Using transfinite numbers in a scifi story

Question

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 ℵ₂^ℵ₃ = ℵ₄, assuming the GCH here. Then the tetration goes to ℵ_ω4, 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 2^ℵ₀, here.

Answer 1

I think the problem here is a frame challenge one. You're in essence asking us to.design your universe, even though the question you ask is about stability.

The mathematics are a framework. Within that framework and the bounds of SciFi (even hard grounded SciFi), there is so much flexibility, that the only answer I can think of to "Is it stable" is "depends on your design". If you then ask what design to pick, you're asking us to choose your universe.

Its so open, that in effect you are asking which fundamental structures to choose. maybe update the question to focus on the problems you are trying to work round by doing so? Right now it reads more like "I just arbitrarily want my universe to be a mathematical universe with infinities, which one shall I choose so it can be that?" (and I appreciate that's probably not how it is, but its how the question comes over)

Answer 2

/ 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.../

I answer your problem of space, time and physical law with Milton and the stuff of stories.

https://www.poetryfoundation.org/poems/45718/paradise-lost-book-1-1674-version

Is this the Region, this the Soil, the Clime,

Said then the lost Arch-Angel, this the seat

That we must change for Heav'n, this mournful gloom

For that celestial light? Be it so, since he

Who now is Sovran can dispose and bid

What shall be right: farthest from him his best

Whom reason hath equald, force hath made supream

Above his equals. Farewel happy Fields

Where Joy for ever dwells: Hail horrours, hail

Infernal world, and thou profoundest Hell

Receive thy new Possessor: One who brings

A mind not to be chang'd by Place or Time.

Your agents have moved either by design, or compulsion or both, and find themselves different in different place. But their minds are "not to be chang'd by Place or Time" and Satan here demonstrates his /ability to directly resolve the CH by intuition/. Perhaps cognition operates by its own rules? A matter for their scientists who can wrap heads around matters of κ and λ. There is a place for those scientists and their musings and speculation. You will need a Satan character, who recognizes what has happened, and recognizes himself as himself, and moves on with his agenda.