My favorite example, that stopped me thinking about the “endpoint” of an infinite ray, is from the late George Gamow’s wonderful book, ONE TWO THREE…INFINITY. His example was to think of a hotel with a finite number of rooms, all of them filled. A new guest shows up, the hotel is stuck, they can’t provide a room.
Now imagine a hotel with an infinite number of rooms, all of them filled. A new guest shows up. The clerk says, “Sure, no problem.” He moves the people from room 1 into room 2, the people from room 2 into room 3, the people from room 3 into room 4, the people from room N into room N+1, etc etc. Lo and behold, every former hotel guest has a room, and now room 1 is open for the new guest.
Even more mind-boggling: suppose an infinite number of guests show up at the hotel, asking for rooms. Again, the clerk says, “No problem.” He moves the people from room 1 into room 2, from room 2 into room 4, from room 3 into room 6, from room 4 into room 8, … from room N into room 2N, etc etc. Now, he has moved all the guests into the even-numbered rooms, and all the odd-numbered rooms are open to accomodate the infinite number of new guests.
You can’t think about “infinite” the way you think about finite.
In terms of the universe itself being finite, think of an intelligent two-dimensional bug in a two-dimensional plane… he thinks it’s infinite in all directions. Suppose instead, he’s on the surface of a very large beach ball. As far as the bug can tell, he’s on a plane, infinite in all directions. Yet, if he walks in a straight line from his point of origin, he goes around like a longitude line, and comes back to where he started – yet he never encountered an edge or a boundary. He has no concept of three dimensions, so he can’t understand how this happened. His universe is finite and but has no boundary. In the same way (analogously) our universe could be bent around in some fourth dimensional way that we can’t visualize, that would make it finite but unbounded.
The more fun thought than a universe shaped like a four-dimensional sphere is a universe shaped like a four-dimensional Moebius band. Heh.
Final comment, on why space and time are “the same.” They aren’t, per se. They are intricately bound together. You can’t describe a point in space without also describing its point in time. You can say that at a certain time, the planet Mars is HERE relative to the sun, and at another time it is THERE. Thus, time can be thought of as a dimension, similar to the three spatial dimensions, used to describe the universe around us.
One of the ways of visualizing four-dimensional space is to think of time as the fourth dimension. I want to visualize a four-dimensional sphere of radius 1, I think of it as starting at time 0 as a point, the point expands (like a balloon blowing up) over the next few seconds, becoming a sphere of radius 1 at time 1, then it starts decreasing in size (like a balloon deflating) until at time 2 it’s back to a point and then disappears. You thus have visualized a sphere that has radius 1 in four dimensions, using time as the fourth dimension. Nothin’ to it.
Using that as a 4-dimensional model of the universe makes some sense, too: at point of Big Bang, the universe is a point, expanding over the billion of years as a sphere, and eventually contracting to a point again (say, for example).
It has a lot to do with why I started thinking about infinity in the first place. People kept telling me that the universe was infinite, and I didn’t buy it. Then I thought that you could travel in one direction forever on Earth and never find an end to it. Add a dimension, and that could be the universe. On Earth, we learned to fly, and then to leave the atmosphere, and now we can leave Earth altogether and look upon it. I wonder if someday we’ll learn to leave the universe in some fashion and see what else there is?