The standard view of spacetime is that it's a manifold based on the real numbers. The view that spacetime is discrete is also sometimes considered. Much more rarely, the idea that spacetime is a "manifold" based on a field like the hyperreals that extends the reals gets considered. But I have never heard anybody wonder whether spacetime might not have a dense but countable structure, like the structure of the rational or algebraic numbers.
I see no philosophical benefits to thinking spacetime might have such a structure. I wonder, though, whether we have any philosophical or empirical reason to think it doesn't. Perhaps there is a consistent first-order axiomatization of the mathematics that physicists actually use. If so, then the downward Loewenheim-Skolem theorem will model it within a countable structure. The algebraic numbers are unlikely to be rich enough, though.
3 comments:
I’ve always hoped that space-time would turn out to have the sort of countably infinite structure you described. I’ve hoped this only because it would be very strange and somehow cool. (This is the only benefit I can see.)
One of the reasons I think it would be strange is that my concept of the real number line seems to be derived from my experience of space-time. But the concept of a dense but countable structure is very difficult to grasp. So the relatively easy and obvious concept would then have no object corresponding to it in reality, whereas the difficult and obscure concept would correspond to some reality that every person in this world experiences all the time. Of course, this isn’t impossible. But it is strange.
I find the real numbers harder to grasp than the rational ones.
Hello.
I came at this post by some chance, and I'm leaving this comment just to say I think there are some consequences to consider space-time as countably dense while not "just discrete".
"Simple" discreteness (i.e. we can represent points in space-time with finite precision) imply the calculus formulations of physical principles will only be approximations to reality - the difference between the trapezoidal rule and actual integrals. As we can see from finite precision computer simulations, limited precision can lead to strange behaviour of physics.
On the other hand, with a countably dense space-time, the numbers we get from integration and differentiation can be reinterpreted to stay valid, via standard connections formed in measure theory; all physical laws we have can still be considered smooth, so we can keep the formulas based on real numbers, that, apart from other epistemological concerns, it is possible to formulate a true theory of physics using calculus.
Now why don't we want space-time to be real? Very peculiar things can happen due to axiom of choice, such as the Banach-Tarski paradox and the ability to predict the future. A countably dense space-time is sufficient to avoid these things.
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