Seems like an attempt to reinvent the _Permutation City_ argument. In any event, as formulated, I don't think it quite works:
> Since there are an infinite number of universes, an infinite number of dimensions, or an infinite number of non-identical universe “cycles”, there are an infinite number of particles and particle arrangements.
The last claim does not follow without additional assumptions about the nature of universes.
Your point about the universes is true. Thanks for the feedback, I'll update the blog post. As an aside, I've never heard of Permutation City, seems interesting.
all these "infinite" have different meanings, in particular cardinality, and using just "infinite" makes it into a senseless babble that can have any meaning one would ascribe to it
Particle arrangements aren't the only property of a universe. Imagine a universe with 1 particle in it. Imagine a second universe with the same particle but +1 volume bigger. Not identical. Now imagine a third universe with the same particle arrangement with +3 volume bigger... And so on for all natural numbers. An infinite number of universes, each distinct, with the same particle arrangement. We can do this same trick for all sorts of universe properties (laws, energies, geometries).
If we define universes as the particle arrangements, then it would then logically follow that infinitely many unique universes implies infinitely many particle arrangements (a ≡ b; a; ∴ b).
(This is common to many Eternal Recurrence theories as well; need some assumption which prevents repeating patterns from 'running out the clock' and not exploring the full range of possibilities.)
> Since there are an infinite number of universes, an infinite number of dimensions, or an infinite number of non-identical universe “cycles”, there are an infinite number of particles and particle arrangements.
The last claim does not follow without additional assumptions about the nature of universes.