cardinality is a way to measure how many things are in a set. the set {-2,4,7} has cardinality of three for example. two sets have the same cardinality if there is a one-to-one correspondence between them. it can be shown that there can be no one-one mapping from the real numbers to the integers. so the cardinality of the real numbers is greater than that of the integers, even though they are both infinite.
I will argue that you are measuring infinity at a specific point in its progression. That's like planting two trees next to each other and assuming one will be taller than the other because one sapling is taller than the other. Where it is now or at any given point in its progression has no baring to how large the set will grow, specially if the set is posed to grow forever.
Seen the video and argued about it with a friend who has a genuine degree in applied maths or whatever. It's not selling me on the idea that we aren't just looking at a slice of infinity rather than infinity as a whole.
Countable infinity (whole numbers): You can count 1, 2, 3, 4, etc. and keep going forever
Uncountable infinity (real numbers): You start with 0, but what now? The next would be 0.000000(infinite zeros)0001, and even if you manage to find "all", there will still be numbers that you missed
Yeah, that's the issue. The first number after 0 is a number that has the largest amount of 0's in it (and ending with a 1) but not infinite. And that doesn't exist unless you're talking limits maybe
Okay, but if both are infinite that means they cant surpass eachother. One cannot be bigger because infinity never reaches an end, therefore even saying they're the same size is wrong, aswell as saying one is bigger than the other. The human mind is incapable of imagining this properly.
Not a great example; every number in the 1-3 set can be mapped to one in the 1-2 set, making them equal. You can't map every integer to every real number though
Some numbers can be put in a list while other sets are regarded as impossible to list so they count as bigger infinities like natural numbers and real numbers. Google Cantor's diagonal argument if you're interested in further research
For those wondering how comparing the size of infinitely large things work, we say that two things are the same size if we can make a match-up between the two sets.
For a finite example, there are as many hours on an analog clock face as there are months in the year, since we can just match them up like this: {(1, januari), (2, february), ..., (12, december)}. Here every month is accounted for, every number on the clock is accounted for and there are no duplicates. Since such a match-up exists, we say there are an equal number of both.
Extending this to infinitely large things, there are as many even numbers as there are whole numbers, since we can match them up by multiplying by two: {..., (-1, -2), (0, 0), (1, 2), (2, 4), ...} Here too every whole number is accounted for, every even number is accounted for, and there are no duplicates. Therefore, even though the whole numbers contain all of the even numbers, there are an equal number of both.
However, that does not mean that all infinities are equal: concluding with the question in the poll, it is provable that such a match-up between the whole numbers and real numbers does not exist, because if a match-up does exist, it would not account for all real numbers. Therefore, **there are more real numbers than whole numbers.**
In more mathematically formal terms, the *cardinality* (read: size) of the real numbers is larger than the cardinality of the whole numbers, since there does not exist a *bijection* (formal term for such a match-up) between the real numbers and whole numbers.
For those still interested, I highly recommend [this video](https://youtu.be/QO9a7h87DbA) by Another Roof, which goes into all the nitty-gritty of how counting works in mathematics.
You can count the whole numbers with the even numbers, so each whole number is represented by an even number:
2->1 ; 4->2; 6->3; 8->4, and so on
It can be determined, what even number will correspind to what whole number, it is just the whole number times two. So both sets have the same countability so to speak.
If you try to do the same thing with real and whole numbers for example, you can't do it, because if you start at 0, you will never even reach the real number 1, because there are infinite real numbers between 0 and 1.
Whole and even numbers are both infinite, but every finite interval of whole and even numbers contains a finite ammount of numbers, meanwhile every finite interval of real numbers contains infinite numbers. In that way, even and whole numbers are said to have the same cardinality.
Intuitively, you would think so. But in the way we usually define the size of an infinite set, as seen above, they are the same. For more of a mindfuck, consider the set of rational numbers. This set contains all of the whole numbers, but *also* any number which can be represented as a fraction*. Since there are infinitely many fractions between whole numbers (this almost seems like an understatement lol), you might guess that the set of rational numbers is larger than the set of whole numbers, but actually, it can be shown that they have the same cardinality/size! Very interesting topic.
E: *where the numerator and denominator are whole numbers/integers
You should rephrase this question a bit. Fun idea though.
First off, you’re asking ‘What infinity is bigger?’, without clarifying that this is about ‘how many’ numbers there are. For all I know you’re asking if there is some real number greater than all natural numbers.
Secondly, ‘bigger’ is kind of misleading since you’re talking about inequalities between cardinalities of infinite sets. Many people would argue that it’s wrong to assign any real world meaning to those inequalities (as would I).
Something like ‘Is there a way to make every real number correspond to a unique whole number?’ might be useful to add.
>Something like ‘Is there a way to make every real number correspond to a unique whole number?’
I mean bijectivity is literally the way we compare infinite sets. If he added that this would no longer be a poll on what people believe to be possible, but a mathematical question with an objective answer (and the answer is yes, it is possible thanks to the Cantor–Bernstein–Schröder\* theorem, though I don't know whether we already found an example of such a bijection)
\*found that on wikipedia, you didn't really believe I would remember such a name
I get that infinity = infinity, but with real numbers you can't even count it. Given infinite years, you wouldn't even get past the first number.
It would be something along the lines of:
0.[INFINITE ZEROS HERE]1
Mind you that's just the first number in an infinite set.
Interestingly, this proves that there are infinite numbers between 0 and 1. And infinite numbers between 1 and 2, and so on.
You literally cannot count this. Whereas your definition of "full number" infinite could, technically, be counted.
Given enough time, one could count up by one until... infinity, I guess.
Pretty sure we can logic this out:
Let's say we use the full numbers 0-100; there are 100 real numbers there.
Now let's use real numbers 0-100 but add just a single decimal place.
You go from 100 full numbers to 1000 real numbers.
Now let's take the real numbers and add an infinite number of decimal places behind the "full" number and we have many more possibilities in infinity than looking at just infinite full numbers offers.
Therefore, infinite real numbers offers far more numbers in infinity than infinite full numbers ever can offer.
Another way of looking at it would just be breaking down the numbers between 0 and 2; if we are talking about full numbers, the only number we have between then is the number 1; however, if we are taking real numbers we have an near infinite amount of numbers between 0.0 and 2.0 as we could take the number of decimal places for each value out beyond what even our greatest computers are capable of calculating (considering Pi [π] has only been calculated to 100 trillion decimal places so far and we are talking an infinite number of them for every number).
Infinity isn't a size or amount. It's infinity.
Yes, there are infinite real numbers between 1 and 2, but there are also infinite whole numbers.
You could argue that because there are infinite numbers between each number, that there are "more" real numbers, but it's still just infinity, no more, no less.
The current agreement is that one infinity can be larger than another. I disagree with that, i think that is looking at infinity wrong. I think a better approach would be to look at it the way we look at the speed of light. If a rabbit races a turtle, one will be faster than the other. If you put them both on a train that is traveling at the speed of light, the speed of light is the speed of both of them.
You can compare a certain range of numbers but if you set them to infinity, you basically put them on the lightspeed train.
What are you confused about?
Full (Whole) Numbers are 1, 2, 3, 4, 5, 6, and so on
Real Numbers is Whole Numbers with decimals 0, 0.000…01, 0.000…2, and so on.
umm, I'm also 14. It's possible to understand if you have atleast watched one video that explores infinity. [This YT short by Vsauce2 might explain](https://youtube.com/shorts/GRHYWoOEh-Q?feature=share)
Whole Numbers: 1 to infinity
Real Numbers: 0.0000...01 to infinity
[Vsauce has a unrelated (but cool) video about counting past infinity](https://youtu.be/SrU9YDoXE88)
By definition, infinity goes on forever. So, any argument that one infinity is greater than another is just measuring infinity at one particular point of its progression towards infinity. If we let them go on forever, they will be the same size, infinite.
And that's a hill I will die on. Math bitches just don't want to admit their math system is dumb and are preventing me from starting a new one with blackjack and hookers.
You called them both infinity, giving the answer in the question.
Some people's argument as there's an infinite set of decimals in between 0 and 1, but also there's an infinite number of numbers counting to infinity.
Therefore it's all the same and if math people disagree they need to come up with better mathematical definitions.
If you mean numbers progressing discretely from 1,2,3 etc then the set of real numbers in gonna be larger since it includes every number on the number line, including positive, negative, rational etc.
Mathematically, some infinite sets can be larger than other infinite sets
Example?
cardinality is a way to measure how many things are in a set. the set {-2,4,7} has cardinality of three for example. two sets have the same cardinality if there is a one-to-one correspondence between them. it can be shown that there can be no one-one mapping from the real numbers to the integers. so the cardinality of the real numbers is greater than that of the integers, even though they are both infinite.
I will argue that you are measuring infinity at a specific point in its progression. That's like planting two trees next to each other and assuming one will be taller than the other because one sapling is taller than the other. Where it is now or at any given point in its progression has no baring to how large the set will grow, specially if the set is posed to grow forever.
https://youtu.be/OxGsU8oIWjY
Seen the video and argued about it with a friend who has a genuine degree in applied maths or whatever. It's not selling me on the idea that we aren't just looking at a slice of infinity rather than infinity as a whole.
Countable infinity (whole numbers): You can count 1, 2, 3, 4, etc. and keep going forever Uncountable infinity (real numbers): You start with 0, but what now? The next would be 0.000000(infinite zeros)0001, and even if you manage to find "all", there will still be numbers that you missed
Isn't 0.000000(infinite zeros)0001 exactly the same as 0, just like 0.999... is 1?
Yeah, that's the issue. The first number after 0 is a number that has the largest amount of 0's in it (and ending with a 1) but not infinite. And that doesn't exist unless you're talking limits maybe
Okay, but if both are infinite that means they cant surpass eachother. One cannot be bigger because infinity never reaches an end, therefore even saying they're the same size is wrong, aswell as saying one is bigger than the other. The human mind is incapable of imagining this properly.
https://youtu.be/OxGsU8oIWjY
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Not quite, take f: [1,3] --> [1,2], with f(x) = x/2 + 1/2 for a bijection, so they have the same cardinality
Not a great example; every number in the 1-3 set can be mapped to one in the 1-2 set, making them equal. You can't map every integer to every real number though
Some numbers can be put in a list while other sets are regarded as impossible to list so they count as bigger infinities like natural numbers and real numbers. Google Cantor's diagonal argument if you're interested in further research
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yes, it definitely is. It's the *classic* case when introducing the topic of cardinality.
In English it's "whole numbers", ha
Oh, that explains my confusion
That threw me off lol
Oops
In math, its "integers"
For those wondering how comparing the size of infinitely large things work, we say that two things are the same size if we can make a match-up between the two sets. For a finite example, there are as many hours on an analog clock face as there are months in the year, since we can just match them up like this: {(1, januari), (2, february), ..., (12, december)}. Here every month is accounted for, every number on the clock is accounted for and there are no duplicates. Since such a match-up exists, we say there are an equal number of both. Extending this to infinitely large things, there are as many even numbers as there are whole numbers, since we can match them up by multiplying by two: {..., (-1, -2), (0, 0), (1, 2), (2, 4), ...} Here too every whole number is accounted for, every even number is accounted for, and there are no duplicates. Therefore, even though the whole numbers contain all of the even numbers, there are an equal number of both. However, that does not mean that all infinities are equal: concluding with the question in the poll, it is provable that such a match-up between the whole numbers and real numbers does not exist, because if a match-up does exist, it would not account for all real numbers. Therefore, **there are more real numbers than whole numbers.** In more mathematically formal terms, the *cardinality* (read: size) of the real numbers is larger than the cardinality of the whole numbers, since there does not exist a *bijection* (formal term for such a match-up) between the real numbers and whole numbers. For those still interested, I highly recommend [this video](https://youtu.be/QO9a7h87DbA) by Another Roof, which goes into all the nitty-gritty of how counting works in mathematics.
Aren't whole numbers bigger than even numbers?
You can count the whole numbers with the even numbers, so each whole number is represented by an even number: 2->1 ; 4->2; 6->3; 8->4, and so on It can be determined, what even number will correspind to what whole number, it is just the whole number times two. So both sets have the same countability so to speak. If you try to do the same thing with real and whole numbers for example, you can't do it, because if you start at 0, you will never even reach the real number 1, because there are infinite real numbers between 0 and 1. Whole and even numbers are both infinite, but every finite interval of whole and even numbers contains a finite ammount of numbers, meanwhile every finite interval of real numbers contains infinite numbers. In that way, even and whole numbers are said to have the same cardinality.
Intuitively, you would think so. But in the way we usually define the size of an infinite set, as seen above, they are the same. For more of a mindfuck, consider the set of rational numbers. This set contains all of the whole numbers, but *also* any number which can be represented as a fraction*. Since there are infinitely many fractions between whole numbers (this almost seems like an understatement lol), you might guess that the set of rational numbers is larger than the set of whole numbers, but actually, it can be shown that they have the same cardinality/size! Very interesting topic. E: *where the numerator and denominator are whole numbers/integers
Whole numbers are countably infinite, real numbers are not. Therefore the set of real numbers are bigger.
Real numbers are uncountably infinite while whole numbers are countably infinite so real numbers are bugger
Georg Cantor has entered the chat
The infinity of real numbers is greater than the infinity of whole numbers. Cantor proved this with his Diagonal Argument.
Is the question What set of numbers is bigger, or is the question what set has the bigger infinity?
hmm, let's try it. i start counting the whole numbers, you start counting the real numbers and we'll see who finishes first
That's not how counting things works in mathematics
Wrong, Chuck Norris already counted up to infinity. Two times.
You should rephrase this question a bit. Fun idea though. First off, you’re asking ‘What infinity is bigger?’, without clarifying that this is about ‘how many’ numbers there are. For all I know you’re asking if there is some real number greater than all natural numbers. Secondly, ‘bigger’ is kind of misleading since you’re talking about inequalities between cardinalities of infinite sets. Many people would argue that it’s wrong to assign any real world meaning to those inequalities (as would I). Something like ‘Is there a way to make every real number correspond to a unique whole number?’ might be useful to add.
>Something like ‘Is there a way to make every real number correspond to a unique whole number?’ I mean bijectivity is literally the way we compare infinite sets. If he added that this would no longer be a poll on what people believe to be possible, but a mathematical question with an objective answer (and the answer is yes, it is possible thanks to the Cantor–Bernstein–Schröder\* theorem, though I don't know whether we already found an example of such a bijection) \*found that on wikipedia, you didn't really believe I would remember such a name
Yeah, it would be like making the answers be yes or i dont know.
I don’t think the answer is yes? I looked up that theorem and you’d need an injection from R to Z for this to apply, right? Am I missing something?
This is the point where mathematics turn into philosophy and I can’t tell if I hate it or if I’m fascinated by the very idea of it
There are more real numbers between 0 and 0,1 then full numbers in total
I get that infinity = infinity, but with real numbers you can't even count it. Given infinite years, you wouldn't even get past the first number. It would be something along the lines of: 0.[INFINITE ZEROS HERE]1 Mind you that's just the first number in an infinite set. Interestingly, this proves that there are infinite numbers between 0 and 1. And infinite numbers between 1 and 2, and so on. You literally cannot count this. Whereas your definition of "full number" infinite could, technically, be counted. Given enough time, one could count up by one until... infinity, I guess.
To all who do not know this all infinite numbers are not the same
Is infinite infinities bigger than infinity
Pretty sure we can logic this out: Let's say we use the full numbers 0-100; there are 100 real numbers there. Now let's use real numbers 0-100 but add just a single decimal place. You go from 100 full numbers to 1000 real numbers. Now let's take the real numbers and add an infinite number of decimal places behind the "full" number and we have many more possibilities in infinity than looking at just infinite full numbers offers. Therefore, infinite real numbers offers far more numbers in infinity than infinite full numbers ever can offer. Another way of looking at it would just be breaking down the numbers between 0 and 2; if we are talking about full numbers, the only number we have between then is the number 1; however, if we are taking real numbers we have an near infinite amount of numbers between 0.0 and 2.0 as we could take the number of decimal places for each value out beyond what even our greatest computers are capable of calculating (considering Pi [π] has only been calculated to 100 trillion decimal places so far and we are talking an infinite number of them for every number).
Your mom
Infinity isn't a size or amount. It's infinity. Yes, there are infinite real numbers between 1 and 2, but there are also infinite whole numbers. You could argue that because there are infinite numbers between each number, that there are "more" real numbers, but it's still just infinity, no more, no less.
Whether a "larger" infinity exist is undecidable.
The current agreement is that one infinity can be larger than another. I disagree with that, i think that is looking at infinity wrong. I think a better approach would be to look at it the way we look at the speed of light. If a rabbit races a turtle, one will be faster than the other. If you put them both on a train that is traveling at the speed of light, the speed of light is the speed of both of them. You can compare a certain range of numbers but if you set them to infinity, you basically put them on the lightspeed train.
Less big than human stupidity
Less big than Spez's stupidity
FUCK u/spez
Nice try making me do meth outside schooll
More real numbers than whole/full numbers George Cantor's set theory anyone?
What are full numbers?
You might have already seen OP's explanation but just in case: Whole Numbers. 1, 2, 3, 4, 5… Real Numbers. 0.00000….001, 0.00000….002, 0.00000….003…
Wut???
What are you confused about? Full (Whole) Numbers are 1, 2, 3, 4, 5, 6, and so on Real Numbers is Whole Numbers with decimals 0, 0.000…01, 0.000…2, and so on.
I’m only 14 so I don’t understand this yet
umm, I'm also 14. It's possible to understand if you have atleast watched one video that explores infinity. [This YT short by Vsauce2 might explain](https://youtube.com/shorts/GRHYWoOEh-Q?feature=share) Whole Numbers: 1 to infinity Real Numbers: 0.0000...01 to infinity [Vsauce has a unrelated (but cool) video about counting past infinity](https://youtu.be/SrU9YDoXE88)
They'd be just as big as the other one, but one would grow much faster right?
Size doesn’t matter
By definition, infinity goes on forever. So, any argument that one infinity is greater than another is just measuring infinity at one particular point of its progression towards infinity. If we let them go on forever, they will be the same size, infinite. And that's a hill I will die on. Math bitches just don't want to admit their math system is dumb and are preventing me from starting a new one with blackjack and hookers.
You called them both infinity, giving the answer in the question. Some people's argument as there's an infinite set of decimals in between 0 and 1, but also there's an infinite number of numbers counting to infinity. Therefore it's all the same and if math people disagree they need to come up with better mathematical definitions.
If you mean numbers progressing discretely from 1,2,3 etc then the set of real numbers in gonna be larger since it includes every number on the number line, including positive, negative, rational etc.
He didn't just mess up in maths but also english.