I was thinking about Zermelo's critique of Cantor's reasoning for the well-ordering principle, how Zermelo characterized it as an appeal to temporal intuition, whereby time itself does the well-ordering, as it were. So my question is: is there something ordinal-like about time, and something cardinality-like about space?
Relations of size are almost always considered spatial relations; though strictly, ten years is the same amount and length of time, yet we do not usually talk about the size of a time period but its length, which can be compared to ordinal height/length in general.
Most importantly, then, consider the surreal numbers. They seem akin to ordinals as well as cardinals without being reducible to them as such (or rather, if there is a reduction procedure, here, it is only to a join function of the ordinal and cardinal transets (AKA proper classes) together). From our vantage of spacetime theory vs. space-apart-from-time theory, if pure space-like relations are cardinality-like and time-like relations are order-theoretic, then is a congealed spacetime-like relation system best "mapped" using surreal numbers? Note that infinitesimals and real numbers (and much else besides) are given in the surreal transet directly, whereas the "construction" of these objects is much more tedious using a stricter ZFC-like basis; and so the Continuum has surreal elements/functions to its name, it seems to me. So then, if the Continuum is the image of spacetime coordination, this too goes to show that surreal numbers are dually expressive of cardinal and ordinal numbering, and then just the same goes for space and time "themselves"?