I think a fairly typical “intuitive” notion is something like:
Pick a space Rn that contains the sets you want to compare (let’s call them A and B). Then consider balls of radius r growing from the origin. There are four possibilities:
There’s as much A as B for almost every r (e.g. comparing positive numbers to negative numbers).
There’s an infinite extent of r for which there’s more A than B in the ball, and also an infinite extent of r for which there’s more B than A (e.g. comparing alternating pairs (0,3,4,7,8,11...) to (1,2,5,6,9,10...))
There’s an infinite extent of more A but not vice versa.
There’s an infinite extent of more B but not vice versa.
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 r”? 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 r on A to the extent of r 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 r=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.
I think a fairly typical “intuitive” notion is something like:
Pick a space Rn that contains the sets you want to compare (let’s call them A and B). Then consider balls of radius r growing from the origin. There are four possibilities:
There’s as much A as B for almost every r (e.g. comparing positive numbers to negative numbers).
There’s an infinite extent of r for which there’s more A than B in the ball, and also an infinite extent of r for which there’s more B than A (e.g. comparing alternating pairs (0,3,4,7,8,11...) to (1,2,5,6,9,10...))
There’s an infinite extent of more A but not vice versa.
There’s an infinite extent of more B but not vice versa.
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 r”? 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 r on A to the extent of r 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 r=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?