Under Reinhardt cardinals in ZF, the cumulative hierarchy exhibits a periodicity in that for large enough $λ$, certain properties of $V_λ$ depend on whether $λ$ is even vs odd. See Periodicity in the cumulative hierarchy (Goldberg and Schlutzenberg), Even ordinals and the Kunen inconsistency (Goldberg), and Choiceless cardinals and the continuum problem (Goldberg).
Are there results or conjectures for periodicity in the low levels of the cumulative hierarchy such as $V_{ω+2n}$ vs $V_{ω+2n+1}$?
Under ZF + DC, to what extent can the theory of $V_{ω+3}$ (resp. $V_{ω+4}$) resemble the theory of reals (resp. sets of reals) under AD?
Reinhardt cardinals are preserved under small forcing, and thus of limited informativeness for 'small' sets. However, for future canonical inner models with Reinhardt cardinals, I see no reason why periodicity would not begin with $V_ω$ and $V_{ω+1}$. While some had suggested (Large Cardinals Beyond Choice (Bagaria, Koellner, Woodin)) that Reinhardt cardinals may bring an anti-inner-model chaos, the implications so far (periodicity, well-orderability of sufficiently closed ultrafilters on ordinals, etc) suggest order rather than chaos. Also, under I0 at $λ$, the theory of $L(V_{λ+1})$ above $V_λ$ resembles that of $L(ℝ)$ under AD.
For another angle, the axiom of determinacy (AD) gives a good theory of reals and natural sets of reals, and we would like to extend the theory to higher types, and get something like The universe of hereditarily natural sets. We may want to proceed by analogy, but under AD the structural properties of reals are very different from those of sets of reals. However, periodicity gives us an opening. (Incidentally, AD also gives some periodicities such as $Σ^1_{2n}$ vs $Π^1_{2n+1}$.)
Some conjectures/extrapolations
Consider a possible universe $M⊨\text{ZF + DC}$ satisfying AD and its appropriate (undiscovered) analogs for higher types, embedded into $V⊨\text{ZFC}$, the universe of all sets, and with $M$ having a canonical theory but also in a sense being close to $V$. $Θ$ is the supremum of ordinals with a surjection from the reals in $M$.
* integers — bounded subsets of $ω$, can be well-ordered.
* reals — subsets of $ω$, unrestricted ($\mathrm{Ord}^ω⊂M$), no uncountable well-ordering of reals, sufficiently high Turing degrees are indistinguishable in that every set (in $M$) of Turing degrees contains or is disjoint from a cone.
* sets of reals — bounded subsets of $Θ$ (their Wadge rank is $<Θ$; under $\text{AD}^+$, sets of reals have $∞$-Borel codes, but without $\text{AD}^+$ it is unclear if sets of reals can be encoded by sets of ordinals), tightly restricted (for example, every subset of $ω_1$ is constructible from a real), prewellordered under Wadge rank.
Extrapolation:
* sets of sets of reals — relatively unrestricted; perhaps encoded by subsets of $Θ$. I think we can put enough of them into $M$ to make $(Θ^+)^M = Θ^+$; I lean towards $V⊨\text{GCH}$ and $Θ^+=ω_2$. Plausibly, every set of ordinals of cardinality $Θ$ is covered by a cardinality $Θ$ set of ordinals in $M$. No Wadge rank-like prewelleordering. Every set (in $M$) of $\mathbf{Δ^2_1}$-reducibility degrees of sets of sets of reals might contain or be disjoint from a cone.
* elements of $V_{ω+4}$ — tightly restricted, structural theory somewhat analogous to sets of reals.
* and so on, with a period two.
The layers thus alternate between order and freedom. Under the axiom of choice, if we want each $ω_i$ to have sufficiently strong reflection properties, two layers are collapsed into one (i.e $V_{ω+2n+1}^M$ corresponds to $V_{ω+n+1}$). The theory of sets of reals under AD does not appear to correspond to the theory of sets of reals under AC and $\operatorname{Col}(ω_1, <\text{inaccessible})$ (and its extension to higher types in Independence through forcing vs generic collapses). Instead, in the latter, projective sets of subsets of $ω_1$ behave somewhat like projective sets of reals.
As their large cardinal strengths increase, models with a canonical theory can capture more and more aspects of $V$ (without an anti-large cardinal assumption). For example, there is an iterate $N$ of the minimal inner model with a proper class of measurable cardinals (assuming its sharp exists) such that uncountable cardinals in $V$ are exactly the measurable cardinals in $N$ and their limits. Woodin cardinals (plus iterability) allow genericity of arbitrary sets over iterates. Plausibly, around supercompact cardinals, appropriate covering properties will appear. Without AC, covering properties need not be anti-large cardinal assumptions (unlike the covering lemma for $L$). For example $L(ℝ)$ includes all reals, but is canonical if it satisfies AD.
However, the above is just my vision, and the answer does not have to be based on it.
