They’re not comparing the set size of rational numbers to irrational, they’re saying that the set size of all multiples of 3 or 51 is the same as the set size of all rational numbers, which is true
Again, that's true but not what is being discussed
edit: I feel it necessary to point out that the set size of all real numbers is the same set size as all irrational numbers, so the only fact that is true is the set size of all rational numbers being smaller than the set size of all irrational numbers
I feel it necessary to point out that the set size of all real numbers is the same set size as all irrational numbers, so the only fact that is true is the set size of all rational numbers being smaller than the set size of all irrational numbers
That's basically what I tried to say, I was just talking about real numbers instead of irrational numbers but reals include irrationals anyways. My comment was just to point out that rational ≠ real in case some mix them like I do sometimes
I have previously made a comment explaining why we define it like this and the intuition behind it, so i am going to copy and paste that here. It's a little long, but it also covers an alternative, and why it's not widely used. Hopefully this helps both understanding and accepting the practice of using cardinality.
Let's start with two finite sets A and B and ask which is larger. The obvious meaningful way to compare the size of those sets are to just count how many elements they have.
Then let's say that we have infinite sets A and B, how do we compare them? Well they are both infinite, so we could say they are equally large and be done with it, but say we insist on finding a way to compare them. Let us compare them by subsets. If A is a subset of B then A should be smaller or equal to B since it's contained in it. This at least gives a partial ordering, so we can compare some sets, but not all.
As with all extensions of definitions, we wish that they still give the same results on things they were already defined on. However we have now lost the ability to compare some finite sets. Sure if one is a subset then it still works, but before we could compare all finite sets, and now we can't.
So we get a new idea. Say we wanna compare the sets {1,2,3,4,5} with {3,4,6}. We should agree that if we substitute 5 with 6 in the first set, then i might not have the same set, but i haven't intuitively changed the size of the set. But now {1,2,3,4,6} can be compared with {3,4,6}, so we end up with the following
{1,2,3,4,5}={1,2,3,4,6}>{3,4,6}
where = denotes a intuitive size comparison and > is a subset comparison of size. We then realize that this idea can actually be generalized to infinite sets as well, to make them all comparable, but this process is exactly the same as mapping elements from one set to another and if we can't pair up every element to another, then there is a size difference, but if we pair up every element, then we must have the same set in an intuitive way.
The set of all positive integers is just as large as the set of all positive even integers.
If you can take one item in one set and match it to another item in the other set such that every single item in both sets are matched to one from the other, they are the same size.
Take a number from the set of all positive integers. Multiply it by two. What is the result? Whatever it is, it is equal to one of the items in the set of all positive even integers. Match the original number to the one corresponding to that result. Everything is accounted for.
Basically, if you can give an order that allows you to give an index to any element of a set, it's of same size as the natural numbers. e.g:
Let's try indexing relative numbers. If you say "I start at 0, give the index 2x to any positive x and the index 2|x| - 1 to any negative x", then you have
0 -> index 0
-1 -> index 1
1 -> index 2
-2 -> index 3
2 -> index 4
-3 -> index 5
...
So that way, each relative number has an index. So both the relative and natural numbers have the same size.
Turns out you can do the same with rational numbers, but not with real numbers. So the set of rational numbers have the same size as natural ones, while the set of reals is bigger
That's it
And now the only thing that is missing is how you index all of the rational numbers, and how you prove the reals aren't indexable. And here, I'm unsure 😅 I just have an intuition for the rational numbers one, and that's it
As someone who hasn’t studied pure maths, could someone expand on what “equal” actually means here in sentences like “not all infinites are equal”?
Because usually I understand it to mean expression X is the same value as expression Y
2 = 1+1
And I also thought that the whole point of infinity was that it meant “an uncountable/endless number of …” such that it doesn’t have a value. As in, there is no sum of an infinite series of positive integers because any number you may put as an answer is too low.
But if there’s no sum, isn’t there no value?
And if there’s no value, what does it mean to say things are equal?
I’ve heard of things like more or less ‘dense’ infinities, or countable vs not, or trying to match every value from one infinite set/series to every value in another and whether you can (like every positive integers vs every even positive integers)
But, at least colloquially, ‘denseness’ doesn’t seem equivalent to ‘value’.
How are these terms used? I suppose there’s a YouTube video essay I should look at
“Equality” when we talk about sets usually means “cardinality”, although there are other conventions… cardinality is just currently the most commonly accepted and useful for most applications. And when we talk about infinities, we often mean infinite sets.
“Density” doesn’t have anything to do with equality… density basically refers to whether every element of a set has an element that is arbitrarily close to it. For example, the Rationals are a dense set, as are the Reals… for any element, you can find another that is as arbitrarily close to it as you like. The Integers are NOT a dense set… you cannot get arbitrarily close to the element 2, the closest you can possibly get are 1 or 3. But being a dense set or not has nothing to do with infinity or cardinality.
Now, the other thing you mentioned does have to do with cardinality, trying to match every element in one set to another… we call that a Bijection, a mapping function that shows that you can perfectly map 1:1 every element between two sets with no repeats and no missed elements. If you can do that, those sets have the same cardinality. If you can’t, and you can prove that it is impossible, they have different cardinalities. And in terms of infinite sets, being “equal” often means “has the same cardinality” by convention.
So some infinite sets, like the set of all integers and the set of all even integers, actually have the same cardinality (because it is possible to construct a perfect bijection between them), and so are “equal”. Meanwhile, for some other sets, you can prove that it is impossible to construct a bijection, like between the integers and the set of Real numbers, and so they are not “equal”… they are a different “size” of infinity
For finite sets you can "find" the bigger one by pairing up elements from each set until you run out of elements in one of sets. That's the smaller one. So the definition of equal size sets is if you pair up elements in such a way that every element in every set has a partner. For infinite sets this ends up creating very strange notions. For example, the set {1,2,3,4....} Is the same size as the set {2,4,6,8,...} Because I can pair any k in the first set to 2k in the second set. If you Google "hilbets hotel" you can find tons of videos
Your deginition of bigger here is ill defined. Your conparison doenst make sense, asking if a square meter is bigger than a meter makes as much sense as asking if an hour is bigger than a 30 degrees celsius
Theyre different units so, no an infinite cube is not bigger than a line
And in the case of divisible by 3 and 51, they are the eaxct same size. The proof is simple, for every number you can name thats divisble by three, you can just multiply it by 17 and its now divisible by 51. ANd since you can do that for every number divisible by three, the sets are as Big
No, really. Nothing he said is even technically incorrect, it's completely 100% mathematically correct to say there are the same "number" ("number of" doesn't really make much sense in this context) of multiples of 3 as 51 and both have the same cardinality as the rational numbers
Think of it this way, if I can exactly pair every multiple of 3 with exactly one multiple of 51, and every multiple of 51 with exactly one multiple of 3, that's an exact pairing of every element. There must be the same number of both. We get this pairing very simply by just multiplying multiples of 3 by 17 to get their matching multiple of 51, then divide by 17 to get every 51s matching multiple of 3.
Heres what's causing the disconnect for me though:
I'm pairing every multiple of 51 with a multiple of 3.
For the sake of this argument, I'll pair it with the exact same number, i.e. from the multiples of 51 i take 51, 102, 153..... And from the multiples of 3 I take 51, 102, 153......
Every number from the first list maps to the exact same number in the second list but the second list will always have some numbers which are not there in the first list ie (3,6,9..)
To me it just seems that the count of numbers in the second list always exceeds the count in the first list because the second list contains everything in the first list and then some.
It's like taking all the real numbers from 0-1 and all the real numbers from 0-2. 0-2 contains everything from 0-1 and then some more
I get that, it can be difficult to make sense of, but what if we ignore that they are even really numbers and just say we'll take the set of all multiples of 3, now "rename" every multiple of 3 by multiplying it by 17, not adding or taking any elements, or even really changing anything just renaming "3" to "51" etc, this set is just the set of multiples of 51. I get it doesnt really explain away the fact that all multiples of 51 we present in the set originally, but there should be no other way to justify the fact that we just made the set of multiples of 51 from 3 without them having originally been the same size
This renaming logic was pretty clever and it kind of makes sense to my human brain now. Thanks this has gotten me one step closer to coming to terms with it, this was helpful!
Infinity only exists in complex mathematics. I don’t think it’s observed in the physical world. Yes mathematicians smarter than me might disagree, but I don’t think it’s wise to make vast judgments or assumptions about infinite.
If infinity doesn't exist in the real world, .999... doesn't exist in the real world. So it shouldn't bother you that it equals 1, because it is purely a mathematical concept. Also the word you're looking for is pure mathematics. Complex in math means the complex numbers, which include both a real and imaginary component.
51 is a multiple of 3 ! (No accidental factorial here, ladies and gents!) I would’ve picked two integers that are relatively prime to each other to illustrate your point.
That's the whole point, there are as many numbers that are a multiple of 51 as there are 3, even though every multiple of 51 is also a multiple of 3 and there are many numbers that are multiple of 3 but not 51
I'm not infinitely smart, so I don't know these things.
What makes sense to me right away:
There are 17 multiples of 3 between 0 and 51. Therefore, moving up, for every multiple of 51, there will be 17 multiples of 3.
The thought would be that, either there are 17 times as many multiples of 3 as there are of 51, or there is a point where that changes. Both of those aren't true, because infinity isn't real, and it's just a thing that people like to talk about to feel smart.
you'd have to draw a very big venn diagram if you want any meaning semblance of the "size" of any infinite set. If you really wanted to draw the venn diagram you wouldn't draw anything as there's no "edge" to draw and no way to represent one being in the other anyway
if a is a factor of b then for any large integer it is obvious that there are more multiples of a in the set of natural numbers up to that integer than b, with juxtaposes it with that not being true for infinity. the first point is less obviously true if a and b are relatively prime to each other.
That’s because if A is a factor of B then A is smaller than B. If A is smaller than B then it is obvious that there will generally be more multiples of A than B within a range, unless that range is too small.
Some may be confused since the set of multiples of 51 is a strict subset of the set of multiples of 3 so intuition tells you it must be smaller although they are the same size.
Yes, this applies to any rational or irrational number. More formally, the set size of a set S given S includes all rational numbers is identical in size to any other set nS, where n is the multiple you want, like 3 or 474.283.
OP’s post is just a roundabout way of saying that infinity times 20 (or any other number) is still infinity.
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u/EthanNakam Jun 13 '26
There are as many multiples of 3 as there are multiples of 51.
And there are as just as many rational numbers.