It seems to me that there is a bit of bait and switch going on here. When you say, 1+2+3+4+5… it implies to me that the series under consideration is the set of all positive integers from 1 on into ∞, and does not include any negative numbers nor any fractions. When the gentlemen in the video started breaking up the set into subsets via the brackets, and adding in negative numbers he changes the car actor on the set initially under consideration into an entirely different set of integers.
Then when he started mixing the 3 different sets of (I did study set theory, back in the day) it reminded me of when doing differential equations and had to solve three equations with three unknowns, by substitution parts of one equation into another as you worked to the solution. But here you have sets not equations, and I am not convinced that 1+2+3+4+5…= – 1/12.
I recall an algebraic proof going around back then that proved 1=2, however, if you substituted numbers for the letters as the proof was worked out you would see that at one time a number had to be divided by zero to reach the results of 1=2. In the part where he talks of Thomson’s Lamp I was reminded of one of Zeno’s paradoxes, “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” as recounted by Aristotle, Physics VI:9, 239b10. Suppose you want to catch a stationary bus. Before you can get there, you must get halfway there. Before you can get halfway there, you must get a quarter of the way there. Before traveling a quarter, you must travel one-eighth; before an eighth, one-sixteenth; and so on.
When I Got my BS in Electronics I took a job right out of school with General Electric in New Berlins, WI, working on integrating all the different components from the various vendors into a working Cat Sacnner (as they were called then). I also soon there after enrolled into Marquitee Universite’s Graduate program in Electrical Engineering. That turned out to be a big mistake as GE started working me 60-70 hours a week to meet the demand for their whole body scanner.
I bring this up to tell you about a homework assignment I got in one of my math classes, prove that between 1 and 2 there is an infinite number of positive numbers. Now intuitively I knew that this was true, but I never, back then, could figure out how to express it. I had to drop out of that program to meet my obligations to GE and my growing family. Bty there were 10 students in that class and if just one of us understood what the professor was lecturing on he would not go over it.
However, last night after reading this post I went to bed and could not sleep for thinking of that math problem from so long ago, and this is what I came up with:
Do you think I solved it?