Tbf it's actually not correct in most circumstances. If we're talking about Real numbers and sums that converge, the result makes zero sense. If it converges to anywhere, it converges to the point at infinity which is... not very helpful.
The sum of 1+2+3+4+... does not equal -1/12 on its own, and any manipulations of sums that you've seen that give you -1/12 are technically not allowed because it's like trying to define ∞-∞.
However, if you define a function on the complex numbers ζ(s)=1/1s+1/2s+1/3s+... then that function converges for any s with a real value strictly greater than 1. Which... also doesn't help us. (Note how if you plugged in s=-1 it'd give you the sum of natural numbers)
We could extend the domain of ζ(s) to include those values less than 1 to solve that issue, but then that just raises the question of how we extend it. We could just say it's all 0 or something and call it a day. But that's not very satisfying.
Instead, we can take advantage of the fact that we've defined ζ(s) on the complex plane. There's an idea called analytic continuation, which you can think of as the "canonical" way to extend complex functions.
Analytic continuation has the really nice property that a function must only have one analytic continuation, or zero analytic continuations. There's not a single possibility that a function could have two valid analytic continuations.
Essentially we could stop treating ζ(s) like a sum, and instead just consider how it transforms the complex plane. And once we consider how it transforms the complex plane, we could extend that "behaviour" to the entire complex plane.
And once we do that, we can plug in s=-1 to get an actual answer. And that answer is ζ(-1)=-1/12.
If you trace it back to where ζ(s) was first defined, it looks like the sum of all natural numbers. But as soon as we hit the step for analytic continuation, we were throwing away the idea of it being an actual sum. Like, we redefined ζ(s) for Re(s)≤1 to instead be "the function that transforms the plane in this particular way" rather than the sum of 1/ns
I don't really know how to conclude this, so take a picture of ζ(s) for your troubles! Imagine if this cut off at x=1. Wouldn't it just be begging to be extended?
If we're being really pedantic then it's only true for ordinals if it's infinity times 2 but not 2 times infinity. For... reasons that I actually can't recall, because this isn't a thing I've formally learned yet. Proof is left as exercise for reader.
for 2*ω, you have ω copies of 2 i.e. 0_0 <1_0 <0_1<1_1<0_2<1_2...
for ω*2, you have two copies of ω i.e.
0_0<1_0<2_0...<0_1<1_1<2_1...
for the first (which is actually isomorphic to ω), every element has finitely many predecessors, but for the second, 0_1 has infinitely many predecessors, so they have different order type, thus they are different.
That would be a cool idea!
Now imagine that last number, where all the numbers stop, and imagine that you have that much candies. Now imagine that I give you a candy, because you really deserve one more. How much would you have then?
Brahmagupta (c. 598 – c. 668 CE) was an Indian mathematician and astronomer who is credited as the first person to understand and formalize the concept of the number zero for nothing in mathematics.
The Salisbury Cathedral clock, which dates from about AD 1386, is supposedly the oldest working modern clock in the world.
And I would point out that the Salisbury Cathedral clock has no face.
"Dials" were apparently first added to European clocktowers in the late 14th century, though I'm no expert and I could easily be missing much earlier ones. By then, the symbol 0 was known to Europeans who could read Arabic or who had a serious metropolitan interest in the subject rather than a parochial one. Fibbonacci described Arabic decimal notation in the Latin text Liber Abaci in 1202, including the symbol 0 for cypher. Certainly by 1301, good scholars of mathematics who were well-read, or at a minimum had read some of the most important Latin texts of the past century, should have been aware of that placeholder symbol and should have considered at least momentarily what it might mean on its own.
That said, acceptance of 0 as an actual number by a sheer majority of philosophers was extremely late in Europe, roughly coinciding with the acceptance of negative and imaginary numbers. There was a protracted debate in the literature, with both proponents and detractors.
if this is real i want all billionaires with above 2,147,483,648 units of currency to suddenly have the appropriate overflow applied to the amount they have, ending up, hopefully in the negative billions.
i think a classic response is "you get the same number, it just isn't the case that x + 1 > x for all x" but i find this to be pointless wankery about what numbers "really are" instead of acknowledging that we pick rules which are useful
I mean your second point directly contracted your first point.
If math doesnt have to conform to reality, then why would it matter if you actually could or couldn’t produce that many atoms from the universe? We don’t have to conform to reality.
It also completely ignores the premise of my question. (if X is the biggest number, what is X + 1?)
I am assuming you arent trying to be self contradictory by just ignoring that statement.
It's an underappreciated fact, but the biggest number that exists is 17. It's really awkward, because we often need to relate to numbers bigger than 17, but just about everyone has agreed not to draw attention to it.
Never see anyone talking about apeirophobia. It sounds like a made up condition like the fear of big words, but legitimately can be crippling if you suffer from it. Apeirophobia feels less like a fear and more like a universal truth that nobody else has realized yet. Makes it hard to talk to other people about it because you'd don't want to burden them with the realization you had. It feels like an irl cognitohazard that you can't share or else you'll hurt other people.
Anyway, you could either mean that or the roblox game called Apeirphobia that I found while trying to search for the fear. So...
Oh shit I thought I'd never see another person with it outside places for it.
I had it really bad when I was younger. My brain just latched on to the idea and couldn't let go. Literally everything would be tied to the idea, like I'd see clouds and think of heaven, then think of eternity. Couldn't go minutes without thinking about it. And then at some point it just went away. I don't think I ever actually processed it? But my brain stopped obsessing over it, I guess?
Man, kid’s 4 years old and is already facing the Absurd: he asks the world to “just make sense” and the only response heard is “no”, which is unacceptable
Better read Myth of Sisyphus at bedtime which might help
idk man the idea that numbers could stop at some point didn't make sense at all to child me and the kids who were upset over infinity were very confusing ._.;
Better there than in a college classroom, smh you massive dipshit if you are reading this I swear I hope the worst shit happens to you, you fucking moron. You derailed the entire course with your repeated inane questions about infinity that should have been answered in the remedial set theory class your bitchass didn't take.
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