Sure, what I just said is logically impossible

Really?

Here's an analogy: Suppose you thought you lived on the unit interval [0,1] in the real line. Then experiments showed that whenever you got to 1, you were magically whisked away back to 0. So a clever mathematical physicist, well versed in topology, comes along and says, "Hey! Why don't we just identify 0 and 1 as the same point? That way, we can say that we're living on a circle, instead of a line segment".

Suddenly, a whole new research program emerges. If we're living on a circle, what's its radius? Is it even a circle at all, or mightn't it be an ellipse? Or something even more exotic? Is there an "extra dimension", i.e. an underlying 2-dimensional plane in which the circle (or whatever) is embedded? And so forth.

(Technically, you could have asked some of these questions under the old paradigm. E.g.: is our line segment really a line, or is it curved? But you wouldn't necessarily have thought to do so! )