Blass's finite-modification classes and parameter-HOD model

Work in the metatheory with a countable transitive $M\models \mathrm{Z}\mathrm{F}+V=L$. Thus $M$ satisfies Choice by the canonical constructible well-order, and this is the only ambient source of Choice in the setup. Force over $M$ with

$P=\mathrm{Fn}(\omega \times \omega ,2),$

the finite partial functions ordered by reverse inclusion. If $G$ is $M$-generic, define the mutually Cohen-generic reals

$a_n=\{k<\omega :(\bigcup G)(n,k)=1\}(n<\omega )\mathrm{.}$

For any real $x\subseteq \omega$, its finite-modification class is

$\delta (x)=\{y\subseteq \omega :x\mathrm{△}y\text{ is finite}\},$

where $\mathrm{△}$ is the symmetric difference of The difference $a\setminus b$, the symmetric difference $a\mathrm{△}b$, and the complement $X\setminus a$ relative to a set $X$. Put

$f(n)=\{\delta (a_n),\delta (\omega \setminus a_n)\}$

and

$S={\bigcup }_{n<\omega }(\delta (a_n)\cup \delta (\omega \setminus a_n))\cup \{f\}\mathrm{.}$

Blass's class $N$ consists of all $x\in M[G]$ such that every member of $\mathrm{TC}(\{x\})$ is uniquely definable in $M[G]$ from $f$, finitely many members of $S\setminus \{f\}$, and finitely many ordinal parameters. This is the convention denoted $\mathrm{HOD}(S)$, or “HOD over $S$,” in the source. It is important that $S$ acts as a reservoir of finitely many parameters, not as one pointwise named parameter: $S$ is definable from the single permitted parameter $f$, while individual definitions may also use only finitely many reals from its displayed union.

The range

$R=\mathrm{ran}(f)=\{\{\delta (a_n),\delta (\omega \setminus a_n)\}:n<\omega \}$

is therefore a canonically enumerated family of pairs in $N$. The later term Blass model refers to this parameter-HOD class $N$. The ordinary HOD coding convention is that of Ordinal definability and HOD; the usual HOD inner-model proof from HOD as an inner model and comparison with L must be relativized to this finite-parameter reservoir before any ZF conclusion about $N$ is used.