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The other day I came across a weird mathematical statement, the Ramanujan sum that states that -1/12 = 1+2+3+4+5...
I can show the proof if anyone wants. I managed to figure out the problem which goes something like this
If you have a series S=1+2+3+4+5...+n and another S1=4+8+12+16...+n which can be written as S1=4(1+2+3+4+5...+n) is it true that 4S=S1?
Well, it is true for every possible value of n that you can try with. On the other hand, if S is infinite, does it make sense to think of S1 as a slightly larger infinity? I think not but if you buy the idea that 4S=S1 then you also have to buy the idea that -1/12=1+2+3+4+5.. as they go hand in hand as far as I can see.It really comes down to how you interpret the meaning of the equal sign. This is also what you get if you feed -1 to the Riemann-Zeta function.
And it made me wonder if there is a twilight zone as it were between truth and paradox i.e, a statement for which the truthfulness can be questioned and for which one may think it's a paradox. It sounds very weird to me so I hope someone can figure this one out. The only thing I could find is that there's a name for statements where the negation of the statement is true as well. Memory doesn't serve me right now so I don't recall the name. If there is such a weird thing, what would you call it?
BTW. My first question is if there's a weird twilight zone between numbers and infinity. These are different concepts. A number, any number, has a position on the number axis. Infinity doesn't, all you can say is that number axis points at infinity. If there is such a place, it would have very weird problems indeed. As far as I can see, if you were to add 1 to this number, you wouldn't be able to tell whether it made any difference. The point where our understanding of algebra would fail. It would be the point where you could entertain the idea that infinity divided by this number would make some kind of sense. Normally, infinity divide by any number is 0 but in this case it would be more like +0.
I hope someone can make sense of these ideas that I got.
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The standard answer would be no. Two infinite sets are regarded as having a size which is the same order of infinity if their members can be placed in one-to-one correspondence with each other. This is true of your two sets, so they are of the same order of infinity. For every member of the set of all integers there is a corresponding member of the set of all multiples of 4.On the other hand, if S is infinite, does it make sense to think of S1 as a slightly larger infinity?
A higher level of infinity would be one in which for every member of the lower set there are an infinite number of members of the higher set.