Brainlet test

Its just R^3 with some metric.

What exactly are you asking?
1. Is true by definition of the metric
2. Is also obviously true
3. Also true.

>only a handful
what he said is common knowledge to EVERYONE who is in college for a non-meme degree.

Either you are underage or really uneducated.

But here is your explanation.

R^3 means that every point is described by 3 coordinates.

The distance between 2 points is given by the euclidean metric that means:
if (a,b,c) and (x,y,z) are points, then their distance is ((a-x)^2+(b-y)^2+(c-z)^2)^(^/2).

You will recognize that formula as Pythagoras theorem and it is the way distance works in the real world.

This space very obviously describes all your criteria and is used in most engineering and physics disciplines to describe real world interactions.

dude it's still normal 3-Space with usual euclidean distance

find me two points in R^3 without a finite length path between them and I'll give you $1

>Any two points must have a path between them that is finite in length.

Hmm, can length ever be truely finite though? I mean, if you can continuely add percision to the measurement when you look closer, is it technically a kind of infinity?

>Hmm, can length ever be truely finite though?
Yes.
Both in a mathematical sense and a real world sense.

> I mean, if you can continuely add percision to the measurement when you look closer, is it technically a kind of infinity?
No. Something finite can be made out of infinitely many parts, at least in mathematics.
In reality everything has to be finite.

Every Lorentzian 4-mfld satisfies the above.

im assuming you want length defined by the metric, but how about volume? do we have to define that with a sigma-algebras over the metric space?

if there are no constraints, just use normal euclidean space with normal volume and define the discrete metric. then volume is infinite and all distances are equal to 1, thus finite, and 3 is automaticly satisfied

also I assumed you want a space where the distance is bounded by some number M, otherwise this whole thing is retarded and almost any space will work.

I like this pic

So...
It's just ordinary space?
That means is an exceptionally retarded post.
It explains something obvious in as confusing a way as possible, and fails to satisfy the finite travel time requirement since it has to be infinitely large to have infinite capacity.

Not sure what OP was thinking here.

You mean any unbounded set right? An open set with finite volume probably doesn't match what OP was looking for. Probably.

Space or imagination take your pick

Ok, for realzies, is the awnser just standard 3d space with no set boundaries?

I mean, even if the space is infinitely large, isn't it still true that you could always pick two arbitrarily defined points, regardless of how far they are away from each other, and determine the travel time between them which will always be finite?

If I'm wrong here please let me know. This question seems silly to me, but I'm just a layman with nothing to prove.

tl;dr I think you might be dumb OP

It becomes a lot clearer why that doesn't work if you think in fractions of the infinite 3D space.
Say that point A is at 0,0,0 of the infinite 3D space, which spans from 0,0,0 to 1,1,1.
Even if point B is at 0,0,0.0001, that 0.0001 is a fraction of infinity, and thus is infinitely far away from point A.

>Not possible in 4D space
But we live in 4D Space OP. You're confusing the three-dimensionality of objects that we perceive with the actual number of dimensions they occupy. To a human viewer in our universe, your "paper" idea would look like a three-dimensional ball flashing in and out of existence for an infinitely small amount of time, as it occupies the slice of the fourth dimension that the viewer is currently observing.

An actual Earth ball, observed by a four-dimensional entity, would look like a long cylinder or snaking shape because its position in the fourth dimension would all be observable at once. Two three dimensional balls colliding in a 4D space would look to a 4D observer like a trace of their motion (some kind of X-shape, for example). Not like two pieces of paper.

What? That's retarded my man. You can't just invent an arbitrary unit for measuring distance and claim there is infinite space inside a length defined as finite by the measuring system. The concept of infinity is inherently uncountable. It would be like if I said I invented a scale that ranges from -1 to 1 and that contains all integers. Have I proven that there are two integers with infinite distance between them, just because I asserted something nonsensical about the set of integers?

I'm not counting time as a dimension, smartass. If I meant time as the fourth dimension I would have said 3+1D, as eveyrone else does.

I mean objects with three spatial dimensions inside space of four dimensions.

It's not an arbitrary unit, the unit is equal to infinity.
The math is exactly the same except that it becomes possible to pick a random number between 0 and infinity, you just have to remember to multiply by infinity afterwards.

Being able to pick a random number between 0 and infinity is necessary to show why two points in infinite space will always be infinitely far apart.
There's infinite numbers between two integers, so two random numbers between 0 and 1 have a 0% chance of being identical. Call them A and B.
The distance between A and B is |A-B|*infinity.
Obviously, it's just going to equal infinity if A and B aren't identical, and they never are.

The whole thing is a little unintuitive because if you try a non-mathematical approach, you get caught up in the fact that there's no cutoff between infinity and not, and thus an infinite volume that is within a finite range seems to exist.

>is an exceptionally retarded post.
It is honestly the best reply possible under the assumption that you are not underage.

> confusing
It is written in well defined and easy to understand terms.

>fails to satisfy the finite travel time requirement
wrong.
Even in an unbounded space the distance between 2 points is always finite,by definition of the metric.

>I said I invented a scale that ranges from -1 to 1 and that contains all integers
[-1,1] and R are isomorphic dude.

This is literally just ordinary space

See That's a proof that any two randomly chosen points in infinite 3D space are infinitely far apart.
Unbounded and infinite space are the same thing, but unbounded makes you think about it in the wrong way; if it's unbounded, that means you can keep moving point B away from point A for as long as you like and at no point will the distance hit infinity, so it sounds like point B will always be within a finite distance; but then if you actually do the math it turns out that point B is always infinitely far away if point B is randomly placed inside an infinite volume.

>randomly chosen points
That's exactly why the space is specified to be standard Gaussian in ; it means that it is endowed with the Gaussian measure over R^3 (informally, a three-dimensional bell curve, formally en.wikipedia.org/wiki/Gaussian_measure).

The "proof" in conflates the notion of a distribution over a set with the cardinality of that set, and ends up assuming a uniform distribution over [0,1] and concluding that the distance between two finite points is infinite. This is basically Zeno's "proof of the impossibility of motion" dressed up in mathematics but without a rigorous definition of infinity, and the paradox has been resolved over 200 years ago with the invention of the foundations of calculus.

The volume is finite, unless the scale of the bell curve is infinite in which case the separation of the points is still going to be infinite.

>infinite 3D space
>spans from ... to ...
Retard

>you are not bound by any physical laws
Well then, fuck you, I do what I want.

Then just choose some bounded subset.
Or some other metric.

Also I cant seriously believe that OP really meant "random" because that makes literally no sense if you dont also give a distribution, as I can just arbitrarily make one up which trivially fulfills it.

The distribution and randomness doesn't matter.
"travelling between two randomly chosen points always takes a finite time" means exactly the same thing as "All possible pairings of points have a path of finite distance between them".

>The distribution and randomness doesn't matter.
I can just choose some distribution where the chance of getting a 0 is 1.

Then 2 randomly chosen points have always distance 0.

But again, just choose a different metric or a bounded subset, that really should be trivial.

>me and someone else in a furry thread found a solution to this same question
>Veeky Forums still can't even understand the question, and think that the problem will go away if they spout jargon they barely even know the meaning of
This is absolutely pitiful

>found a solution
R^3 with some metric is a solution....

Jesus christ
That doesn't even mean anything, I doubt you could pass a Turing test.

>That doesn't even mean anything
?
Of course it does, you just dont understand it, because you have no education in mathematics.
R^3 is just the regular 3 dimensional space, a metric is a way to measure distance.

The question OP asked, is literally hundreds of years old, just like the solution, which is R^3 with the euclidean metric.

It is used LITERALLY everywhere from physics to engineering.

So inserting your elaborations into your/ prior sentence, your answer is "3 dimensional space with some way to measure distance".

>hey how can I make this space have infinite volume yet be easily navigable
>"give it some way to measure distance"
This is exactly the shit I'm talking about.
This is why you'd fail a Turing test.

>"3 dimensional space with some way to measure distance"

No I mean the R^3, there are arbitrarily many 3 dimensional spaces, but I want a specific one, namely the one which is ALWAYS used in physics and engineering and the one every high school student knows about.

>This is exactly the shit I'm talking about.
You are?
What is wrong with what I said, I even mentioned a specific way to measure distance which directly relates to our understanding of distance.

>This is why you'd fail a Turing test.
That makes no sense. I am not a computer I can not take a Turing test. This is about as sensible as asking a Computer to take an IQ test.
And just because YOU dont get the meaning doesnt mean there isnt one.

But what am I expecting from a furry.
Please tell me your """""solution""""".

Why bother writing the solution?
You don't even understand the question, if you think that regular ordinary space provides a satisfactory solution.
Finite space satisfies the first and third conditions, infinite space satisfies the second and third conditions.

In what way does an infinite space NOT satisfy the first condition?

And what even IS an infinite space.

Not all pairs of points in infinite space are a finite distance apart.
The first condition requires that they are.

>Not all pairs of points in infinite space are a finite distance apart.
Wrong.
Show me 2 points in the regular 3 dimensional space which are NOT a finite distance apart.

With points I mean a vector (a,b,c) which describes x, y and z coordinate.

If using a global coordinate system, have point A at 0,0,0, and point B at ∞,0,0.

If you use a local coordinate system where the infinite space spans from -1,-1,-1 to 1,1,1, the distance is infinite if point B is anywhere except 0,0,0.

This, any non-retard can tell that the idea of two points even in an infinite space being infinitely distant is nonsense

We can create a distribution that can reach to infinity with almost any function that produces points with limit infinity. Tan90, 1/0, etc.

>∞,0,0.
hahaha you are kidding, right?
Infinity is NOT A NUMBER therefore not a valid coordinate.
I was talking about R^3 NOT R*^3, thats why I said "regular".

>infinite space spans from -1,-1,-1 to 1,1,1
This is also hilarious.
You dont have slightest Idea what you are talking about. That space is CLEARLY bounded.
And you gave no answer to how you measure distance, but still claimed to have some form of distance.

An extended metric space may have pairs of points at infinite distance. For example we say two vertices of a graph have infinite distance if there is no path from one to the other

Those aren't functions and lack distributions.
I see where you're going though; things like the bell curve to reach off to infinity; but they lack infinite area, or in this case volume.

Sure, infinity is a number.
It's the biggest number you can think of plus one.
It's just a hassle to deal with mathematically, especially with vectors, which is why working in the local coordinates of infinite space is preferrable.

But if you still aren't convinced that the distance can be infinite, say that I put point A and B next to each other, then moved point B as far away as the infinite space would allow.
Is the space going to stop me from moving the points infinitely far apart? Certainly not, whatever finite limit is suggested, point B can be moved past it.

>moved point B as far away as the infinite space would allow
>whatever finite limit is suggested, point B can be moved past it
The contradiction in your argument undermines any point you were trying to make.

What contradiction?

>Sure, infinity is a number.
AHHAHAHAAHHA.
No its fucking not. By no definition of the real numbers is infinity EVER part of it.
>It's the biggest number you can think of plus one.
Lmaoing at your education.
>It's just a hassle to deal with mathematically
That certainly is true, thats why no definition of the reals includes infinity, of course there is R* but that isnt even a field.

>Is the space going to stop me from moving the points infinitely far apart? Certainly not, whatever finite limit is suggested, point B can be moved past it.
That is called "unbounded" but was never a question I asked.

Your claim was: "Not all pairs of points in infinite space are a finite distance apart"
Sure you can find points arbitrarily far away from each other, but between every 2 of them is a finite distance.

Then prove it, that in an infinite space you cannot have points that are infinitely far apart.

lmao.
That is true by definition.

Lets say (X,d) is a metric space, by definition the image of d() are the real numbers, which doesn't include infinity.

qed.

Assuming by "infinite space" you mean "unbounded metric space". If you dont you have to give me your definition first.

Then if infinite distance is impossible, there must be a clear finite limit to the distance.
What is it, if not infinity?

>Then if infinite distance is impossible, there must be a clear finite limit to the distance.
lmao.
You dont get it do you?
"unbounded" means the distance between 2 points can be arbitrarily large, but the distance between any 2 given points will ALWAYS be finite.

This REALLY are the basics of calculus.

Take a series of REAL NUMBERS they are all "smaller" then infinity, but they can still get arbitrarily big and diverge towards infinity.
This means, that just because every point has a finite distance to another point, the distance can not grow arbitrarily large.
Just like you can have a divergent series of real numbers.

>"the distance between 2 points can be arbitrarily large"
>"the distance can not grow arbitarily large"
>arbitrarily large does not include infinity, even though it doesn't stop you from incrementing by 1 an infinite number of times
I can't think of anything else to say, you're just making bold, unfounded statements and not actually doing the logic behind anything.
Really, what stops me from moving the points infinitely far apart?
At no distance am I no longer allowed to move them further, I can just increment the distance by one infinity times instead of incrementing the distance by infinity one time, since your rules apparently don't allow the latter.

>"the distance between 2 points can be arbitrarily large"
>"the distance can not grow arbitarily large"
How do you NOT understand the context. In the second part I was saying the contraposition of the first, that is just basic logic.

>I can't think of anything else to say, you're just making bold, unfounded statements and not actually doing the logic behind anything.
As I said, these are the BASICS of calculus, something you learn in the first year of university.

>Really, what stops me from moving the points infinitely far apart?
How would you mathematically do that?
You can not add or multiply infinity.

>At no distance am I no longer allowed to move them further, I can just increment the distance by one infinity times instead of incrementing the distance by infinity one time, since your rules apparently don't allow the latter.
There is a difference between THE LIMIT and a point in the series.
As I said, you will understand this if you learn calculus.

>Really, what stops me from moving the points infinitely far apart?

Nothing, but where are you placing the points so that we can measure the distance between them?

Or are you implying that we should take a measure while you're still moving the points? What use would that be

|x-y| cannot be infinity if x and y are finite. It's as simple as that. If you think otherwise I suggest you give an example of x and y.

Since all of you seem to accept the idea that the maximum distance between the points is indefinite but think treat infinity as a banned number, I'll rephrase condition A so that infinity doesn't have to be considered.
>the maximum time to travel between two points in the space is both definite and finite

Also, it sounds like your university professor fed you bullshit, I was never taught to avoid infinities and infinity occurs in calculus all the time.

>>the maximum time to travel between two points in the space is both definite and finite
Just say "the space is unbounded" it is the applicable mathematical term.

>Also, it sounds like your university professor fed you bullshit, I was never taught to avoid infinities and infinity occurs in calculus all the time.
Infinity is NOT a real number. If you were taught that you were taught bullshit. Especially because R*(=R U {infinity} U {-infinity}) exists, which is not even a field.
If you say "lim f(x)= infinity", you mean that the series diverges, saying it "equals infinity" is just a short and more intuitive way, although not completely rigorous.

>Just say "the space is unbounded" it is the applicable mathematical term.
I obviously meant to say "bounded"

>Just say "the space is unbounded" it is the applicable mathematical term.

A space can be unbounded and finite.

Isn't that just normal 3D space? Assuming the universe is in an infinitely massive void.

Holy shit furries are fucking retards.
Take an Euclidean space with 3 dimensions, with coordinates (x,y,z) where x, y and z can be any real number.
The space is obviously infinite.
Now, find a pair of coordinates an infinite distance apart. Just try.

you realize there's a canonical inclusion of R^3 inside R^4? so your last point is retarded. just have 4D space.

I believe you're thinking of [math]\mathbb{R}^3[/math].

A fine example of the Dunning-Kruger Effect here.

The space i create has the following properties:
>>travelling between two randomly chosen points always takes a finite time
>>it can fit an infinite number of 3D objects inside
>>it is realistically possible for normal three-dimensional interactions to occur (so you can't just say "just have 4D space" because attempting normal 3D interaction would be like holding two infinitely-thin pieces of paper and having them touch edges)

>A space can be unbounded and finite.
What the fuck is a "finite space".

A space with finitely many elements?