.

(I've always liked that proof).

Grey Wolf

[georgie_leech]

Can you explain the difference between 0.999999... and 1?

Caveat: a difference that doesn't also apply to 1+1 and 2, or 1/2 and 0.5

[SpoonR]

I started to get lost in complex calculus when they were talking about "and now we draw a circle, and a line out to infinity, and then go around the complex plane at infinity so you come back in from negative infinity." :smallconfused:

But the simplest way to understand infinity, number 8 laid down to take a nap.

[BeerMug Paladin]

Again, it's philosophy vs math.
I am arguing from the former. 0.999... is not 1. However, mathematically it is. I KNOW that. That's what I meant when I said "I'll treat it as 1".

Again, though, my initial argument was that people seem to have a problem grasping the very concept of infinity (or nothingness, on the other end of the scale). I find that confusing. No, I cannot fathom how incredibly large (and becoming larger all the time) the Universe is. But I can understand it in a philosophical sense. Just imagine something without an end.

There. Done.
That wasn't so hard, was it?

I'm not sure how to make sense of this sentiment.

[georgie_leech]

I'm not sure how to make sense of this sentiment.

Philosophical sense is apparently code for "wrongly," given that 0.infinite-9's is 1 but they insist it isn't.

[NichG]

Well, I don't think its so mysterious how 0.999... and 1 could be considered different. They're, literally, different symbol sequences, as if I were to write 1, 1.0, and 1.00. It's just that you can have different symbol sequences that mathematically represent the same object.

But there is something of infinity here as well in that generally we think that by constructing a positional notation system like a digital representation of a number, equality can be determined by comparing the numbers symbol by symbol and then if any symbol disagrees (not counting leading or trailing zeros), the numbers are different. E.g. for numbers written as finite sequences I can say: x != y if x(i) != y_(i) for any i. That means, among other things, that I can seemingly check equality between random two numbers with an amortized O(1) cost rather than O(N) in their length.

But infinite sequences violate that idea.

[danzibr]

Can you explain the difference between 0.999999... and 1?
Hmm, 0 is a difficult thing to fully explain.

[Jay R]

But there is something of infinity here as well in that generally we think that by constructing a positional notation system like a digital representation of a number, equality can be determined by comparing the numbers symbol by symbol and then if any symbol disagrees (not counting leading or trailing zeros), the numbers are different.

There's your problem. You have taken an English description of how to manipulate the symbols and tried to act as if it is a mathematical principle. It works at one level but not at other levels.

Similarly, we were taught that multiplication means taking so many groups of so many things. 3 x 4 means three groups of four items, which is twelve items. That's an English language approximation, and it helped us to learn to multiply. But it's only an approximation, and doesn't always apply. You cannot take pi groups of -1/6 items, but you can multiply pi time -1/6.

Similarly, we originally defined exponents as multiplying a number times itself a certain number of times.

43 = 4 x 4 x 4 = 64

You can't multiply a number by itself half a time, or negative three times. But we can prove that

91/2 = 3, and 2-3 = 1/8

So the English language approximation of the mathematical reality is insufficient. To understand multiplication, or infinite sequence of digits, we have to drop the English approximation and apply mathematical logic.

[NichG]

There's your problem. You have taken an English description of how to manipulate the symbols and tried to act as if it is a mathematical principle. It works at one level but not at other levels.

Similarly, we were taught that multiplication means taking so many groups of so many things. 3 x 4 means three groups of four items, which is twelve items. That's an English language approximation, and it helped us to learn to multiply. But it's only an approximation, and doesn't always apply. You cannot take pi groups of -1/6 items, but you can multiply pi time -1/6.

Similarly, we originally defined exponents as multiplying a number times itself a certain number of times.

43 = 4 x 4 x 4 = 64

You can't multiply a number by itself half a time, or negative three times. But we can prove that

91/2 = 3, and 2-3 = 1/8

So the English language approximation of the mathematical reality is insufficient. To understand multiplication, or infinite sequence of digits, we have to drop the English approximation and apply mathematical logic.

It's not even an English language thing, since it works for any finite symbol sequence under the conditions I specified (removing leading and trailing zeros). For those cases it'd be mathematically true as well that you can determine equality with such an algorithm. It's just that the construction doesn't work when you violate it's preconditions, and taking the limit to infinity is a way to do that (because equality on dense and sparse measures means something different).

[Rockphed]

It's not even an English language thing, since it works for any finite symbol sequence under the conditions I specified (removing leading and trailing zeros). For those cases it'd be mathematically true as well that you can determine equality with such an algorithm. It's just that the construction doesn't work when you violate it's preconditions, and taking the limit to infinity is a way to do that (because equality on dense and sparse measures means something different).

What does .999... look like in other bases if you just do the translation? Let's look at base 16:

It is less than 1, but greater than 15/16, so it is

0.F16 + (0.0624999....)

0.0624999.... is greater than 15/256, so the next place is also F, leaving 0.00390624999

And it will keep going forever, leaving us with 0x0.FFF...

[gomipile]

What does .999... look like in other bases if you just do the translation? Let's look at base 16:

It is less than 1, but greater than 15/16, so it is

0.F16 + (0.0624999....)

0.0624999.... is greater than 15/256, so the next place is also F, leaving 0.00390624999

And it will keep going forever, leaving us with 0x0.FFF...

And in base 2 it is 0.1111111...

In base 8, it's 0.7777777...

In base 3, it's 0.2222222...

And so on.

Of course, all of these are 1 in each of those bases, also.

[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 (https://en.wikipedia.org/wiki/Go_and_mathematics) 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 (https://en.wikipedia.org/wiki/Go_and_mathematics) 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 (https://waitbutwhy.com/2014/11/1000000-grahams-number.html), I still don't know what it is! Small "eeks" aren't going to cut it. This requires big ones! :eek::eek::eek::eek::eek::eek::eek::eek::eek::eek: :eek::eek::eek::eek::eek::eek::eek::eek:

[Grey_Wolf_c]

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


https://www.youtube.com/watch?v=qbInsYok8x8

Grey Wolf

[Peelee]

I refuse to accept Graham's pile of pumpkins (https://www.smbc-comics.com/comic/the-largest-number-2).

[JCarter426]

Okay, I had no idea what Graham's number actually was. Now that I know what it is (https://waitbutwhy.com/2014/11/1000000-grahams-number.html), I still don't know what it is! Small "eeks" aren't going to cut it. This requires big ones! :eek::eek::eek::eek::eek::eek::eek::eek::eek::eek: :eek::eek::eek::eek::eek::eek::eek::eek:
Allow me to introduce you to Graham's number's bigger sibling TREE(3) (https://www.youtube.com/watch?v=3P6DWAwwViU).

[Lord Torath]

Allow me to introduce you to Graham's number's bigger sibling TREE(3) (https://www.youtube.com/watch?v=3P6DWAwwViU).See, now TREE(3) just makes me go :smallconfused:? 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 :smallconfused:? 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 :smallconfused:? 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? :smallconfused:

[Rockphed]

...is akin to what now? :smallconfused:

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? :smallconfused:

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 (https://www.google.com/search?q=HELIUM+TANK&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiH6f6A08XeAhXLzFMKHY-xC4IQ_AUIFCgC&biw=1443&bih=708&dpr=1.3). 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 (http://www.giantitp.com/comics/oots0855.html) and Panel 7 (http://www.giantitp.com/comics/oots0979.html). :smallbiggrin:

[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 (http://www.giantitp.com/comics/oots0855.html) and Panel 7 (http://www.giantitp.com/comics/oots0979.html). :smallbiggrin:
https://i.pinimg.com/originals/5b/d5/a9/5bd5a94ae231b4ab10f0a856d57f9b6f.jpg

[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 :smallbiggrin: My favourite is Plaid Paint

[keybounce]

I want to thank this thread for introducing me to "Wait but why?". I now have something else to read since "What-if" stopped updating.