This thread is dedicated to speculation about an actual infinite-dimensional space, in which there are an infinite number of macroscopic, orthogonal coordinate axes. I realize that such a thing may be controversial, because it implies the acceptance of completed infinites (otherwise the space would be ill-defined -- you can never finish constructing it!), but for the sake of speculation, we'll just take it as a given. What would the consequences of such a space be? What would life be like in such a space (making the HUGE assumption that life can possibly exist in such a space)?

Here are some simple consequences that I've come up with so far:

- The ∞-cube has an uncountable(!) number of vertices. This is a consequence of the fact that there's a bijection between its vertices and the real numbers between 0 and 1, which in standard set theory is an uncountable set.
- A creature that lives in this space either must have an infinite number of sensory organs, or be content to live with the crippling limitation that it can never be fully aware of its surroundings. Why? Because if the creature has eyes similar to ours, i.e., it has directed vision, then it would take an infinite amount of time for it to survey its surroundings -- since it needs to look at least in the direction of each of the coordinate axes, plus or minus a few, but there are an infinite number of axes. So the only way an ∞-dimensional creature can be fully aware of its surroundings at any given moment is that it has an infinite number of eyes (or equivalent sensory organ).
- The ∞-cube has a non-finite outradius. Implying that while it may take only one unit of time to walk from one end of the cube to another, it takes an infinite amount of time to cross from one corner of the cube to the opposite corner.
- The volume of the ∞-sphere is zero, or at most infinitesimal. This means that you can store an infinite number of marbles in your toy box.
- Actually, if your toy box is bigger than your toy, then you can almost certainly store an infinite number of toys in it.
- It takes an eternity to build a house, because you need to stack an infinite number of blocks together -- at least one per axis. In fact, it takes an uncountable amount of time to build such a house. That's infinitely more infinite than infinity.
- However, it only takes a countable amount of time to build a tent -- if the tent is in the shape of an ∞-cross pyramid. Unfortunately, such a tent would have infinitesimal volume, so you wouldn't be able to fit inside it without your limbs sticking out for predatory beasts at night to bite.
- If creatures in ∞-dimensional space lived on a planet (a ∞-sphere), they would be living in total darkness pretty much all of the time, except for maybe one point on the planet that's directly under the sun. This is because with an infinite number of lateral dimensions, any light from the sun would fade to nothing as soon as you leave that single point that receives direct sunlight; so practically 100% of the planet's surface (minus an infinitesimal number) would be in total darkness.
- In fact, light from any light source would almost instantaneously dissipate into nothing as soon as it leaves the source, because there are an infinite number of directions to disperse in, so any finite amount of light would immediately fade to 0 in all except countably few directions. So the only way for the sun to illuminate the planet is if the sun emitted a non-finitely-large amount of light. I.e., the sun is either completely invisible, or must be infinitely bright!

What can you guys come up with about living in an infinite-dimensional space?