Questions to which somebody probably knows the answer, but not me

9 April 2020

These are two questions that I can’t find a good answer to, even though I’m pretty confident they exist.

Are there other mathematical universes?

There are several axiom systems that can be used to formalize our mathematical knowledge. The most famous one is Zermelo–Fraenkel set theory, or ZFC, but there are others, such as Von Neumann–Bernays–Gödel set theory, or type theory. These all seem focused on defining a concept for sets, and from sets it’s possible to derive all the rest of mathematics. Not all of these theories give you exactly the same theorems, but the parts where they differ seem to mostly concern pathological infinites; broadly, you get the same results from any of them. (I’m not completely certain how “same result” is defined here, because it seems you wouldn’t literally get the same results from two axiom systems, but let’s assume you can define some sort of morphism between formulas in any pair of such axiom systems that preserves the provability of theorems).

The question is: are there axiom systems that give rise to non-trivial mathematics that cannot be made (at least substantially) isomorphic to ZFC?

I suspect the answer is no, and that there is some sort of concept akin to Turing completeness for foundational mathematical theories, where any axiom system complex enough can emulate other ones, but I’m far from certain.

See also Foundations of mathematics on Wikipedia

Why are entropic time and relativistic time parallel?

The theory of general relativity claims we live in a 4-dimensional universe, with three space dimensions, and one time dimension. These are distinguished by their metric sign; the metric sign of time is the opposite of that of space, and that’s basically the only difference as far as relativity is concerned, although it obviously has many implications, such as the speed of light being finite.

But there is another aspect to time: entropy. The second law of thermodynamics says that the entropy of a closed system can only increase, and so the entropy of the entire universe is increasing. So, if you look at an average region of spacetime of some fixed size centered around a point at $(x, y, z, t)$, it’ll have similar entropy to one centered at $(x’, y’, x’, t)$, but a lower entropy than one centered at $(x, y, z, t’ > t)$.

This relates to the position of the Big Bang: the Big Bang happened at one given time at all points in space, and was an entropy minimum. As you get furter from the 3D slice of spacetime where the Big Bang happened, entropy increases. This 3D slice of spacetime happens to be all of space at one point in time, so the perpendicular direction is time, and that’s where entropy increases.

The question is: why does entropy increase exactly along the relativistic time dimension? Or, I think equivalently, why did the Big Bang happen at one given time, in all points of space, and not, for instance, at all times, and along a 2D plane of space, with entropy increasing along the axis of space perpendicular to that 2D plane, instead of along the time axis? That’s just as valid of a 3D subspace of spacetime, after all.

Relativistic time and space dimensions are very similar, but they are not exactly the same. I assume some property of the time dimension provides the answer to the question, but I don’t know what it is.