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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:
New speed rating:
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)
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