This is a linkpost for https://medium.com/@london-lowmanstone/comparing-infinities-e4a3d66c2b07
I think a fairly typical "intuitive" notion is something like:
Pick a space that contains the sets you want to compare (let's call them A and B). Then consider balls of radius growing from the origin. There are four possibilities:
Hmm. My intuition says that your A and B are "pretty much the same size". Sure, there are infinitely many times that they switch places, but they do so about as regularly as possible and they're always close.
If A is "numbers with an odd number of digits" and B is "numbers with an even number of digits" that intuition starts to break down, though. Not only do they switch places infinitely often, but the extent to which one exceeds the other is unbounded. Calling A and B "pretty much the same size" starts to seem untenable; it feels more like "the concept of being bigger or smaller or the same size doesn't properly apply to the pair of A and B". (Even though A and B are well defined, not THAT hard to imagine, and mathematicians will still say they're the same size!)
If A is "numbers whose number of digits is a multiple of 10", and B is all the other (positive whole) numbers, then... I start to intuitively feel like B is bigger again??? I think this is probably just my intuition not being able to pay attention to all the parts of the question at the same time, and thus substituting "are there more multiples of 10 or non-multiples", which then works the way you said.
I like this a lot! I'm curious, though, in your head, what are you doing when you're considering an "infinite extent of "? My guess is that you're actually doing something like the "markers" idea (though I could be wrong), where you're inherently matching the extent of on A to the extent of on B for smaller-than-infinity numbers, and then generalizing those results.
For example, when thinking through your example of alternating pairs, I'm checking to see when =3, that's basically containing the 2 and everything lower, so I mark 3 and 2 as being the same, and then I do the density calculation. Matching 3 to 2 and then 7 to 6, I see that each set always has 2 elements in each section, so I conclude that they have an equal number of elements.
Does this "matching" idea make sense? Do you think it's what you do? If not, what are your mental images or concepts like when trying to understand what happens at the "infinite extent"? (I imagine you're not immediately drawing conclusions from imagining the infinite case, and are instead building up something like a sequence limit or pattern identification among lower values, but I could be wrong.)
I'm not really imagining matching. I'm imagining the scope of points that I'm looking at sweeping outwards, and having different sides "win" by having more points in-scope as a function of time.
But I think if you prompt someone to imagine matching, you can easily pump intuition for sets being the same size if they alternate which is more dense infinitely many times.
So then because the winner alternates at an even rate between the two sets, you can intuitionally guess that they are equal?
Consider the following sets:
Would it mean that C is smaller than B, but equal to A (which is equal to B)?
If the color of the number is considered to be an intrinsic property of the number, then under the Bruce Framework, yes, |C|<|B| and |C|=|A| and |B|=|A|.
I'm posting this here because I find that I don't get the feedback or discussion that I want in order to improve my ideas on Medium. So I hope that people leave comments here so we can discuss this further.
Personally, I've come across two other models of how humans intuitively compare infinities.
One of them is that humans use a notion of "density". For example, positive multiples of three (3, 6, 9, 12, etc.) seem like a smaller set than all positive numbers (1, 2, 3, etc.). You could use the Bruce Framework here, but I think that what we're actually doing something closer to evaluating the density of the sets. We notice that 3 and 6 and 9 are in both sets (similar to the Bruce Framework), but then we look to see how many numbers are between those "markers". In the positive numbers set, there are 3 numbers between each "marker" (3, 4, 5 and then 6, 7, 8), whereas in the set of positive multiples of three there is only 1 number between each "marker" (3 and then we immediately go to 6). Thus, the cardinality of the positive numbers must be 3 times bigger than the cardinality of positive multiples of three.
If you expand this sort of thinking further, you get to a more "meta-model" of how humans intuitively compare sets, which is that we seem to build simple and easy functions to map items in one set to items in the other set. Sometimes the simple function is "are these inherently equal", as in the Bruce Framework. Other times it's "obvious" function like converting a negative number to a positive number. Once we have this mapping of "markers", we then use density to compare the sizes of the two sets.
I'm not 100% sure if density is the only intuitive metric we use, but from the toy examples in my head it is. What are your thoughts? Are there any infinite sets (numbers or objects or anything) where your intuitive calculation doesn't involve pairing up markers between the sets and then evaluating the density between those markers?
People in general are pretty good at understanding 1 to 1 matching for finite sets, and have to readjust their intuition for infinite sets. Thus first thing to help them break their intuition is going through the logic of https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel After someone realizes the universe is not what they expected it to be, you help them build a better intuition. If you are lucky, you get to the Cantor's diagonalization argument. Mind you, the argument requires a certain level of intelligence, it is just completely beyond some people.
Yep, absolutely! It was actually through explaining Hilbert's Hotel that Bruce helped me come up with the Bruce Framework.
I do think it is odd though that the mathematical notion of cardinality doesn't solve the Thanos Problem, and I'm worried that AI systems that understand math practically well but not theoretically well will consider the loss of half an infinite set to be no loss at all, similar to how if you understand Hilbert's you'll believe that adding twice the number of hotels is never an issue.
Mathematicians have come up with a way of comparing infinities that doesn’t make sense to most people. I want to help.
So, I’ve written this to do the following:
The reason why I think this work is important is because computers tend to be based in mathematics. If there’s an artificial intelligence (AI) that is talking to humans about infinity, I want to make sure that you know how the AI is thinking. Also, if an AI (or a mathematician) happens to read this article, I want them to be able to understand how regular people think about infinity.
If you’re new to comparing infinities, I’d recommend reading this whole article. If you’re a mathematician or an AI, I’d recommend skipping to “The Bruce Framework” section.
How You Compare Infinities
Which is bigger, the amount of all positive whole numbers (1, 2, 3, 4, 5, …) or the amount of all positive even numbers (2, 4, 6, 8, …)?
If you’re not a mathematician, my guess is that you think the amount of all positive whole numbers (1, 2, 3, 4, 5, …) is bigger than just the evens (2, 4, 6, 8, …).
Why is this? Well, it’s pretty obvious. If you were to take away all of the evens from all of the positive whole numbers, you’d have some left over, namely, the positive odd numbers (1, 3, 5, 7, …). So if one set of things is literally contained within another set, and there are elements left over, it just makes sense to say the other set is bigger.
How Mathematicians Compare Infinities
Which is bigger, the amount of all positive whole numbers (1, 2, 3, 4, 5, …) or the amount of all positive even numbers (2, 4, 6, 8, …)?
If you’re a mathematician, you’ll say that they’re both the same size! What?!
Mathematicians really care about having clear rules about how they define things. So, they came up with rules for how to determine whether two sets of things have the same size.
Let’s call the first set A and the second set B. Each thing inside of a set we call an “element” of that set. (So for a set of numbers, each number in the set is an element of that set.)
Mathematicians say that A is smaller than or equal to B when you can take every element in A and assign it a partner element in B, such that no element in A has the same partner. There may be some elements in B left over — that’s okay! As long as every element in A is assigned a partner in B, and no two elements of A have the same partner in B, then mathematicians say that the size (technically, the “cardinality”) of A is smaller than or equal to B.
So let’s try this out with our example. Let’s compare the positive even numbers (2, 4, 6, 8, …) to the positive whole numbers (1, 2, 3, 4, 5, …). As long as we can come up with a rule to assign each even number to a whole number, then we’ll know that the cardinality of the even numbers is less than or equal to the cardinality of the whole numbers.
So, let’s make a really simple rule: I’m going to assign every even number to the whole number with the same value, because every even number is a whole number. That is, the 2 from the even numbers is going to be paired with the 2 from whole numbers. And the 4 from the even numbers is going to be paired with the 4 from the whole numbers. This pairing is valid because no two even numbers have the same whole number partner. Thus, there are a less than or equal “amount” of even numbers as there are whole numbers.
You can even come up with really weird pairings to show this. For example, we could pair all even numbers under 1001 to the whole number with the same value, and then pair all even numbers above 1001 to the whole number whose value is one higher than themselves. So this would mean that 104 the even number would be paired with 104 the whole number, but 1006 the even number would be paired with 1007 the whole number. Since numbers below 1001 are paired with even numbers, and numbers above 1001 are paired with odd numbers, no even number is paired to the same partner. Thus, this pairing also shows that the cardinality of the positive even numbers (2, 4, 6, 8, …) is less than or equal to the cardinality of the positive whole numbers.
But what about comparing in the reverse direction? What if we tried to find partners for the whole numbers (1, 2, 3, 4, 5, …)? Well, there’s pretty simple rule for pairing: assign each whole number to an even number that has double the value of the whole number! That means that 1 the whole number is assigned to 2 the even number. And that 7 the whole number is assigned to 14 the even number. This is another valid pairing, because no whole number has the same partner. Thus, the “amount” of whole numbers is less than or equal the the “amount” of even numbers.
This seems really weird, but I think it can help to think about mathematicians just coming up with definitions that work for them. To mathematicians, the goal is to see if we can assign elements in the sets to each other. It’s not “counting” or figuring out the “amount” or “size” of something in the way we regularly do. It’s new way of comparing sets. That’s why mathematicians call it cardinality, and not “size”. It’s a different concept.
Now, you may have already realized this, but since we’ve mathematically proven that the cardinality of whole numbers (1, 2, 3, 4, 5, …) is less than or equal to the cardinality of even numbers (2, 4, 6, 8, …), and we’ve also proven that the cardinality of even numbers (2, 4, 6, 8, …) is less than or equal to the cardinality of whole numbers (1, 2, 3, 4, 5, …), it follows that the cardinality of the two sets are equal!
And this is what mathematicians mean when they say things like “there are just as many even numbers as there are whole numbers”. They don’t mean that there are actually just as many numbers. What they mean is that you can pair up the numbers in both directions. For every whole number, you can find a unique even number to be its partner, and for every even number, you can find a unique whole number to be its partner. Mathematicians often don’t want to take the time to explain exactly what they mean when they say stuff to a general audience; plus it sounds cool and mysterious to say “there are just as many even numbers as there are whole numbers”, so that’s often what they do. Even mathematicians want to seem cool.
The Bruce Framework
What is the Bruce Framework?
I was talking with my younger brother Bruce, and throughout our conversations, we were able to come up with a basic mathematical framework for how non-mathematicians compare infinites.
Remember how what bothered us for the even number and whole number case was that the whole numbers contained the even numbers? Well, the idea behind the Bruce Framework is to make sure that when you do your pairing, you at least match elements with the same value to themselves. That is, if 2 is an even number and a whole number, no matter which direction you’re trying to find partners, 2 should always be paired with 2.
Mathematically, the way to define this is as follows. When comparing the sets A and B, define A*=A-B and B*=B-A. (Subtracting one set from another set means to remove all the elements that are in the one set from the other set.) Then, |A|≤|B| if and only if |A*|≤|B*|. (Note that “|X|” means “the cardinality of X”.)
Mathematically, this does exactly what we described above. By subtracting the sets from one another, we make sure that elements that are the same in both sets are just removed from the mathematical pairing process, because they can immediately be matched to themselves. Then, with the elements we’re left with, we can use the mathematical definition of cardinality.
I like to think of the Bruce Framework as solving what I call the “Thanos Problem”. In the Avengers series (don’t worry, I don’t spoil anything) there is a character called Thanos, who believes that the world will be better if half the people were eliminated, because overpopulation makes the lives of those who are living worse.
Imagine that the number of people in the universe is infinite, in that you can travel in any direction and eventually you will find another person. However, the number of people is countably infinite in that you can assign a (whole) number to everyone.
According to mathematicians, if Thanos got his wish, and half of the people in the universe were eliminated, the “amount” of people in the universe would be exactly the same. (Mathematically, we would say “the cardinality of the set of people in the universe would remain the same”.)
Why? Because this is exactly the same as the whole number and even number case! Imagine we assign everyone a number, and then eliminate all of the people who have an odd number. What we’re left with is the people with an even number. And, according to mathematicians, the cardinality of the set containing all of the people is equal to the cardinality of the set containing all of the people with even numbers. (Because we can match each person to an even person by doubling their number, and that matching works in both directions.)
The Bruce Framework resolves this by forcing the mathematic definition of cardinality to match each person to themselves. Let A be the set of all people, and B be the set of all people assigned even numbers. Then, under the Bruce Framework, the set of all people has an equal cardinality to the set of people with even numbers if and only if the set of people who were assigned odd numbers has an equal cardinality to the empty set. Basically, we’ve already matched all the even people to themselves, and now we need to match the odd people to… no one.
According to the mathematician definition of cardinality, you can’t match the odd people to no one (the empty set), and so we get the result that the cardinality of B is less than or equal to A, and that the cardinality of A is not less than or equal to B. Thus, the cardinality of A (the set of all the people who survived the elimination) is less than the cardinality of B (the set of all people who existed before half the people were eliminated). So, under the Bruce Framework, if Thanos eliminates half the people in the universe, he is considered, to have reduced the “amount” of people in existence.
Overall, the Bruce Framework works to mathematically determine how non-mathematicians evaluate infinity in particular cases. Having an AI use this notion of infinity would at least stop the AI from believing that there are just as many people after destroying half the universe as there were before.
What’s Wrong with the Bruce Framework?
So, if the Bruce Framework does a better job at predicting how humans tend to evaluate infinities, why isn’t it what mathematicians have adopted?
The issue is that the Bruce Framework isn’t transitive. If it were transitive, this would mean that if |A|>|B| and |B|=|C|, then it would follow that |A|>|C|.
However, the Bruce Framework is not transitive. This means that when using the Bruce Framework, you can have two sets that are equal in size, but a third set that’s bigger than only one of those two (and not the other). Isn’t that weird?
Let’s run through an example. Let’s compare the sizes of the positive whole numbers (1, 2, 3, 4, 5, …), the positive even numbers (2, 4, 6, 8, …), and the negative whole numbers (-1, -2, -3, …). The set of positive whole numbers will be A, the positive evens will be B, and the negative whole numbers will be C.
We just went through the Thanos Problem under the Bruce Framework, which showed that |A|>|B|. That is, the number of positive whole numbers is greater than the number of evens, under the Bruce Framework.
Next, let’s compare the positive whole numbers to the negative whole numbers under the Bruce Framework. Since there are no elements in common between positive numbers and negative numbers, evaluating the cardinality of the sets under the Bruce Framework is exactly the same as the regular mathematical framework. Using the regular definition of cardinality, the cardinality of the two sets is equal because we can just pair each number to the the negative of itself. That is, 1 is paired with -1, and -1 is paired back with 1. The same goes for 2 and -2, 3 and -3, etc. Thus, we have unique pairings for every number going both directions, so the cardinality of the two sets is equal.
Finally, let’s compare the negative whole numbers to the positive even numbers. Now, my guess is that intuitively, you think there are more negative whole numbers than there are positive even numbers. But this is where the Bruce Framework breaks down. Since there are no elements in common, once again, we resort to the regular mathematical definition. Here, we can map each negative number to -2 multiplied by itself, and that mapping works in both directions. For example, -1 is paired to 2, and 2 is paired back with -1. The same goes for -2 and 4, -3 and 6, etc. Thus, we have unique pairings for every number going both directions, so the cardinality of the two sets is equal.
This means that under the Bruce Framework, we have a situation where the cardinality of the set of positive evens is equal to the cardinality of the set of negative whole numbers, but the set of positive whole numbers is only bigger than one of them (and not the other)! In our case, |A|>|B| and |B|=|C|, but |A|=|C|!
This lack of transitivity is the main reason why mathematicians don’t adopt other methods of comparing the “size” of infinite sets. There are other frameworks I have developed (with the help of others) that improve on the Bruce Framework in terms of making predictions of how people compare sets (remember that |B|=|C| seemed like a bad result), but none of them are transitive.
And this is why I’m not arguing that these frameworks should replace the usual notion of cardinality. I think that losing transitivity would be too much for mathematicians to swallow. But I do think that the current method of measuring the cardinality of sets is also too much for non-mathematicians to swallow as well. So there’s a tradeoff between the two approaches, which I hope this piece has illuminated.
Summary
Overall, mathematicians use an unintuitive way of measuring the cardinality (or “size”) of sets that have infinite amounts of things inside them. The Bruce Framework helps reduce some of the weirdness of the usual approach by requiring that when matching elements between sets that have common elements, an element always matches to itself. However, the Bruce Framework has the unintuitive result of making the cardinality of a set non-transitive, while the usual mathematical framework preserves the expected transitivity.