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 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?