Considering the scenario whereby space pre-exists and extends an infinite distance.
Assumption: All of space already exists in all directions, i.e., it is actualised; it actually exists.
Fact 1: Any region of space supposed to exist infinitely far away cannot have positional coordinates relative to our own position, because if it did, it would simply mean it is at a finite distance away.
Fact 2: Space must exist as continuous, i.e., unbroken.
Fact 3: Continuity of space only exists if all locations have positional coordinates relative to each other.
In the above scenario, it would be impossible to travel to any region of space that is infinitely far away – such supposed regions would have no relative position to ourselves, and therefore simply cannot exist.
A paradoxical story – relative positional error via a potential head-on collision: Spaceship exploration paradox
The logic above proves that space cannot be infinite, i.e., it reveals a contradiction; paradox; impossibility. It also proves that for something to exist, it must be part of a totality.
Two travellers (A and B) are each located on a different planet that are themselves infinitely far apart.
Each traveller sets off in their spaceship headed towards the other; travelling at the same speed as each other; they pass halfway and wave "hello".
Therefore, the midway point is: infinity / 2 = infinity.
That means when they pass at halfway... the distance remaining is still infinity.
New travellers on the same mission set off from each planet; they pass “A” and “B” at 1/4 the initial total distance.
The remaining distance is yet again – infinity... and so on, i.e., the new crossover point is always half the previous distance, but which is also infinity.
Repeat this process an infinite number of times... and guess what? ... every new midway point is still infinity.
That is to say, even after travelling for an infinite amount of time, they never arrive at the other planet.
A continuum of distance exists as follows:
Between any 2 locations there exists an unbroken series of increments (steps).
Each step exists one after the other; all the same physical size (and above 0).
Each step has a numerical position in the continuing sequence that exists between the two locations.
None of the steps between the two locations exist without being represented by a number – not any.
Infinity is not a number – it does not exist as a numerical value.
Therefore, there does not exist any step in the sequence that resides at location infinity... because infinity is not a number, and every step is represented by a number.
If a piece of string supposedly spans infinitely far in a straight line in opposite directions away from you, and you cut the string and hold the two ends, how long is each piece?
Answer: it’s impossible to have been infinitely long in the first place.
If space pre-exists independent of matter (so ignoring relativity theory, if need be) as some physicists suggest might be possible, and it extends an infinite distance, but without even a single massive particle in existence, and then 1 massive particle popped into existence, would it have positional coordinates?
Answer: no; it would be infinitely lost.
If a 2nd particle popped into existence would that have a position? And now indeed also the 1st particle?
Answer: both particles would have a position if it was possible to travel from one to the other. However, since space is presumed to be infinite, what exactly in the popping-into-existence process determines whether or not the 2nd particle is at a finite distance away from the 1st particle? Theoretically particles could pop into existence for eternity, none of them ever having a position.
If a turnip was infinitely far away in space and you travelled at 99% the speed of light (causality) for 100 million years in an attempt to get closer to the turnip, would you be any closer to the turnip?
Answer: no; you don’t have relative positional coordinates to the turnip, therefore it’s impossible for you to get any closer.
If a potato was infinitely far away in space, would you be able to set off in the correct direction in an attempt to eventually arrive at the potato?
Answer: no; such a direction exists as a line (curved or not) which spans from your location to the potato’s location relative to you, and therefore you won't be able to set off in the correct direction because the potato doesn’t have a relative position to you.
If you fired several peas from a pea-shooter via one powerful puff (they're loaded into the tube one after the other) such that each pea travelled an infinite distance in a perfectly straight-line from its exact starting position, would the peas still form a line (i.e., one pea after the other) when they arrive at their destination?
Answer: no; each pea's relative position to the others will no longer exist, and so they’d all exist in one location, and they’d also be infinitely lost relative to your position.