In set theory, are there approaches to fine structure that give fine-structural models that do not satisfy the axiom of choice?
We can build fine-structural models above a given set (such as $\mathbb R$), treating the set as a black box (for example, see "Scales in $K(\mathbb R)$" by John Steel). However, in that case, we get choice modulo something that we do not treat fine-structurally.
That will not suffice to get fine-structural models with Reinhardt cardinals, as Reinhardt cardinals are not consistent with just small violations of choice. (A possible alternative is to treat the models as canonical extensions of their HODs, but plausibly their HODs will omit too much structure or overcomplicate things.) We might possibly even need choice-less fine-structural models to get to supercompact cardinals (Weak extender models for supercompactness without choice question), with ZFC models obtainable by forcing.
Canonical models of AD have the feel of being fine-structural models, with Wadge ranks giving an exquisitely precise structure. Their HODs (at least so far) are fine-structural models, and perhaps there is a way to treat the AD models themselves as fine-structural. (Fine-structural roughly means being built one small (or in a sense minimal) canonical step at a time, especially with condensation-like and other properties enabling a well-understood theory.)
A possible approach: One possibility is that instead of putting the extender sequence inside the model we are constructing, we just add (at appropriate points) the appropriate extenders to the model. If the model can recover the sequence, it becomes a conventional Mitchell–Steel model (possible alternative: HOD-like model). If it cannot, it may become a model of AD, until it grows enough to recover the sequence from parameters. At higher strengths, it may have multiple regions where it cannot recover the sequence, perhaps eventually enabling periodicity in the cumulative hierarchy.
Examples with AD
To illustrate how adding extenders might lead to models of AD, here are some models of AD based on extender sequences. I would be interested to hear more about similar constructions.
Under large cardinal axioms, I believe there is no minimal (in terms of which reals it contains) inner model elementarily embeddable into $L(\mathbb R)$. However, there is minimal canonical such model; its reals are precisely those computable from some $x_i$ where $x_0=0$ and $x_{i+1}=M_\omega^\#(x_i)$. The construction generalizes above an arbitrary real $r$ (setting $x_0=r$) and is single-valued (unlike the derived model construction) and depends only on the Turing degree of $r$, and among such constructions (if they are universally Baire) I suspect is minimal on a cone of Turing degrees. ("Such constructions" means given $r\in\mathbb R$, return countable $R\subseteq\mathbb R$ that depends only on the Turing degree of $r$, with $r\in R$ and $L(R)\models\text{AD}$.)
There should also be the minimal canonical inner model elementarily embeddable into the minimal inner model of $\text{AD}_\mathbb R$ containing all the reals. I suspect it is obtained as follows. Set $M_{0,\text{adr}} = 0$, and $M_{i+1,\text{adr}}(x)$ as the sharp of the minimal iterable inner model above $x$ with a limit $\delta$ of Woodin and of $<\delta$-strong cardinals, and with the model closed under $M_{i,\text{adr}}$ below $δ$. Set $R$ to the set of reals computable from some $M_{i,\text{adr}}(0)$, and the model to the minimal inner model $M$ with $\mathbb R^M=R$ and $∀i \, M_{i,\text{adr}}(R)\in M$. $M_{i+1,\text{adr}}(R)$ should have Wadge rank $\theta_i$ in $M$.
A few more notes:
- The earliest (rudimentarily closed) point at which AC (wellordering principle) would fail in the suggested construction should correspond to $L_1(R)$ where $R$ is the minimal elementary submodel of the $V_{ω+1}$ (assuming projective determinacy).
- The earliest point at which a canonical fine-structural inner model cannot parameter-free recover its extender sequence is $L[M_1^\#]$; if we do not require the model to contain all ordinals, the failure likely first happens around local Woodin cardinals.
- There might also be other approaches to fine-structure. For example, at least for smaller large cardinal strengths, I think we can use indiscernibles in place of measures (Inner model theory using indiscernibles question), and some natural ZFC models (such as $L[\text{Card}]$ under large cardinal axioms) are of that type.
