Infinite speed revision

[GarrixianXD]

Uncountable and higher-order infinities are discussed a lot when it comes to tiering and the attacking potency of characters, but not exactly in terms of speed.

As far as I know, moving an uncountably infinite distance or any distance beyond it but still having your movement bound by linear time, and is still considered infinite speed since it doesn't necessarily qualify for the criteria for immeasurable speed.

Now, the issues rather lie in that, since we cramped all sets and orders of infinities, into a single rating, which pretty much ignores the vast differences between countable and uncountable infinities and the significance of it.

To start, let's take an example of any character who can move at a finite speed. That character has all types of self-sustaining abilities, type 1 immortality and infinite stamina. If this character moves for an infinite duration, then that character can travel an infinite distance, and the units of those distances can be counted throughout his time of travel {1 m, 2 m, 3 m... } (those elements can be progressively counted). However, if the character were to move an uncountably infinite distance, it is impossible even with an infinite period given to that character's travelling journey, since it contains far too many elements to be counted, and this is one of the major differences that should be considered.

With this said characters who can move uncountably infinite distances should be far faster than those who can move up to a countable infinite distance. Logically, the difference would be so vast that characters with the speed of the former view characters with the speed of the latter, as completely frozen.

Another detail that is worth consideration is the analogy between the ability to move an infinite distance in a finite period of time, and the ability to move a finite distance within zero time. Now, nothing wrong with either of those descriptions but this line in the article is greatly oversimplified, giving the impression to many users that moving an infinite distance within a finite time might be equivalent or similar to moving any distance within zero time. Though, really, those two capabilities aren't even remotely similar; a motion in exactly zero time is far vaster and greater than traversing an infinite distance. I'll go into detail as to why:

Now... it should be noted that there is rather a difference between dividing by exactly zero and dividing an infinitesimal real number that is extremely close to zero.

The speed formula goes by v(x) = dx/dt

If the time is approaching infinitely close to zero, then the speed will be approaching infinity, so I suppose you can give a direct infinite speed rating for it. It'll be at a countable magnitude as well, and comparable to that of moving an infinite distance in any given finite time. Though it isn't the same as time being directly zero in exact, and I think we all know how division by zero works... yeah, undefined. Motion in exactly zero time is just impossible in any given way, even if the character were to have actual infinite speed.

I'll also be bringing up the concept of hyperreal numbers.

The hyperreal numbers bring a piece of key information to this section of the revision thread: any number divided by a non-zero infinitesimal (ε), is an infinite hyperreal number (ω).

The infinitesimal is a non-zero positive number that is smaller than any non-zero real number. Infinite hyperreal numbers are rather uncountable since the cardinality of all hyperreal numbers is at least uncountable, if not greater than that of the cardinality of all real numbers, considering hyperreal numbers are an ultraproduct of real numbers.

Based on this inference, if dividing any number by an infinitesimal already stretches far beyond the margin of countability, then let alone dividing by zero, is a number even smaller than the infinitesimal. In conclusion, any action made in infinitesimal and zero time is far greater and more impressive than traversing an infinite distance in a given finite time.

Basically:

∞/x = countably infinite, x∈(all finite numbers)

x/ε = uncountably infinite, x∈(all finite numbers)

x/0 = undefined; impossible, x∈(all finite numbers)

To cover this potential concern, I came up with the proposal to add a new speed rating and revise the infinite speed rating:

Old and new infinite speed rating:

Old: Infinite Speed (Able to travel any finite distance in zero time, or move an infinite distance within a finite amount of time. Teleportation does not count. For further information, see the "Further Explanations"-section below)

New: Infinite Speed (Able to travel any finite distance within a timeframe endlessly approaching zero, or move an infinite distance within a finite amount of time. Teleportation does not count. For further information, see the "Further Explanations"-section below)

New speed rating:

Inaccessible Speed (Able to travel a distance of any magnitude within an infinitesimal timeframe or exactly zero time. It can also be achieved by moving an uncountably infinite distance within either a finite or countably infinite amount of time, or any greater distances within the time of a period of its subsets)

 

[GarrixianXD]

I don't want to see any derailings here. The only reason I didn't immediately make this a staff discussion is because I'm giving an easier route for blue names to share their opinions and input towards this topic.

Edit: Seems like its a staff discussion topic now. Get permission to comment first, from now on.

@DontTalkDT @Ultima_Reality @Agnaa Thoughts?

 

[Huesito88]

Any examples for an actual character preforming a "Inaccessible Speed" feat for your new preposed rating?

 

[Phsccarvalho]

Any examples for an actual character preforming a "Inaccessible Speed" feat for your new preposed rating?

Maybe Zeus (RoR)?

Since one of the interpretations is that the zeros continue, therefore infinity.

Although I think it would also include the OP's new infinite speed proposal of being infinitely close to 0. Since we don't know if there was any number that wasn't 0 in Zeus's attack

 

[GarrixianXD]

Edit: Seems like its a staff discussion topic now. Get permission to comment first, from now on.

^^Just a heads up.

 

[Agnaa]

Now, the issues rather lie in that, since we cramped all sets and orders of infinities, into a single rating, which pretty much ignores the vast differences between countable and uncountable infinities and the significance of it.

There's a big difference, but I don't think there's a large amount of characters which embody it.

To start, let's take an example of any character who can move at a finite speed. That character has all types of self-sustaining abilities, type 1 immortality and infinite stamina. If this character moves for an infinite duration, then that character can travel an infinite distance, and the units of those distances can be counted throughout his time of travel {1 m, 2 m, 3 m... } (those elements can be progressively counted). However, if the character were to move an uncountably infinite distance, it is impossible even with an infinite period given to that character's travelling journey, since it contains far too many elements to be counted, and this is one of the major differences that should be considered.

This is incoherent. Distances are continuous, and so, form a set with uncountably many elements. Uncountable infinity is named such since it's constructed as a more dense form of numbers, filling up arbitrarily long decimals with arbitrary digits, meaning that any attempt to count them will always leave a gap. If you don't reach uncountable infinity through sheer density like that (which even a movement of one planck length would cross), but just by being so utterly long in one direction, I don't know if that same counting issue exists.

Don't get me wrong, such a distance would still be astronomically longer, but it seems like your understanding of this is flawed.

Now... it should be noted that there is rather a difference between dividing by exactly zero and dividing an infinitesimal real number that is extremely close to zero.

The speed formula goes by v(x) = dx/dt

If the time is approaching infinitely close to zero, then the speed will be approaching infinity, so I suppose you can give a direct infinite speed rating for it. It'll be at a countable magnitude as well, and comparable to that of moving an infinite distance in any given finite time. Though it isn't the same as time being directly zero in exact, and I think we all know how division by zero works... yeah, undefined. Motion in exactly zero time is just impossible in any given way, even if the character were to have actual infinite speed.

"Moving in a timeframe approaching zero, in much the same way an integral would, rather than just a tiny but ultimately finite and ordinary length of time" seems like it would be astonishingly rare in fiction.

Plus, we do have the worry about how common the misconception you're talking about is. If fiction generally takes those as equivalent, we may have to do so too, unless they elaborate a lot.

I'll also be bringing up the concept of hyperreal numbers.

The hyperreal numbers bring a piece of key information to this section of the revision thread: any number divided by a non-zero infinitesimal (ε), is an infinite hyperreal number (ω).

The infinitesimal is a non-zero positive number that is smaller than any non-zero real number. Infinite hyperreal numbers are rather uncountable since the cardinality of all hyperreal numbers is at least uncountable, if not greater than that of the cardinality of all real numbers, considering hyperreal numbers are an ultraproduct of real numbers.

Well yeah, they're uncountable since they're a superset of the real numbers. But they don't have a greater cardinality, as your link specifies:

Therefore the cardinality of the hyperreals is 2^ℵ0.

Which is the same as the cardinality of the reals.

Based on this inference, if dividing any number by an infinitesimal already stretches far beyond the margin of countability, then let alone dividing by zero, is a number even smaller than the infinitesimal. In conclusion, any action made in infinitesimal and zero time is far greater and more impressive than traversing an infinite distance in a given finite time.

That's not really how cardinality works. It's not the size of any particular element that matters, but how many elements are in the set. The set {999} has a smaller cardinality than the set {0, 1}, because the latter has two elements and the former only has one.

I'm not sure what size-equivalent any particular hyperreal number has, but you have not provided enough information to demonstrate that they have a size beyond that of countable infinity.

This suggestion is not well-formed enough to be accepted in its current state. I would want to see more theoretical backing, and more practical backing of verses making this distinction (I'd want at least twelve different verses to qualify before accepting it as its own rating, even if it is theoretically valid).

 

[DontTalkDT]

Definitely should be a staff discussion. Moved it.

That aside, I disagree.

Let's start with uncountably infinite distance: That's not a thing. Distance is always finite by definition. Length can be infinite, but that is in a limes sense. The idea of cardinality doesn't apply to distance, just as it doesn't apply to measures in general. (length is just the 1D Lebesgue measure).

Next, hyperreal numbers aren't used in physics and neither in areas of mathematics that concern themselves with notion of speed. Forget the use of them. You would pretty much have to invent the field of hyperreal measurement theory to do that. Well, if I had to bet someone has tried that already, because someone has tried everything, but I have never heard of it. Hyperreals are just not very practical constructs because they are typically defined via the axiom of choice. It's always worth remembering that people can construct virtually anything that isn't self-contradicting in mathematics, but just because it's constructible it doesn't mean we should care.
I should also note that one shouldn't carelessly connect them to cardinals. The symbols are sometimes used, but fundamentally, hyperreals only have higher infinities in terms of ordering, not cardinality.

the analogy between the ability to move an infinite distance in a finite period of time, and the ability to move a finite distance within zero time. Now, nothing wrong with either of those descriptions but this line in the article is greatly oversimplified, giving the impression to many users that moving an infinite distance within a finite time might be equivalent or similar to moving any distance within zero time.

They are the same thing, if done at a constant speed. Well, in the usual setting of physics and measurement theory. (i.e. real numbers)
If a character moves an infinite distance in finite time, how long did they take to move 100m? 0s. In the real numbers, which are the de facto used description used for these things, that is the only viable answer, as any number greater 0 would result in finite speed. This also relates to the fact that infinitely small and zero in the real numbers are basically the same thing, meaning that in reverse, moving a finite distance in zero time is actually the same as moving a finite distance in infinitesimal time.
That is reason number 2 why hyperreal numbers won't help you: Any mention of zero seconds would include infinitesimals as part of the statement, since nobody talks in hyperreal numbers.

Not convinced regarding hyperreals yet? You bring up a concept of a derivative here as the definition for speed. For a start, how is the derivative defined on hyperreal functions exactly? If you look into that, to my limited knowledge in the field, I believe you will find that it's defined with the standard part function that projects the result on the real numbers. I.e. no infinite results actually exist. It, in practice, ends up identical to the regular derivative.
I have tried to find a use of derivatives without the standard part function in hyperreals to no success. Could be because I'm not in the field, though. My current best guess: It's probably not done or at least not without some modifications to the concept. Why? Consider the function f(x) = H*x^2 where H is an infinite hyperreal. If we define with an infinitesimal ∆x the derivative without standard part function as f'(x) = (f(x+∆x) - f(x) )/ ∆x = (H (x+∆x)^2 - H x^2) / ∆x = (H x^2 + 2 H x ∆x + (∆x)^2 - H x^2)/ ∆x = (2 H x ∆x + (∆x)^2 )/ ∆x = 2 H x + ∆x
Whoops, we get a definition of the derivative that for real values doesn't match what we usually get and is dependent on the choice of the infinitesimal (i.e. not well-defined).
Now, one could get the idea to fix this via a hyperreal sequence of infinitesimals converging towards zero in a hyperreal sense. It seems obvious that one would first need to extend the idea of convergence of a series to have any hope of success, as otherwise no non-trivial convergence by the usual criteria is possible, as the series won't be able to reach the infinitesimal neighbourhood of any point. But at this point we are stacking non-standard definitions on top of non-standard definitions to make this somewhat work. Which you can likely do, just that virtually nobody would.

But the funny thing is, after all of that you haven't actually solved the problems:
Consider the actual distance over time function. Trying the map into unto the hyperreals... well, we get that any moving of an infinite distance in finite time in fact moves greater infinite distances (greater in hyperreal not cardinal sense) and the 0 time function still does not exist. Also not sure if the prior one can be said to be continuous. So really, what have you won? You moved from two cases which you can address at best as a limes process, to a case that doesn't really match what we have and a case you still can't map mathematically.
Ultimately, the theory didn't really get better.

 

[Flashlight237]

Wait, don't we already have a rating like the proposed Inaccessible rating called "Immeasurable?"

 

[Agnaa]

Wait, don't we already have a rating like the proposed Inaccessible rating called "Immeasurable?"

No.

 

[Agnaa]

Closing at the OP's request.

 

[Antvasima]

Thank you to everybody who helped out here.