You’ve misunderstood “some infinities are bigger than others.” Both of these infinities are the same size. You can show this since each person on the bottom track can be assigned a person from the top track at 1 to 1 ratio. An example of infinities that are different sizes are all whole numbers and all decimal numbers. You cannot assign a whole number to every decimal number.
Matt parker does a good video on this. I can’t remember the exact title but if you search “is infinite $20 notes worth more than infinite $1 notes” you should find it.
By assigning people to it it becomes countable. You can’t assign uncountable numbers like that. It’s both in the text of the meme and in the illustration
An example of infinities that are different sizes are all whole numbers and all decimal numbers.
not sure what you mean by this, if you mean fractions you are wrong. Rational numbers and natural numbers can have a 1 on 1 assignment, look up cantors diagonalization. If you meant real numbers then you are right.
Decimals are how you represent numbers, not the numbers themselves.
By assigning a person to a decimal value and implying that every decimal has an assigned person the meme is essentially counting all the decimals. This is impossible, the decimals are an uncountable infinity. It’s like saying. Would you rather the number of people the trolley hits to be 7 or be dog.
What the meme has done is define the decimals to be a countable infinity bigger than another countable infinity. They’re both the same infinity.
Ah I see why they worded it the way they did. I would argue that’s just the limitation of the illustration, considering the text words the premise correctly, but fair!
But you actually can assign a unique person to every number, you just need an infinite number of people. You literally mathematically can’t do that for uncountable infinities.
Really? Isn’t the point that when you assign a natural number to every real number you can always generate a “new” real number you haven’t “counted” yet, meaning the set of real numbers is larger which is also is the point of the image.
No, thats not what I mean and that’s not the case. Even though there are infinite natural numbers, you can count them all. More accurately you can define a process that eventually will count them all. This is entirely different from decimal numbers which there is no process you can define that will exhaust all decimals. In this way the decimals are uncountable.
When talking about infinities this makes the infinity that contains all decimals larger than the infinity that contains only whole numbers.
My disagreement with the meme is that assigning an individual to each decimal is essentially a process of counting and this is a fundamental contradiction. As such the comparison to the set of natural numbers is nonsensical and the implication that there are less people assigned to the smaller infinity is incoherent.
actually you can show that the naturals, integers and rationals all have the the same size.
for example, to show that there are as many naturals as integers (which you do by making a 1-to-1 mapping (more specifically a bijection, i.e. every natural maps to a unique integer and every integer maps to a unique natural) between them), you can say that every natural, n, maps to (n+1)/2 if it is odd and -n/2 if it is even. so 0 and 1 map to themselves, 2 maps to -1, 3 maps to 2, 4 maps to -2, and so on. this maps every natural number to an integer, and vice-versa. therefore, the cardinality (size) of the naturals and the integers are the same.
you can do something similar for the rationals (if you want to try your hand at proving this yourself, it can be made a lot easier by noting that if you can find a function that maps every natural to a unique rational (an injection), and another function that maps every rational to a unique natural, you can use those construct a bijection between the naturals and rationals. this is called the schröder-bernstein theorem).
it turns out that you cannot do this kind of mapping between the naturals (or any other set of that cardinality) and the reals. i won’t recite it here, but cantor’s diagonal argument is a quite elegant proof of this fact.
now, this raises a question: is there anything between the naturals (and friends) and the reals? it turns out that we don’t actually know. this is called the continuum hypothesis
You can’t count the decimals, op is counting the decimals and insisting that they are more of those counted decimals than in the integers. This inherently doesn’t make sense and is improper use of what infinities are and what they can represent.
You’ve misunderstood “some infinities are bigger than others.” Both of these infinities are the same size. You can show this since each person on the bottom track can be assigned a person from the top track at 1 to 1 ratio. An example of infinities that are different sizes are all whole numbers and all decimal numbers. You cannot assign a whole number to every decimal number.
Matt parker does a good video on this. I can’t remember the exact title but if you search “is infinite $20 notes worth more than infinite $1 notes” you should find it.
The bottom rail represents the real numbers, which are “every decimal number”.
No it’s doesn’t because the bottom rail is a countable infinity, the decimals are an uncountable infinity. Go watch the video it explains it.
I know it is drawn countably infinite, but it represents the reals, also wise guy, i studied basic logic in university, it covers this topic.
By assigning people to it it becomes countable. You can’t assign uncountable numbers like that. It’s both in the text of the meme and in the illustration
i don’t know why you’re trying so hard to away explain why the meme doesn’t work, but sure.
the real numbers on the bottom rail include all decimals, no?
not sure what you mean by this, if you mean fractions you are wrong. Rational numbers and natural numbers can have a 1 on 1 assignment, look up cantors diagonalization. If you meant real numbers then you are right.
Decimals are how you represent numbers, not the numbers themselves.
I’m not talking about fractions, I’m talking about the reals because that it what op referred to
There are more reals than naturals, they do not match up 1 to 1, for exactly the reason you mentioned. Maybe you misread the meme?
By assigning a person to a decimal value and implying that every decimal has an assigned person the meme is essentially counting all the decimals. This is impossible, the decimals are an uncountable infinity. It’s like saying. Would you rather the number of people the trolley hits to be 7 or be dog.
What the meme has done is define the decimals to be a countable infinity bigger than another countable infinity. They’re both the same infinity.
Yeah, but if you can line up the elements of a set as shown in the bottom track, then they’re, at most, aleph 0.
Ah I see why they worded it the way they did. I would argue that’s just the limitation of the illustration, considering the text words the premise correctly, but fair!
One person for every decimal isn’t possible even with infinite people. That is the point I’m making.
Neither is assigning a person to every natural number, so I’m not sure what point you’re trying to make?
But you actually can assign a unique person to every number, you just need an infinite number of people. You literally mathematically can’t do that for uncountable infinities.
Really? Isn’t the point that when you assign a natural number to every real number you can always generate a “new” real number you haven’t “counted” yet, meaning the set of real numbers is larger which is also is the point of the image.
No, thats not what I mean and that’s not the case. Even though there are infinite natural numbers, you can count them all. More accurately you can define a process that eventually will count them all. This is entirely different from decimal numbers which there is no process you can define that will exhaust all decimals. In this way the decimals are uncountable.
When talking about infinities this makes the infinity that contains all decimals larger than the infinity that contains only whole numbers.
My disagreement with the meme is that assigning an individual to each decimal is essentially a process of counting and this is a fundamental contradiction. As such the comparison to the set of natural numbers is nonsensical and the implication that there are less people assigned to the smaller infinity is incoherent.
I don’t think we should take the visuals of the hypothetical shit post literally.
If they say there’s one guy for every real number, let them
It’s just an illustration… how else would you draw it?
My argument applies to the text of the post too. You literally can’t assign people (count) the decimals.
Diagonally
Natural numbers < whole numbers < rational numbers < real numbers
Okay, to clarify, I mean the “is partial set of” instead of “is smaller than”.
Your saying it would be correct for “whole numbers” and “decimal numbers”. But that’s exactly what OP said “natural” and “real”
actually you can show that the naturals, integers and rationals all have the the same size.
for example, to show that there are as many naturals as integers (which you do by making a 1-to-1 mapping (more specifically a bijection, i.e. every natural maps to a unique integer and every integer maps to a unique natural) between them), you can say that every natural, n, maps to (n+1)/2 if it is odd and -n/2 if it is even. so 0 and 1 map to themselves, 2 maps to -1, 3 maps to 2, 4 maps to -2, and so on. this maps every natural number to an integer, and vice-versa. therefore, the cardinality (size) of the naturals and the integers are the same.
you can do something similar for the rationals (if you want to try your hand at proving this yourself, it can be made a lot easier by noting that if you can find a function that maps every natural to a unique rational (an injection), and another function that maps every rational to a unique natural, you can use those construct a bijection between the naturals and rationals. this is called the schröder-bernstein theorem).
it turns out that you cannot do this kind of mapping between the naturals (or any other set of that cardinality) and the reals. i won’t recite it here, but cantor’s diagonal argument is a quite elegant proof of this fact.
now, this raises a question: is there anything between the naturals (and friends) and the reals? it turns out that we don’t actually know. this is called the continuum hypothesis
I clarified my above comment
You can’t count the decimals, op is counting the decimals and insisting that they are more of those counted decimals than in the integers. This inherently doesn’t make sense and is improper use of what infinities are and what they can represent.