I have been bandying about some exotic mathematics ideas as the basis for a hard scifi setting, and I wonder what technology/alien life would "make sense," here, given the physics arising from the mathematics...
The core premise is that the universe has an infinitary logic assigned to it, where infinite sequences in the logic ground laws of physics (as something akin to, if not identical with, exceptionless regularities). Following Smolin's lead, the idea is that at t = 0, κ & λ for the logic ℒ(κ,λ) are 0,0, with the initial expansion over the t-interval 0:1 being a consequence of ℒ(κ,λ) shifting to ℒ(ω,ω), such that a further shift in time (in the future) to ℒ(ω1,ω) results in accelerated expansion, so that the next shift has another cosmological effect that changes the laws of physics again, etc. The equation I assigned to these shifts technically has κ & λ go to ω4 & something else next, at the start of the story.
Things I need to avoid: (a) an immediate Big Rip; (b) photon-induced overheating (the speed of light changes during shifts, becoming intrinsically faster, so more photons hit things in shorter periods of time). Things I need to have: (a) some means for aliens to use a shift to travel between parallel universes; (b) interstellar travel made possible by the increased speed of light.
I also want aliens who engage in mathematics primarily involving exotic hyperoperations, e.g. via hyperoperators with negative indexes. For technical reasons, these would not be the inverses of the positive-index operators. My "assumption" is that these aliens (also "escapees" from another world destroyed by a Big Rip) would come from a universe where the very function that shifts the values of ℒ(κ,λ) there uses negative hyperoperations to compute κ & λ.
So another problem: how to describe the transformation of aliens from one sphere with different laws of physics, into beings compatible with the laws of physics of the world they've escaped into? I could have it that you can only shift into a world whose ℒ(κ,λ)-signature "lines up with" the signature of the world you're leaving.
I've considered trying to make most of the physics involved in these processes depend on inflatons, such that either (a) these particles reemerge during shifts (embody them, even) or (b) some of these particles did not decay after the initial expansion but can be somehow "mined" from within black holes, and so the occurrent shift is part of what allows "black hole mining" (the specific idea I have is that cosmic strings exist and can be "plugged into" black holes to form the "mining" conduit; see e.g. this article).
More higher-level mathematics stuff: have κ be the number of time dimensions and λ the number of space dimensions. Then spacetime being perceived as continuous is an elaborate consequence of the value of 𝔠 according with the Continuum Hypothesis. I was thinking that the ability to directly resolve the CH by intuition might even be one of the results of the shift for cognitive agents...
Is the universe I am trying to describe, too unstable to survive?
EDIT: here's the start of the shift equation:
{{{{0 ↑0 ℵ0} + ℵ1} × ℵ2}ℵ3} ↑↑ ℵ4}
So ℵ2ℵ3 = ℵ4, assuming the GCH here. Then the tetration goes to ℵω4, which is supposed to be the next shift. So technically κ goes to ω4 at the start of the story, which is "how" it is easier to "just see" the truth of 2ℵ0, here.