RJG wrote:P1. Finite objects have boundaries.
Lagascienza wrote:This will not always be true. It will not be true if we are talking about a whole universe with a particular curvature. All objects within any 4D universe will indeed have boundaries. But a 4D universe with closed curvature would not itself have boundaries.
Not so. If the 4D universe has closed “curvature”, then it will logically have a boundary, that defines its shape. A shape without a boundary is an oxymoron.
Lagayscienza wrote:3D objects like a Klein bottle are analogous – they are finite but have no boundary on their curved surface.
If they have a curved surface, then they have shape. If they have shape, then they have boundaries. A curve without a boundary is an oxymoron.
Lagayscienza wrote:Now try to imagine the the space of a 4D universe with closed curvature. (If you don't know what a Klein bottle looks like just do a web search using the term "Klein bottle". I am not permitted to post images here. Keep in mind that 4D space is almost impossible to imagine and it is impossible to draw the geometry of the Universe on a piece of paper, it can only be described mathematically. But the boundaryless surface of a 3D Klein bottle is analogous.
The surface of the Klein bottle is
not “boundaryless” in the sense that the Klein bottle has
no boundaries. It is only “boundaryless” in the sense of
endless travel within the Klein bottle.
Endless travel and
no boundaries are two different animals.
Falsely conflating one sense (meaning) of the word “boundaryless” (or “unbounded”) to imply another sense of the word is the trickery (equivocation fallacy) being played on all of us with the deceptive phrase “finite but unbounded”.
Lagayscienza wrote:With my comments above in mind, here is my logic:
A 4D universe contains all of space.
Logically, only an 'infinite' universe can contain ALL of space. A finite universe cannot logically contain all space; it cannot contain the space that contains its boundaries.