This question is motivated from a previous question, but is in itself independent of it.
So, I understand that it is not possible to embed $\omega_1$ or any uncountable ordinal into the real line, since every cut in the reals has countable cofinality. So one cannot undertake an uncountable process in a real-number of seconds, if every step is to take some nonzero real-number amount of time. But is it possible to generalize the concept of real line, in the sense that we could embed $\omega_1$ or any uncountable ordinal into a finite segment of it? I know that the reals fill the real line, so you cannot add more numbers into it, but I am not completely sure if you cannot define a more general "line" so that the number of a finite segment of points within it can have any arbitrary cardinality. And if this were possible, is there any reason why we could not think of physical time or space as having these generalized properties rather than those of the real line?
As rightly suggested by one of the comments, the actual question should be if whether or not there is a generalized concept of a valued ordered space. The body of the question apparently talks about metric spaces, but my intention is to see if it is possible to generalize this concept (and, in addition, if such concept could be also be, in principle, a model of physical space or physical time)
Caveats: For instance, take a Hilbert space (which is a metric space) with an orthonormal basis indexed by B. The Hilbert dimension is the cardinality of B (which may be a finite integer, or any countable or uncountable cardinal number). Do this mean that a neighborhood of a point in Hilbert space also have a set of points of cardinality B? If we define a "line" on a Hilbert space of uncountable cardinality B, will this line have also a number of points of cardinality B?