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A model of Smolin's postulate

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PostPosted: Tue Jan 19, 2021 9:59 pm    Post subject: A model of Smolin's postulate Reply with quote

I'm referring to Lee Smolin and his idea of changing laws of physics.

So there is an article in the Stanford Encyclopedia of Philosophy on something called infinitary logic. The idea is to have a language allowing conjunctions, variables, and quantifiers with ranges indexed by transfinite numbers. The format reads (roughly) as L(k,l), where k,l are infinite ordinals like omega or perhaps very tall ordinals associated with very large cardinals.

Let us suppose that the universe has an L assigned to it. At t = 0, k,l are 0. But due to the passage of time, these values increase. The initial expansion of space then reflects this shift, in fact space is correlated with l while time itself corresponds to k. I'm writing this on a phone so unfortunately I can't justify this particular correspondence right now as it requires an equation I can't write via phone, but anyway the idea is that by now, time's ordinal is the initial ordinal for aleph-2. The rules of dimensional perception say that we can see complete enclosures in the next dimension down from the one we're "on," like we can see all sides of polygons at once and if we were four-dimensional we could observe polyhedra likewise. So if the Continuum is aleph-1, then we can "see" continuity due to us being in time that is higher-dimensional.

Now, this model is supposed to explain the accelerated expansion of space as a result of a shift in L for our universe. Modulo Smolin's postulate, the model predicts, therefore, an infinite sequence of shifts, each of which "sort of" alters the "laws of physics."

At worst, this could make for a good hard scifi novel called THE SHIFT Razz

EDIT: according to the equation, btw, the next switch would be to aleph-4, not aleph-3, and the one after would be to aleph-(omega-4). So perhaps events much more extreme than accelerated expansion. Does this make the model into a testable hypothesis? I hope so!

EDIT 2: Here's the equation, partly:

{{{{0 ↑<sup>0</sup> ℵ<sub>0</sub>} + ℵ<sub>1</sub>} ℵ<sub>2</sub>}<sup>ℵ<sub>3</sub></sup>} ↑↑ ℵ<sub>4</sub>}
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