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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?

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Wolphram jonny
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The surreal numbers. They include the hyperreal numbers as well as the ordinals. The surreal numbers are the largest possible ordered field. Because they are a field, for any uncountable ordinal $\omega_{\alpha}$ you can define 1/$\omega_{\alpha}$, which lies within [0,1]. So the interval [0,1] contains a set of points that are larger in number than any cardinality (it would have the "cardinality" of the proper class of all ordinals). Regarding the issue of why physical space is considered a real field and not a surreal field, I have no idea!

Wolphram jonny
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  • To your last sentence, the surreal numbers are an invention of the 1970's or so; modeling the physical reality in $\Bbb R^3$ is something that started several centuries ago. – Asaf Karagila Apr 08 '13 at 12:07
  • @Asaf what I meant was if there is any obvious reason about why it would or would not make sense to replace the real line by the surreal line in geometry or physics. Would it change anything? would it be plain wrong? is it totally inconsequential? – Wolphram jonny Apr 08 '13 at 17:05
  • It would be somewhat troublesome, because we can't do analysis over the surreal numbers properly in $\sf ZFC$, because we would have to talk about classes of classes. I don't know whether or not it would make too much difference. Think that if you switch to a class-friendly set theory, e.g. $\sf MK$, maybe even $\sf MK$ with $2$-classes; then you will run into other limitation. – Asaf Karagila Apr 08 '13 at 17:31
  • Also, all physical measurements are only of finite precision. The surreal numbers provide for infinitesimals, things smaller than any finite real but not zero. There would be no hope of measuring something down to infinitesimal distances or times due to the quantum (Planck) limits, even if they existed. Also, all of our measuring technologies are limited in precision anyways -- to a finite amount of precision. So there would seem no advantage to using $\mathbf{No}$ over $\mathbb{R}$ anyways. – The_Sympathizer Sep 15 '13 at 11:10
  • Also, another big stumbling block to analysis over $\mathbf{No}$ is that $\mathbf{No}$ is not Dedekind-complete. Much of what makes real analysis, and so also calculus, work is the Dedekind completeness property of the reals. And physics uses calculus extensively, even in non-classical physics. – The_Sympathizer Sep 15 '13 at 11:14
  • To get an idea of how weird things become without Dedekind-completeness, consider measuring lengths. One might think of the "length" of a segment as being the distance between its endpoints. But on a surreal line, there are "segment-like" classes which include between any two points in them every surreal between those two points, yet are not the entire line, yet both have no endpoints and are not the entire line, due to the lack of completeness. So we cannot reasonably assign them a length! – The_Sympathizer Sep 15 '13 at 11:21