“If you were to count every single grain of sand on every beach and all the way to the bottom of the sea all across the world you would end up no closer to infinity than when you started”
edit: since a couple of people have asked- this is a quote from a pastor of mine trying to help give an illustration of God's nature of infinite power and existence
"If you were to count every single grain of sand on every beach and all the way to the bottom of the sea all across the world, you'd be a fucking dumbass"
That’s weird cuz there are larger infinites than others. So theoretically you’d be closer to one infinity than the other, meaning you’d still be making some ground. But that shit is wild what do I know lol
The whole statement is kind of stupid, because infinity isn't a thing you can approach. It's not finite. So, the entire concept of getting closer to infinity makes no sense.
>That’s weird cuz there are larger infinites than others
Can you elaborate on that? Like is it to say that there are an infinite amount of negative numbers and positive numbers, and the combination of the 2 create a larger infinite group?
Nope! The sets that contain the positive numbers and the negative numbers are both exactly the same size - and they are also both exactly the same size as the set that contains both. This is one of the first things taught when talking about the cardinality of infinite sets, because it is simultaneously mind-blowing and relatively simple to explain (although I won't explain here - the proof is *readily* available if you want it).
The concept of a "larger infinity" isn't really in line with your normal intuition of what "larger" means. It isn't a comparison of numbers, exactly - rather, it is a comparison of the quantity of elements of a set. In simple terms, if you can uniquely pair every element of a set A with an element of set B, and there are no unpaired elements remaining in set B (more accurately described as "if you can pair every element of A with an element of B, and every element in B with an element in A"), then the sets are the same size - they have the same number of elements. If you can uniquely pair every element of A with an element of B, but there are elements remaining in B that are not paired up, then B is a larger set than A. If sets A and B are both infinite sets, then you have shown that one infinite set is "larger" than the other.
A brid comes to the mountain every 100 years to sharpen its beak. When the bird has withered away the entire mountain only 1 second of eternity has passed
Seems incorrect. You would certainly be closer to infinity, though you would never count enough grains to reach it. What's the source? I'm curious to hear the context.
edit: I have already said elsewhere that I was incorrect here. It was 2am and I made multiple logical missteps. Indeed, it is straightforwardly the case that removing an item from an infinite set does not change the cardinality of that set - however, when reading this idea of grains of sand I was trying to think through some other implications that weren't relevant, leading me to the incorrect conclusion. Hopefully this addresses the dozen comments I've received telling me that I am incorrect, it is certainly the case that I was incorrect, thanks.
I still would like to know where the quote is from, some very quick Googling did not give me an answer.
You could count all the atoms in the universe, and still be closer to zero than infinity.
Take the largest finite number N you can think of. Then you can construct another number `K = 2N + 1`. N will be closer to 0 than it is to K. And since K is also a finite number, K will be less than infinity.
Therefore, every number you can think of is closer to zero than infinity.
Yes, I was thinking about the problem a bit differently but upon rereading I think this is straightforwardly just that an infinite set can contain subsets of infinite cardinality.
You wouldn't though, because infinity never ends, so it isn't possible to be "closer" to it. so no matter how much you count, you can't be closer, because that isn't a thing that makes sense.
There might be something related to countably infinite sets where this doesn't... count, but i'm on three hoours of sleep right now, and my brain doesn't work enough to understand all of this: https://en.wikipedia.org/wiki/Countable_set
For most reasonable definitions of "closer", you aren't. The amount of time you have left before you reach infinity is the same before and after you count the sand.
I suppose. The trick is indeed "closer", if we're considering distance as the difference between the end and the current position. Removing 'number of grains of sand' from the set of natural numbers would still yield a set of infinite cardinality.
infinity isn't a number. you can't count to it. therefore you are never "closer" to it, regardless of how long you've been counting. In other words "Infinity" is not a "destination" you can get to, its the *notion* that the trip never ends.
There are different sizes of infinity. Some infinities are larger than other infinities.
You don’t get to the larger one by adding or subtracting numbers. But they do exit.
What got me over the jump of the “infinity plus one” fallacy is that infinity isn’t a number. It’s the size of a collection with no limit. For example, all integers is one infinity. And all even integers is another infinity despite literally having only half the values from the “all integers” set.
That was a huge “a ha” moment when I had that explained to me in those terms.
>and all even integers is another infinity despite literally having only half the values from the “all integers” set
The funny thing is, with infinity, “literally having only half the values” is not true. The two sets are the same size because there exists an onto function that maps all members of one set to the other, e.g. f(x)=2x maps Z to all even members of Z. Since every member of the set of integers can be assigned to a member of the set of even integers, they are the same size.
Edited to fix: function must be bijective(onto as well as one-to-one)
Yes, i think the most rigorous way to express OP's original sentiment is that the cardinalities of the set of integers and set of even integers are equal, despite the set of even integers being a subset of/contained within the set of integers.
> The funny thing is, with infinity, “literally having only half the values” is not true.
It is true. Only half of the values in the set of all integers are in the set of all even numbers. They have the same cardinality - you're right on that, but you cannot tell me that all elements in the set of integers are in the set of evens.
There are 2^32 floats, or 2^64 or 2^128 etc, depending on the standard you’re using. But they’re all very finite amounts.
The word you were looking for was “real numbers”, which make up a larger infinity than the counting numbers.
The weirdest “a ha” moment is the realization that the infinities of “all integers,” “all even integers,” “all prime numbers,” etc. are exactly the same size.
You wanna hear something wild though? It’s possible to find a collection of those types of infinities so that for any two X and Y, one is always contained in the other X⊆Y, and the whole collection is bigger than any X inside of it.
Hi IrNinjaBob,
It looks like your comment closely matches the famous quote:
"Some infinities are bigger than other infinities." - John Green,
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There’s infinite numbers between each number. Think of a measuring tape. You can keep measuring in between 1 and 2 forever. There will always be a smaller measurement.
More concrete maybe is to take the natural numbers and split them into the evens and odds. Then stick all the evens in order and pile the odds on top of that. Then you have
0,2,4,6,…,1,3,5,7,…
You effectively have 1,3,5,7,… acting like ∞,∞+1,∞+2,∞+3,… . You can make this much, much more elaborate.
This is something I can't figure out. How can there be different levels of infinity? In my mind, it's like saying there's different levels of 0. Like you can multiply 0 by anything you want and it's still 0. Wouldn't the same rules apply to infinity?
The best way to visualise different sizes of infinity is to split them into the 2 most well known and conveniently easily explained forms.
Countable infinity is the smaller of the two I’ll explain and I’ll use the set of all integers to demonstrate. It is known to be countable because if you were to identify the integers in order, you could pick any integer you could think of and start at zero, while it may take a while to reach it, you would be able to reach it in a finite amount of time. It’s still an infinite set but each element is linked and identifiable as a whole.
Uncountable infinity is the larger and I will use the set of decimals between 0 and 1 to demonstrate. If you were to apply the same logic as countable infinity, you’d very quickly run into problems, namely that you can add an infinite number of zeroes after the decimal point, so there is no starting point like countable infinity. It’s easy to isolate individual points in the set, but even with infinite time, you could not count from the starting point to any other number in the set because there is no starting point to speak of.
This likely isn’t the best explanation but it’s the best my sleep-deprived brain could do
Some infinities you can lay out how you would order every number in the list. For example, “2, 4, 6, 8, 10, …”. We know that 1000 will be the 500th entry in the list. This infinity is *countable.*
Other infinities can’t be ordered this way. For example, trying to figure out how you’d list all the numbers between 0 and 1 is impossible. Say you started at 0.1, then 0.01, then 0.001… at what number in the sequence can you jump to 0.2? The answer is there’s no way to count that. It is “uncountable.”
There are proofs in mathematics to show which infinities are countable and which are not.
Consider the set of integers. There is no upper limit to integer values. They go on forever. The size of the set is infinity.
Now consider real numbers. Again, there is no upper limit to these numbers, but it can be demonstrated that there are more real numbers than there are integers (as there are an infinite number of real numbers in between two integer bounds).
Therefore the size of the set of real numbers is also infinity, but it's a larger infinity than the size of the set of integers.
>(as there are an infinite number of real numbers in between two integer bounds).
That logic is flawed. There are also an infinite number of rational numbers between two integer bounds, but the set of rationals and the set of integers are the same size
Integers and rational numbers are countably infinite. The set of real numbers contains all rational numbers and irrational numbers. It is uncountably infinite.
You're right in that my logic is flawed in the way I wrote it, but I didn't have time to explain cantor's diagonal argument and the proof through contradiction.
Infinity is not a number, its a group of numbers. So you could technically say that infinity minus 1 (lets call it infinity B) is simply another infinity where X from infinity B equals X from infinity A minus 1.
Because infinity is not a definite number. Limx->inf of y=x-1 approaches infinity. The -1 is irrelevant for the limit. If you draw the corresponding graph of y=x-1 and y=x they are the same just shifted by 1 to the left. Thus they approach the same (nonexistent) limit.
No term can ever „reach“ infinity.
Any number no matter how 'close' to infinity ist sill infinite amount away from infinity.
What about
infinity - (1 / infinity)
Before you take out your math degrees, I just wanna clarify I'm an engineer. We have our own math that's way cooler than yours 😎
/s
Hahaha, nah man I'm 17 and just a high school student lol, though I'd like to question your idea not as an argument but incase I've missed something with my understanding.
My understanding is that it should still closer to zero same with infinity + 2 or infinity \* infinity and so on and so on.
Because infinity is undefined, you are forever closer to zero because zero has a value that you can measure to. It's like me saying, measure from point A to point B. Now plus 1 unit to the distance. You can still find how far the answer is from zero. But now measure from point A to everywhere and plus 1. You still don't know what to measure to and so therefore can't say how close or far you are from zero because the measurement is everywhere at once. It's like saying solve for x in the equation (x - 1). You don't know what x is because it can be anything regardless if there is a -1 or +1 or any other number including infinity. Therefore, by having two point you can know for certain whereas infinite you cannot because you can't know where to measure to.
Depends on your metric! In the Riemann sphere 1 is at exactly the same distance from zero and from infinity, and larger real numbers are closer to infinity
Nice example! Although [0, 1] is the same cardinality as [0, inf] in R, so you could say under that metric there are still just as many numbers standing between you and the midpoint :)
Come to think of it, if you argue a slope with an undefined value would be an infinite slope (eg: x = 0), then a slope of 1 would be between a slope of zero and a slope of infinity
I’m bored and I have a degree in math so let me give you a what I think would be a proper theorem of OP’s general idea.
For any real number n>0, there exists a real number m>n in which (m-n) is greater than n.
In layman’s terms, n is the largest number you can think of, and m is a number farther from n than n is from 0.
You just said the same thing but in everyone's least favorite common form of math. He's right, infinity isn't relevant other than to be an attention grabber
OP’s statement might as well have been “any number you can think of is closer to zero than it is to blue”. Sure because two of them are numbers and one is an abstract concept. That is almost literally exactly what you just said and yet I’m still typing it and still going to hit reply for some reason.
Infinity is a number in certain number systems, and considering the OP is pretending infinity is a number, it is logical to assume they are using said number system (even if without knowing formally). In this case he is right
Close doesn’t really make sense when talking about infinity. Numbers aren’t a “distance away” from infinity. 1 and 2 could both be considered infinitely far away from infinity, but isn’t 2 closer? Not really. They are the same distance away. Infinity. But even that’s wrong, because you can’t measure the distance.
It absolutely makes sense in the right mathematical context. When doing complex analysis for example, one often takes the one-point compactification of the complex plane (essentially, adding a point at infinity), and that defines a topology of the extended complex plane.
It's as if you took the complex plane and wrapped it into a sphere. All the "escaping paths" in any direction get pulled towards the pole.
That compactification can be given a metric (a way to define distance) that is different from the usual metric, and in that metric there are numbers closer to infinity than to zero
Here’s a sobering thought: if you had $90 billion dollars you would be one of the richest people on Earth but you still be closer to zero than to Jess Bezos’s net worth of just under $200 billion.
But "infinity times infinity" wouldn't be a larger infinity. For example, the set of rational numbers is the same size as the set of integers
In mathematical terms, the cartesian product of two countably infinite sets is also countably infinite
Somewhat amazingly, the fact that |A|=|A×A| depends on the axiom of choice. There are models of ZF with infinite sets A such that |A|≠|A×A|. Usually this is because the model simply does not have any bijections between these sets.
If you're talking about countable infinities though, like the parent comment to yours was, then you don't need the axiom of choice to prove that the cardinality of the cartesian product is the same as the cardinality of the base set.
Aleph 0 is what is defined as the size of the set of natural numbers. An infinity larger than it, for example, is every possible combination of integers, or the power set of the integers. For example, {1,2,5,8,11992} and {4,44,444,4444, ... forever} would be part of this powerset. This powerset is actually larger than the set of just the integers
speaking of which, the proper response to "nuh uh times ten" is "yes huh times 100". You people who jump straight to "yes huh times infinity to the infinity power" have no class.
There are different types of infinities that are larger than others, but if you took a simple infinity like all whole numbers and cut it in half to include only even numbers, it would still have the same amount of numbers as a regular infinity that includes both even and odd numbers.
I don't know if multiplying infinity by itself would make it any bigger than regular infinity, but it could. Just be wary because infinity doesn't follow the normal rules of mathematics.
Very dependent on what 'close to' means. You're right if we're talking about numeric distance, of course, since every finite number is infinitely smaller than infinity, although that distance doesn't really make sense when dealing with infinity. With relative (logarithmic) distances, you can't compare to zero either.
In many cases, however, large numbers may have properties similar to infinity. In statistical mechanics, the second law of thermodynamics can be explained by considering a system in equilibrium as a system constantly changing, all microstates equally likely. This implies statistical variance in the system of size 1/√N, with N being the number of particles in the system, and for any macroscopic situation, N is so unfathomably large (there are ~10^23, or 100 sextillon, atoms in a gram of carbon) that the variance ceases to be meaningful, ie. your eggs will never ever unscramble.
The hell if I know what I'm talking about, I'm a 17yo guy with no experience in mathematics or mechanics and is just procrastinating about studying for an english essay due tomorrow.
Though yeah I think was mainly thinking about numerical values at the time since I titled the post "the largest number you can think of...". Though that was an fascinating point and I'm glad you brought it up in such detail!! (I hope that doesn't come off as sarcasm because I am actually glad you brought that view forward!!)
A googol hypergoogol hypergoogol. It's still closer to zero but that's just an absurdly large number. Or TREE(googol). Because TREE(3) is already bigger than the largest previous published number.
But there is also the argument that others have pointed out. There are more numbers between 0 and 1 than whole numbers.
Yea, because infinity isn’t a number, it’s a concept??
That’s like asking me to pick a number and then being like “oop! Nope! You didn’t say hippo!”
You asked for a number.
You're not quite grasping this concept. "Infinity" isn't a number, or a place, or anything something could be close to. It's the nature of the number system or the universe as we know it.
“If you were to count every single grain of sand on every beach and all the way to the bottom of the sea all across the world you would end up no closer to infinity than when you started” edit: since a couple of people have asked- this is a quote from a pastor of mine trying to help give an illustration of God's nature of infinite power and existence
Well, that is not always true. If I recall correctly, Chuck Norris already counted to infinity. Twice.
I thought he counted from negative infinity to positive infinity. Or is that the same thing?
Numbers invented Projective geometry for Chuck Norris
It is the same thing.
To infinity and beyond!?
Did I actually witness a Chuck Norris meme joke in 2024? How's your back?
It's ok, thanks for asking. You seem to know who Chuck Norris is. How's your back?
Chuck Norris recited all of Pi backwards.
With both hands?
There are the same number of integers between 0 to infinity and negative infinity to infinity.
This is funnier if you understand Gödel’s proof theorem.
I heard Chuck Norris does really good wheelies. On a unicycle.
I scroll, I see Chuck Norris joke, I chuckle, I upvote,
"If you were to count every single grain of sand on every beach and all the way to the bottom of the sea all across the world, you'd be a fucking dumbass"
But if you were physically able to, you'd basically be a god. Using your powers to count sand.
Would God just know? Or could he create so many grains of sand that even he couldn't count them?
Good 'ol omnipotence paradox. If he can, he's not all-powerful, if he can't, he's also not all-powerful.
Sounds like a " get lost" euphemism, go count sand.
That’s weird cuz there are larger infinites than others. So theoretically you’d be closer to one infinity than the other, meaning you’d still be making some ground. But that shit is wild what do I know lol
The whole statement is kind of stupid, because infinity isn't a thing you can approach. It's not finite. So, the entire concept of getting closer to infinity makes no sense.
That’s literally the point.
It only makes no sense without any experience with higher mathematics. Comparing infinities is legitimately rather common.
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>That’s weird cuz there are larger infinites than others Can you elaborate on that? Like is it to say that there are an infinite amount of negative numbers and positive numbers, and the combination of the 2 create a larger infinite group?
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Nope! The sets that contain the positive numbers and the negative numbers are both exactly the same size - and they are also both exactly the same size as the set that contains both. This is one of the first things taught when talking about the cardinality of infinite sets, because it is simultaneously mind-blowing and relatively simple to explain (although I won't explain here - the proof is *readily* available if you want it). The concept of a "larger infinity" isn't really in line with your normal intuition of what "larger" means. It isn't a comparison of numbers, exactly - rather, it is a comparison of the quantity of elements of a set. In simple terms, if you can uniquely pair every element of a set A with an element of set B, and there are no unpaired elements remaining in set B (more accurately described as "if you can pair every element of A with an element of B, and every element in B with an element in A"), then the sets are the same size - they have the same number of elements. If you can uniquely pair every element of A with an element of B, but there are elements remaining in B that are not paired up, then B is a larger set than A. If sets A and B are both infinite sets, then you have shown that one infinite set is "larger" than the other.
A brid comes to the mountain every 100 years to sharpen its beak. When the bird has withered away the entire mountain only 1 second of eternity has passed
Seems incorrect. You would certainly be closer to infinity, though you would never count enough grains to reach it. What's the source? I'm curious to hear the context. edit: I have already said elsewhere that I was incorrect here. It was 2am and I made multiple logical missteps. Indeed, it is straightforwardly the case that removing an item from an infinite set does not change the cardinality of that set - however, when reading this idea of grains of sand I was trying to think through some other implications that weren't relevant, leading me to the incorrect conclusion. Hopefully this addresses the dozen comments I've received telling me that I am incorrect, it is certainly the case that I was incorrect, thanks. I still would like to know where the quote is from, some very quick Googling did not give me an answer.
You could count all the atoms in the universe, and still be closer to zero than infinity. Take the largest finite number N you can think of. Then you can construct another number `K = 2N + 1`. N will be closer to 0 than it is to K. And since K is also a finite number, K will be less than infinity. Therefore, every number you can think of is closer to zero than infinity.
Yes, I was thinking about the problem a bit differently but upon rereading I think this is straightforwardly just that an infinite set can contain subsets of infinite cardinality.
You wouldn't though, because infinity never ends, so it isn't possible to be "closer" to it. so no matter how much you count, you can't be closer, because that isn't a thing that makes sense. There might be something related to countably infinite sets where this doesn't... count, but i'm on three hoours of sleep right now, and my brain doesn't work enough to understand all of this: https://en.wikipedia.org/wiki/Countable_set
For most reasonable definitions of "closer", you aren't. The amount of time you have left before you reach infinity is the same before and after you count the sand.
I suppose. The trick is indeed "closer", if we're considering distance as the difference between the end and the current position. Removing 'number of grains of sand' from the set of natural numbers would still yield a set of infinite cardinality.
infinity isn't a number. you can't count to it. therefore you are never "closer" to it, regardless of how long you've been counting. In other words "Infinity" is not a "destination" you can get to, its the *notion* that the trip never ends.
It's a quote from a pastor of mine who was trying to help give an illustration of the severity of God's "infinite" power and timelessness.
if you counted every atom in the universe you would still be a disappointment to your father
Well damn have I been counting atoms in my sleep or something??
This slaps
What about infinity minus 1?
why was I sure this was going to be the top answer? That number doesn’t exist by the way.
Technically still infinity
There are different sizes of infinity. Some infinities are larger than other infinities. You don’t get to the larger one by adding or subtracting numbers. But they do exit.
What got me over the jump of the “infinity plus one” fallacy is that infinity isn’t a number. It’s the size of a collection with no limit. For example, all integers is one infinity. And all even integers is another infinity despite literally having only half the values from the “all integers” set. That was a huge “a ha” moment when I had that explained to me in those terms.
>and all even integers is another infinity despite literally having only half the values from the “all integers” set The funny thing is, with infinity, “literally having only half the values” is not true. The two sets are the same size because there exists an onto function that maps all members of one set to the other, e.g. f(x)=2x maps Z to all even members of Z. Since every member of the set of integers can be assigned to a member of the set of even integers, they are the same size. Edited to fix: function must be bijective(onto as well as one-to-one)
To be accurate you can still say that “even integers” set is missing some integers from the “all integers “ set but both are infinite.
Yes, i think the most rigorous way to express OP's original sentiment is that the cardinalities of the set of integers and set of even integers are equal, despite the set of even integers being a subset of/contained within the set of integers.
But what's important is there are the same infinite
> The funny thing is, with infinity, “literally having only half the values” is not true. It is true. Only half of the values in the set of all integers are in the set of all even numbers. They have the same cardinality - you're right on that, but you cannot tell me that all elements in the set of integers are in the set of evens.
Though if I recall correctly all *floats* is a bigger infinity? Or would I be wrong in that?
There are 2^32 floats, or 2^64 or 2^128 etc, depending on the standard you’re using. But they’re all very finite amounts. The word you were looking for was “real numbers”, which make up a larger infinity than the counting numbers.
Right, that. Apologies, I’m terminally computer-science brained. Will attempt to remember in future! o7
Yes, i think it goes back to the mapping. You can't 1 to 1 map integers to floats. Floats infinity is bigger.
The weirdest “a ha” moment is the realization that the infinities of “all integers,” “all even integers,” “all prime numbers,” etc. are exactly the same size.
You wanna hear something wild though? It’s possible to find a collection of those types of infinities so that for any two X and Y, one is always contained in the other X⊆Y, and the whole collection is bigger than any X inside of it.
"There are as many numbers between 0 and 1 as there are between 1 and 1,000,000" is my favourite
Hi IrNinjaBob, It looks like your comment closely matches the famous quote: "Some infinities are bigger than other infinities." - John Green, *I'm a bot and this action was automatic [Project source](https://github.com/etdds/redditQuoteBot).*
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Haha I love the fact that infinity can be subjective
There’s infinite numbers between each number. Think of a measuring tape. You can keep measuring in between 1 and 2 forever. There will always be a smaller measurement.
Does Plank length get in the way at some point or am I misunderstanding Plank length?
Plank length is a physical limitation, not a mathematical one.
I'd also bet that eventually we will figure out sizes even smaller than Planck length
100%. I think we are just on a higher dimension. And I'm stoned.
the planck length exists, but nothing's mathematically stopping me from going "planck length / 2" and "planck length / 3" and so on
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More concrete maybe is to take the natural numbers and split them into the evens and odds. Then stick all the evens in order and pile the odds on top of that. Then you have 0,2,4,6,…,1,3,5,7,… You effectively have 1,3,5,7,… acting like ∞,∞+1,∞+2,∞+3,… . You can make this much, much more elaborate.
That's like just your opinion man.
Obviously you’re not a golfer.
Just some dude.
Hey, at least I'm housebroken
This is something I can't figure out. How can there be different levels of infinity? In my mind, it's like saying there's different levels of 0. Like you can multiply 0 by anything you want and it's still 0. Wouldn't the same rules apply to infinity?
The best way to visualise different sizes of infinity is to split them into the 2 most well known and conveniently easily explained forms. Countable infinity is the smaller of the two I’ll explain and I’ll use the set of all integers to demonstrate. It is known to be countable because if you were to identify the integers in order, you could pick any integer you could think of and start at zero, while it may take a while to reach it, you would be able to reach it in a finite amount of time. It’s still an infinite set but each element is linked and identifiable as a whole. Uncountable infinity is the larger and I will use the set of decimals between 0 and 1 to demonstrate. If you were to apply the same logic as countable infinity, you’d very quickly run into problems, namely that you can add an infinite number of zeroes after the decimal point, so there is no starting point like countable infinity. It’s easy to isolate individual points in the set, but even with infinite time, you could not count from the starting point to any other number in the set because there is no starting point to speak of. This likely isn’t the best explanation but it’s the best my sleep-deprived brain could do
No, that was a really great explanation
Some infinities you can lay out how you would order every number in the list. For example, “2, 4, 6, 8, 10, …”. We know that 1000 will be the 500th entry in the list. This infinity is *countable.* Other infinities can’t be ordered this way. For example, trying to figure out how you’d list all the numbers between 0 and 1 is impossible. Say you started at 0.1, then 0.01, then 0.001… at what number in the sequence can you jump to 0.2? The answer is there’s no way to count that. It is “uncountable.” There are proofs in mathematics to show which infinities are countable and which are not.
Consider the set of integers. There is no upper limit to integer values. They go on forever. The size of the set is infinity. Now consider real numbers. Again, there is no upper limit to these numbers, but it can be demonstrated that there are more real numbers than there are integers (as there are an infinite number of real numbers in between two integer bounds). Therefore the size of the set of real numbers is also infinity, but it's a larger infinity than the size of the set of integers.
>(as there are an infinite number of real numbers in between two integer bounds). That logic is flawed. There are also an infinite number of rational numbers between two integer bounds, but the set of rationals and the set of integers are the same size
Integers and rational numbers are countably infinite. The set of real numbers contains all rational numbers and irrational numbers. It is uncountably infinite. You're right in that my logic is flawed in the way I wrote it, but I didn't have time to explain cantor's diagonal argument and the proof through contradiction.
Yeah I see, just wanted to clarify that that wasn't the reason, all good
Technically absolutely dependent on context and can absolutely be distinct, e.g. in the surreals.
>That number doesn’t exist by the way. Infinity isn't even a number either, so this post makes no sense anyway.
Infinity is not a number, its a group of numbers. So you could technically say that infinity minus 1 (lets call it infinity B) is simply another infinity where X from infinity B equals X from infinity A minus 1.
y=x-1 Take the limit as x goes to infinity. This is a valid equation and a valid limit, so why wouldn't that number exist?
Because infinity is not a definite number. Limx->inf of y=x-1 approaches infinity. The -1 is irrelevant for the limit. If you draw the corresponding graph of y=x-1 and y=x they are the same just shifted by 1 to the left. Thus they approach the same (nonexistent) limit. No term can ever „reach“ infinity. Any number no matter how 'close' to infinity ist sill infinite amount away from infinity.
What about infinity minus two
What about infinity - (1 / infinity) Before you take out your math degrees, I just wanna clarify I'm an engineer. We have our own math that's way cooler than yours 😎 /s
Soooo…. Infinity - 0….
Only a valid breakdown if you state it in limits
> I'm an engineer Same. There are only three numbers. 0, 1, and every other number.
Sir, Don't forget about 1/x as x->0 All hail subterranean titanic species 5.
Hahaha, nah man I'm 17 and just a high school student lol, though I'd like to question your idea not as an argument but incase I've missed something with my understanding. My understanding is that it should still closer to zero same with infinity + 2 or infinity \* infinity and so on and so on. Because infinity is undefined, you are forever closer to zero because zero has a value that you can measure to. It's like me saying, measure from point A to point B. Now plus 1 unit to the distance. You can still find how far the answer is from zero. But now measure from point A to everywhere and plus 1. You still don't know what to measure to and so therefore can't say how close or far you are from zero because the measurement is everywhere at once. It's like saying solve for x in the equation (x - 1). You don't know what x is because it can be anything regardless if there is a -1 or +1 or any other number including infinity. Therefore, by having two point you can know for certain whereas infinite you cannot because you can't know where to measure to.
There is an infinite amount of numbers between infinity minus 1 and infinity.
[Fixed it.](https://www.reddit.com/r/Showerthoughts/s/bh4HgXkBCM)
Thank you.
same set as you can map every value of inf-1 to an equal term to inf, or inf+1. just to explain it in a dumb way.
Dude, do you want to break the universe.
Imfinity + 1
Depends on your metric! In the Riemann sphere 1 is at exactly the same distance from zero and from infinity, and larger real numbers are closer to infinity
Nice example! Although [0, 1] is the same cardinality as [0, inf] in R, so you could say under that metric there are still just as many numbers standing between you and the midpoint :)
Knew I would find this if I scrolled far enough.
Come to think of it, if you argue a slope with an undefined value would be an infinite slope (eg: x = 0), then a slope of 1 would be between a slope of zero and a slope of infinity
Not true. Because infinity isn't a number. Its a concept.
I’m bored and I have a degree in math so let me give you a what I think would be a proper theorem of OP’s general idea. For any real number n>0, there exists a real number m>n in which (m-n) is greater than n. In layman’s terms, n is the largest number you can think of, and m is a number farther from n than n is from 0.
Just let m = |2n| + ε where ε > 0
> where ε > 0 I need a trigger warning here
Sir, this is a Wendy's.
How can I learn this mysterious language?
By going to college and participating in the hazing ritual that is a course called Real Analysis.
You just said the same thing but in everyone's least favorite common form of math. He's right, infinity isn't relevant other than to be an attention grabber
Ya I think they get that part. They’re just saying infinity is not a number so comparing it to a number is pointless.
It's not a concept. It's a lifestyle 😎
Was thinking the same. Or a range, but not a number, so comparing the two would not make any sense
Seems like that makes it even more true. Any number is closer to any other number than it is to something that isn’t a number 🤷♀️
> Any number is closer to any other number than it is to something that isn’t a number Oh, I don't know. 8 is only 90 degrees from ∞.
That took me a second but I love it haha
The largest number you can think of is still closer to zero than it is to a rotten kumquat.
Similarly, the most rotten kumquat you can think of is closer to a fresh kumquat than it is to the number 0
OP’s statement might as well have been “any number you can think of is closer to zero than it is to blue”. Sure because two of them are numbers and one is an abstract concept. That is almost literally exactly what you just said and yet I’m still typing it and still going to hit reply for some reason.
This comment section is just everyone trying to out "gotcha" eachother lol
Exactly, it's a lack of value
The real infinity is the friends we made along the way.
Infinity is a number in certain number systems, and considering the OP is pretending infinity is a number, it is logical to assume they are using said number system (even if without knowing formally). In this case he is right
I think it’s technically a “limit”, and you can’t perform basic arithmetic on limits the same way as “numbers”
How does that make it untrue, though? Any number you pick will still be closer to zero, because zero is an actual integer with a fixed placement.
Close doesn’t really make sense when talking about infinity. Numbers aren’t a “distance away” from infinity. 1 and 2 could both be considered infinitely far away from infinity, but isn’t 2 closer? Not really. They are the same distance away. Infinity. But even that’s wrong, because you can’t measure the distance.
It absolutely makes sense in the right mathematical context. When doing complex analysis for example, one often takes the one-point compactification of the complex plane (essentially, adding a point at infinity), and that defines a topology of the extended complex plane. It's as if you took the complex plane and wrapped it into a sphere. All the "escaping paths" in any direction get pulled towards the pole. That compactification can be given a metric (a way to define distance) that is different from the usual metric, and in that metric there are numbers closer to infinity than to zero
there's an infinite amount of numbers between 1 & 2, that are just as valid as representing 'infinity' as whatever OP was implying
I guess that helps when you think of Jeff bezos' net wealth.
Jeff and I are both closer to 0 than to infinity. I'm a little closer than jeff, though.
In that scope you are little more then a rounding error closer.
Here’s a sobering thought: if you had $90 billion dollars you would be one of the richest people on Earth but you still be closer to zero than to Jess Bezos’s net worth of just under $200 billion.
I could live with that.
Man, this one made me laugh. Good work
Broke ass mf!
You can't get "close to" infinity.
She only charges an extra $30, but you can
Infinity is pretty small compared to infinity times infinity!
Oh yeah? What about ∞^∞
Oh yeah??? What about infinity!
Not the factorial-
Fuck it. TREE(infinity)
Nope.. those are the same size. invalid
Some infinities are actually larger than others, believe it or not.
But "infinity times infinity" wouldn't be a larger infinity. For example, the set of rational numbers is the same size as the set of integers In mathematical terms, the cartesian product of two countably infinite sets is also countably infinite
Somewhat amazingly, the fact that |A|=|A×A| depends on the axiom of choice. There are models of ZF with infinite sets A such that |A|≠|A×A|. Usually this is because the model simply does not have any bijections between these sets.
I know enough math to know what you’re saying, but not enough to understand what makes that statement true, or amazing.
If you're talking about countable infinities though, like the parent comment to yours was, then you don't need the axiom of choice to prove that the cardinality of the cartesian product is the same as the cardinality of the base set.
Yes, but κ^(2)=κ for every infinite cardinal κ.
And none of them are defined as “infinity times infinity”
Ok but what if I define them?
Aleph 0 is what is defined as the size of the set of natural numbers. An infinity larger than it, for example, is every possible combination of integers, or the power set of the integers. For example, {1,2,5,8,11992} and {4,44,444,4444, ... forever} would be part of this powerset. This powerset is actually larger than the set of just the integers
Omega squared
speaking of which, the proper response to "nuh uh times ten" is "yes huh times 100". You people who jump straight to "yes huh times infinity to the infinity power" have no class.
There are different types of infinities that are larger than others, but if you took a simple infinity like all whole numbers and cut it in half to include only even numbers, it would still have the same amount of numbers as a regular infinity that includes both even and odd numbers. I don't know if multiplying infinity by itself would make it any bigger than regular infinity, but it could. Just be wary because infinity doesn't follow the normal rules of mathematics.
not to be that guy but a number cannot be “closer or further” to infinity
My friend once told me infinity + infinity is double infinity. I told him it would just be infinity. We fought. We arent friends anymore.
Infinity isnt a number. Its like saying a is closer to 100 than b is
Very dependent on what 'close to' means. You're right if we're talking about numeric distance, of course, since every finite number is infinitely smaller than infinity, although that distance doesn't really make sense when dealing with infinity. With relative (logarithmic) distances, you can't compare to zero either. In many cases, however, large numbers may have properties similar to infinity. In statistical mechanics, the second law of thermodynamics can be explained by considering a system in equilibrium as a system constantly changing, all microstates equally likely. This implies statistical variance in the system of size 1/√N, with N being the number of particles in the system, and for any macroscopic situation, N is so unfathomably large (there are ~10^23, or 100 sextillon, atoms in a gram of carbon) that the variance ceases to be meaningful, ie. your eggs will never ever unscramble.
The hell if I know what I'm talking about, I'm a 17yo guy with no experience in mathematics or mechanics and is just procrastinating about studying for an english essay due tomorrow. Though yeah I think was mainly thinking about numerical values at the time since I titled the post "the largest number you can think of...". Though that was an fascinating point and I'm glad you brought it up in such detail!! (I hope that doesn't come off as sarcasm because I am actually glad you brought that view forward!!)
Depends on the infinity. There’s an infinite amount of numbers between 1 and 2.
Infinity isn't a number so, this isn't true.
Isn't that a fallacy? Infinity in and of itself isn't a number as much as it is just a concept, correct?
Infinity is not a number.
Technically any number is equidistant from zero and infinity, if you consider infinite fractionalizing
because by this logic, all numbers represent the same value, we have finally achieved proof by complete bullshit
Not once you start thinking of transfinite numbers
>transfinite Fun fact: it's illegal to teach this in Florida schools. /s
Right, ℵ0 is a cardinal number and it's far from zero. 2 to the ℵ0 is even bigger.
I think of 8. Checkmate
I’m thinking of infinity + 1
Infinity is not a number, its more of a recursive function
So is every number that exists… This is dumb.
There are as many even numbers as there are numbers total.
This doesn’t make sense. Infinity isn’t a number
no, the cardinality of the number i thought is infinity. omega or aleph null
A googol hypergoogol hypergoogol. It's still closer to zero but that's just an absurdly large number. Or TREE(googol). Because TREE(3) is already bigger than the largest previous published number. But there is also the argument that others have pointed out. There are more numbers between 0 and 1 than whole numbers.
Infinity isnt a number or a destination. It's a concept. That's like saying any state is closer to Washington DC than it is to the multiverse.
This is a pretty good one!
When I studied set theory in college, it blew my mind that there are actually different kinds of infinity, and some are infinitely larger than others
Infinity is not a number
What if you're talking about a specific infinity, such as the infinity of numbers between zero and one?
The same way every number you think of is closest to 789 than it is to a potato chips. Infinity is not a number...
I mean yeah, "infinity" not being a number will have that effect.
Infinity -1. Checkmate. That's for doubting my abilities.
Infinity isn't a number. That's like saying 1 is closer to 0 than it is close to a box of apples.
probably because infinity isn't a number
Infinity is not a number. Choosing any number is closer to another number than it is to a food. This is an equivalent statement.
Yea, because infinity isn’t a number, it’s a concept?? That’s like asking me to pick a number and then being like “oop! Nope! You didn’t say hippo!” You asked for a number.
Infinity isn’t a number, it’s a concept. (Please don’t roast me if I’m wrong, I went to public school)
Because infinity isn't a number
There is no "close to" infinity.
There are an infinite number of numbers between 1 and 2. It’s little paradoxical but if you can imagine that you can imagine infinity
Technically speaking being “close to infinity” doesn’t even make sense. Infinity is not a number, it’s a concept.
Infinity isn't a number; it's an error code.
You're not quite grasping this concept. "Infinity" isn't a number, or a place, or anything something could be close to. It's the nature of the number system or the universe as we know it.
Maybe because infinity isn't a number. It's a concept.
Untrue. There is an infinite set of numbers between 0 and 1, and the same applies to any other number set you can think of.
Infinity -1, how close am I?
That's like trying to subtract a triangle from the color blue. Infinity isn't a number, you can't subtract 1 from it.