# Space as being "cardinal-like"?

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"?

• Conway found surreals actually form a field including both the reals and ordinals during his surreal application in games such as Go, so it could be used to model space and time. Once you're in a certain dimension of space it's ordinal like, unless you have fractal space where it's unclear how cardinals can fit in. OTOH non-recursive limit ordinals such as the smallest countable Church–Kleene ordinal cannot be represented by any computable function , thus its physics application is doubtful... Aug 21, 2022 at 4:12
• Also note it's a common conclusion that many different infinite ordinals can correspond to a same cardinal such as the famous bijective ω+1 and ω share same cardinality aleph0, so how space conceived through one dimension can be cardinality-like? Aug 21, 2022 at 4:40
• I do suppose that for "conservation of energy" reasons, the "universe"(?) tends to use the smallest forms of infinity for its deep principles(?). However, I also do think there is a dynamic that semiperiodically alters the infinitary logical signature of a world, with the L(kappa, lambda) correlating kappa, for the connectives, to order-theoretic time, and then correlating lambda, for the quantifiers, to cardinal space, so that the cardinality of spacetime changes/evolves as time passes, either because physical forcing ups the Continuum, or because spacetime is made to go beyond the Continuum. Aug 21, 2022 at 4:45
• Cardinals are well-ordered just like ordinals, and decidedly not like space. "Temporal intuition" suggests just linear order, not necessarily well-order, as plays on infinite divisibility, like Zeno's, show. And well-orders beyond the countable are highly artificial. So the time/space divide goes more along the lines of linear order/no order than ordinal/cardinal. And even if Conway's construction is preferable to ZFC construction that concerns only the construction, not the objects constructed. Whether we should stop at reals for spacetime continuum or go on is orthogonal to that. Aug 21, 2022 at 8:49
• @Conifold, cardinals are only well-ordered under various provisos; the choiceless hierarchy gives us cardinals with no normal/regenerate ordinal counterparts, but we have to use stuff like Scott's trick to have the rank function go through, so that the choiceless cardinals can be compared without assuming ordinal height differences. So by itself, the concept of cardinal numbers does not require well-ordering. Aug 21, 2022 at 12:27