As far as I know, the axiom of the continuum (or: the initial instance of the infinitary powerset axiom scheme) is consistent with, but independent on, the axiom of choice. So it seems as if there should be a possible "version" of the continuum, the choiceless continuum (CC), which is not well-orderable. The CC is thus far similar to amorphous sets/cardinals, which subsist as topological determiners of their own; so there should be a set of topologies for/from the CC.
Question: is there anything contradictory about a continuum that at some times is subject to the choice axiom and at other times is not? Call this an order-theoretically oscillating continuum, or orc for short. I envisage at least two ways this might play out:
- Some temporal continuum starts in one state (well-ordered or coamorphous(?)) during the initial expansion, but that expansion's special incidence is occasioned by the continuum switching states, whereafter the continuum is fixed in the other state.
- The orc is a proper orc, i.e. the oscillation is constant (and we have a pragmatic reason to use the word "oscillation"). In this event, though, if the oscillation was spaced out enough, would there be any way for us to empirically distinguish the choiceful from the choiceless periods?
Observation: I can dimly imagine a sort of (3):
- The switch is stretched out over an infinite period of time (in a sense, the world's continuum is re-computed for 1 × 2 × 3 × 4 × ... = ℵ0! = 2ℵ0 so that the "second" time the continuum is arithmetically generated, it switches to or from a choiceful/choiceless state). Or worse, if the continuum passes to an aleph-state at countable eternity, it is not as the continuum proper but the hypercontinuum of 2ℵ0!, which after another infinitely long time turns into another coamorphous value, etc.
Subproblem: if the number of possible particles in the universe changed when its continuum changed cardinality, it seems as if the conservation of substance would be undermined even if the number of particles lost/gained per period was incommensurate with the continuum's cardinality per period.
(Co)amorphous infinitesimals: hyperreal infinitesimals are reciprocals of hyperreal infinite numbers, and surreal infinitesimals are reciprocals of transfinite ordinals. The base surreal example is 1/ω, although there are then 1/ω1 or 1/ω/ω, etc., all the way to 1/ORD. If (co)amorphous cardinals are infinite, do they have reciprocals? It seems as if there is a continuum composed of infinitesimals alongside the one composed of real-valued functions (whether this spawns another kind of orc, one which switches from infinitesimal to real-valued composition, IDK...).