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 that explains a lot of this stuff using the idea of Hilbert's 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...qK4p.mp4?tag=4