# Earlier core-model and Foundation reductions

Verbatim excerpt from the supplied research archive, transcript 4d718fd7-8d0d-4767-a35d-5f10b946a297.md, original lines 2453–2915 (23 September 2026, 05:13–06:07 UTC). This is the earlier research result invoked in Sections 8–9 of the revised manuscript. It is preserved as source material, not as instructions for the present task.

The corrected core-model package occurs in the 05:57 UTC response; the preceding response records the formulation that was then corrected. The Foundation-removal argument occurs at 06:07 UTC.

Full transcript SHA-256: c7f7418e8f9b9fd841bec032d6d0b37609fc92321f5efdcc8cbae5922d1973a2

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### User — 2026-09-23 05:13 UTC

T is conservative over T - SVC. Let V \models T - SVC + \exists \kappa (V\_{\kappa} \models \sigma). This should be a simple exercise of core model theory. Rough sketch: First collapse V\_{\kappa+1} to be countable, let r be a real coding this structure in V[H].

Case 1: If there is a sharp for the minimal inner model M\_r containing r and a measurable of Mitchell rank >0, then iterate out that measurable from M to get an inner model N with r and a proper class of measurables. Then HOD\_{V\_{\kappa+1}}^N \models T and V\_{\kappa} has full power set.

Case 2: If there is no such sharp, then HOD\_{V\_{\kappa+1}}^{K\_r} \models T and V\_{\kappa} has full power set.

---

### Assistant — progress update — 2026-09-23 05:14 UTC

I’ll check the core-model dichotomy and what the relative HOD construction preserves—particularly the proper class of measurables, DC, and the original power set of \(V_\kappa\).

---

### Assistant — progress update — 2026-09-23 05:18 UTC

The main point to verify is the passage back from a mouse over the new coding real \(r\) to relative HOD. Collapse homogeneity applies directly to the whole extension; using it inside \(K_r\) or the sharp-generated model also needs a comparison argument showing independence from the chosen code.

---

### Assistant — progress update — 2026-09-23 05:22 UTC

There may be a short route through that comparison step: the collapse is weakly homogeneous using finite permutations of the enumeration, which give Turing-equivalent coding reals. I’m checking whether the canonical mouse construction is invariant under those recodings. Separately, DC follows once the resulting model lies inside \(V\): enlarge the captured rank to one of uncountable cofinality, so a countable sequence of its parameters is itself an allowed parameter.

---

### Assistant — 2026-09-23 05:28 UTC

Your sketch gives the desired removal of SVC **provided we make the relativized core-model input explicit**. In particular, DC can be handled; it is not a further obstruction. I would enlarge the rank being captured and sharpen the formulation of the two cases.

Write
\[
T_0=\mathrm{ZF}+\mathrm{DC}+\text{“a proper class of measurables.”}
\]

### The core-model input

The precise package needed is a uniform assignment \(r\mapsto N_r\) such that:

\[
r\in N_r,\qquad
N_r\models\mathrm{ZFC}+\text{“a proper class of measurables”}
+\bigl(V=\mathrm{HOD}_{\{r\}}\bigr),
\]
and the **underlying membership universe** \(N_r\) is unchanged by the finite recursive recodings of \(r\) induced by permuting a collapse enumeration.

Your dichotomy is the intended way to obtain this package:

- In the mouse case, use an iterable \(r\)-mouse whose top measure has positive Mitchell rank. Here the relevant threshold is a **measure of rank at least \(1\)**—two coherent measures at the same critical point, or \(o(\mu)\ge2\) in cardinal notation. Iterating the upper measure through the ordinals leaves measurables behind. This is precisely the construction described by [Welch](https://mathoverflow.net/a/164002).
- Otherwise, use the relativized core model and its measure-absorption theorem. The choiceless ambient universe is not itself where one invokes ZFC core-model machinery: one works through \(\mathrm{HOD}_{\{r\}}\) and suitable set-generic extensions. The unrelativized version of exactly this absorption argument appears in the proof of [Hayut–Karagila, Theorem 3.11](https://arxiv.org/pdf/2212.07973).

Thus I would formulate the split using the appropriate **\(r\)-sword mouse**, rather than the somewhat ambiguous “sharp for the minimal inner model.”

### Finishing the argument, including DC

Suppose \(V\models T_0\), and fix the rank \(V_\kappa^V\) whose full second-order theory we want to preserve. Choose
\[
\lambda>\kappa+1,\qquad \operatorname{cf}^V(\lambda)>\omega,
\qquad A=V_\lambda^V.
\]
Collapse \(A\) to be countable, let \(r\) code its membership structure, and obtain \(N_r\) from the package above.

Put
\[
W=\mathrm{HOD}^{N_r}_{A\cup\{A\}},
\]
where parameters from individual elements of \(A\), as well as \(A\) itself, are allowed.

**1. Homogeneity gives \(W\subseteq V\).**

For \(\operatorname{Col}(\omega,A)\), finite permutations of the \(\omega\)-domain already witness weak homogeneity: move the domain of one condition away from that of another.

These permutations transform \(r\) into recursively interdefinable codes \(r_\pi\). By the recoding invariance of the canonical construction,
\[
N_{r_\pi}=N_r.
\]
Consequently, each fixed definition over \(N_r\) using ordinals and ground-model \(A\)-parameters is invariant under this weakly homogeneous group. The usual hereditary rank induction gives \(W\subseteq V\).

Conversely, \(A\subseteq W\). Therefore
\[
V_\lambda^W=V_\lambda^V,
\qquad
\mathcal P(V_\kappa)^W=\mathcal P(V_\kappa)^V.
\]

This is where recoding invariance matters; homogeneity of the collapse alone would not justify using an arbitrary \(r\)-dependent inner model.

**2. \(W\models\mathrm{SVC}(A)\).**

The relative-HOD presentation gives, for every nonempty \(Y\in W\), a surjection
\[
f:\alpha\times A\twoheadrightarrow Y
\]
in \(W\). Finite \(A\)-parameters can be coded into one member of \(A\).

**3. Ground-model DC gives DC in \(W\).**

Here is the useful general lemma:

> If \(W\subseteq V\models\mathrm{ZF}+\mathrm{DC}\), \(A=V_\lambda^V\subseteq W\), \(\operatorname{cf}^V(\lambda)>\omega\), and \(W\models\mathrm{ZF}+\mathrm{SVC}(A)\), then \(W\models\mathrm{DC}\).

To prove it, let \(R\in W\) be serial on \(Y\), and take \(f:\alpha\times A\twoheadrightarrow Y\) in \(W\). Starting from \(y_0\), let
\[
\xi_n=\min\{\xi<\alpha:\exists a\in A\;
y_n\,R\,f(\xi,a)\}.
\]
Use DC in \(V\) to choose corresponding \(a_n\) and set
\[
y_{n+1}=f(\xi_n,a_n).
\]
Since \(\operatorname{cf}(\lambda)>\omega\),
\[
s=\langle a_n:n<\omega\rangle\in A\subseteq W.
\]
Inside \(W\), the parameter \(s\) reconstructs the entire chain: the ordinal coordinates are always the least possible ones. Thus the chain belongs to \(W\).

**4. A proper class of measurables survives in \(W\).**

There is a convenient argument avoiding any assertion that measurability is downward absolute to relative HOD.

Let \(D=\mathrm{HOD}^{N_r}\). Since \(N_r\models V=\mathrm{HOD}_{\{r\}}\), Vopěnka’s theorem makes \(N_r\) a **set-forcing extension** of \(D\). See [Theorem 6.2.7 in Goldberg’s account](https://math.berkeley.edu/~goldberg/Papers/UABook.pdf). Lévy–Solovay then gives a proper class of measurables in \(D\).

Moreover,
\[
D\subseteq W\subseteq N_r.
\]
For sufficiently high \(D\)-measurables \(\mu\), a ground measure \(U\in D\) extends to \(W\) by
\[
U^W=\{B\in\mathcal P(\mu)^W:
\exists C\in U\;(C\subseteq B)\}.
\]
This set belongs to \(W\) and is the restriction of the small-forcing lift in \(N_r\). Hence \(W\models T\).

### Consequence

With that core-model package, your argument proves the structural-\(\Pi_2\) conservativity of \(T\) over \(T_0\). Combined with the preceding Prikry result, it yields
\[
T_0+\mathrm{SVC}+\mathrm{KW}_1^{*,\omega}
\quad\text{structural-}\Pi_2\text{-conservative over}\quad T_0.
\]

The remaining item I would explicitly import or prove is the **uniform, recoding-invariant relativized core-model package**, including its internal \(V=\mathrm{HOD}_{\{r\}}\) property. I have not fully checked that package here. The relative-HOD, power-set, measurable-tail, and DC deductions above are justified once it is supplied.

---

### User — 2026-09-23 05:42 UTC

Check the details of the core-model package.

---

### Assistant — progress update — 2026-09-23 05:42 UTC

I’ll check the package clause by clause: existence and measure absorption, invariance under recoding, and internal definability. I’ll distinguish what the standard theorems supply from what still needs an argument.

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### Assistant — progress update — 2026-09-23 05:46 UTC

The measure-absorption step has a direct precedent in a choiceless setting. I also found a useful simplification: the measurable-tail argument only needs \(N_r\) to be a set-forcing extension of its HOD. Definability of its extender sequence from a set parameter can supply this, without the stronger claim \(N_r\models V=\mathrm{HOD}(r)\).

---

### Assistant — progress update — 2026-09-23 05:51 UTC

The audit has narrowed to one delicate point: passing between the relative core model computed in \(\mathrm{HOD}(r)\) and the surrounding choiceless universe. Finite recoding of \(r\) is handled by change of base for mice, but that alone does not justify the iterability and forcing-absoluteness needed in the no-sword case.

---

### Assistant — progress update — 2026-09-23 05:54 UTC

There is a cleaner way to handle that point: run the dichotomy inside \(B=\mathrm{HOD}(r)\), which satisfies ZFC. Then use relative Vopěnka forcing to bring each ambient measure restriction into a ZFC extension of \(B\). This reduces the choiceless step to the ordinary forcing-absoluteness and maximality theorems for the relative core model.

---

### Assistant — 2026-09-23 05:57 UTC

Yes—the core-model package can be supplied, but I would formulate it slightly differently. The important changes are to **run the dichotomy inside \(\mathrm{HOD}(r)\)** and to replace the unnecessarily strong requirement \(N_r\models V=\mathrm{HOD}(r)\) by definability from some set parameter.

Here is the precise package and its verification.

### 1. Work inside a ZFC model

Write \(\mathcal V=V[H]\) for the collapse extension, and put
\[
B_r=\mathrm{HOD}^{\mathcal V}_{\{r\}}.
\]
This is a model of ZFC containing \(r\). All the following mouse constructions are initially performed **inside \(B_r\)**.

This avoids having to assume that a mouse which \(B_r\) regards as iterable is fully iterable in the surrounding choiceless universe.

Use the classical dichotomy:

- If \(B_r\) has an \(r\)-sword, take its least such mouse in the canonical mouse ordering and iterate its upper measure through the ordinals.
- Otherwise, take the core model over \(r\) below \(r\)-sword:
  \[
  N_r=K(r)^{B_r}.
  \]

Here “sword” means the configuration with an upper measure of Mitchell order \(1\), not merely \(o(\kappa)>0\). The upper measure can be iterated away while leaving lower measures at unboundedly many former critical points. This is the standard construction described by [Welch](https://mathoverflow.net/a/164002).

In particular, in the sword case,
\[
r\in N_r\subseteq B_r,\qquad
N_r\models\mathrm{ZFC}+\text{“a proper class of measurables.”}
\]

### 2. The no-sword case captures the ambient measurables

Fix an ambient measurable \(\mu\), witnessed by a \(\mu\)-complete ultrafilter \(U\), and let
\[
u=U\cap\mathcal P(\mu)^{B_r}.
\]
We do **not** assume \(u\in B_r\), nor that \(U\) is normal.

Since \(\mathcal P(\mu)^{B_r}\) is well-orderable in \(B_r\), code \(u\) by a set of ordinals \(c\). Relative Vopěnka gives
\[
C=\mathrm{HOD}^{\mathcal V}_{\{r,c\}}=B_r[g]
\]
for a set-generic extension of \(B_r\). The fixed-parameter version follows by replacing OD with \(\mathrm{OD}(r)\) in [Vopěnka’s Boolean-algebra proof](https://math.berkeley.edu/~goldberg/Papers/UABook.pdf), Theorem 6.2.7.

The relevant classical core-model facts, relativized to the fixed real \(r\), are:

1. Set forcing cannot create \(r\)-sword.
2. Consequently,
   \[
   K(r)^{B_r}=K(r)^C.
   \]
3. Countably complete normal ultrafilters over this core model are absorbed by it.

For completeness, the first fact has the usual product-forcing proof: alleged least swords in two mutually generic extensions remain iterable in the product extension; comparison identifies them, and the intersection of the mutually generic extensions puts the common mouse back in the ground. At this classical mouse level, the required countable iterability is \(\Pi^1_2\), so Shoenfield absoluteness applies. [Mitchell explicitly records this complexity in Proposition 2](https://arxiv.org/pdf/math/9210202).

Now form the well-founded ultrapower of \(B_r\) by \(u\) and derive a normal \(B_r\)-measure \(D\) on \(\mu\). Its restriction to the common core model is countably complete in \(C\), and maximality gives
\[
D\cap K(r)\in K(r).
\]
Thus \(\mu\) is measurable in \(K(r)\).

This is precisely the absorption mechanism used in the ZF argument of [Gitik–Hayut–Karagila, Lemma 4.2 and the ensuing proof](https://arxiv.org/pdf/2401.02757). Their printed argument is unrelativized; here we use its fixed-real version. All relevant embeddings fix \(r\). The underlying ultrafilter-maximality theorem is Mitchell’s, also recorded in the introduction to [Schimmerling–Steel](https://arxiv.org/pdf/math/9702206).

Since this works for every ambient measurable, the **one fixed model** \(K(r)^{B_r}\) has a proper class of measurables.

Importantly, this is \(K(r)^{B_r}\), not the unrelativized \(K^{B_r}\), which need not contain \(r\).

### 3. Finite recoding invariance

Suppose \(s=r^\pi\) results from a finite permutation of the collapse enumeration. Then \(r\) and \(s\) are uniformly mutually recursively definable, so
\[
B_r=B_s.
\]

One still needs change of base for mice; equality of the ambient HOD models alone is insufficient.

For these particular recodings, replace the bottom predicate \(r\) by its fixed recursive definition from \(s\), and conversely. This canonical rearrangement preserves the extender structure, iterability, and comparison ordering, and commutes with iteration. Hence:

- the least \(r\)-sword rearranges to the least \(s\)-sword;
- their upper-measure class iterates have the same pure universe;
- in the no-sword case, the relative core constructions likewise have the same pure universe.

Therefore
\[
N_{r^\pi}=N_r.
\]

This is equality after forgetting the different bottom-predicate presentations—not a claim that every low-level \(J\)-presentation is literally identical. The relevant rearrangement formalism appears in [Steel, §4](https://math.berkeley.edu/~steel/papers/sporejul07.pdf), especially the finite-change argument in the proof of Theorem 4.5. We need only that local argument, not that theorem’s AD hypothesis.

### 4. The internal definability requirement can be weakened

The presentation \(N_r=L[E,r]\) does not by itself justify
\[
N_r\models V=\mathrm{HOD}(r).
\]
We should not use that shortcut.

Instead, in the usual short-extender presentation, apply the fixed-real version of [Schlutzenberg’s Theorem 1.1](https://doi.org/10.1017/jsl.2024.27). The extender sequence is definable from the bottom segment
\[
m=N_r\!\restriction\omega_1^{N_r},
\]
including its real predicate. Consequently,
\[
N_r\models V=\mathrm{HOD}_{\{m\}}.
\]
The relativization keeps the bottom real fixed throughout; it requires no additional large-cardinal hypothesis.

Using AC in \(N_r\), code \(m\) by a set of ordinals \(t\). Then
\[
N_r\models V=\mathrm{HOD}_{\{t\}}.
\]
Vopěnka, now applied **internally to \(N_r\)**, yields
\[
N_r=D[g],\qquad D=\mathrm{HOD}^{N_r},
\]
for set forcing.

That is exactly what the measurable-tail argument needs.

### Consequence for our construction

For
\[
W=\mathrm{HOD}^{N_r}_{A\cup\{A\}},
\]
we have \(D\subseteq W\subseteq N_r\). Lévy–Solovay gives a proper-class tail of measurables in \(D\); their canonical small-forcing lifts restrict to measures belonging to \(W\).

Thus the corrected package supplies all the core-model ingredients:

\[
\boxed{
\begin{gathered}
r\in N_r\subseteq\mathcal V,\\
N_r\models\mathrm{ZFC}+\text{PC measurables},\\
N_{r^\pi}=N_r,\\
N_r\text{ is a set-forcing extension of }\mathrm{HOD}^{N_r}.
\end{gathered}}
\]

The fixed-real relativizations should be stated explicitly in a write-up, but they are not additional hypotheses. **DC is not needed for this core-model package; ground-model DC enters later, in the argument that \(W\) satisfies DC.**

---

### User — 2026-09-23 05:59 UTC

Apply the setup of my Mathoverflow post to get rid of Foundation in the base theory.

---

### Assistant — progress update — 2026-09-23 05:59 UTC

I’ll check how your post handles the non-well-founded part of the starting model, then see whether the measurable-cardinal and DC arguments survive that reduction.

---

### Assistant — progress update — 2026-09-23 06:03 UTC

The key simplification is to use the well-founded kernel at the end. Once the labelling construction gives \(\mathrm{KW}_1^{*,\omega}\) in the model with atoms, fibers of a surjection from \(\kappa^\omega\) give pure-set labels for the original structure. The kernel then inherits DC and the measurable cardinals, and the labelled structure remains full for second-order quantification.

---

### Assistant — 2026-09-23 06:07 UTC

Yes. Using the atomization setup of your post, we obtain the stronger statement:

\[
\boxed{
\begin{gathered}
\mathrm{ZF}+\mathrm{DC}+\mathrm{PCMeas}
+\mathrm{SVC}+\mathrm{KW}_1^{*,\omega}\\
\text{is structural-}\Pi_2\text{-conservative over}\\
(\mathrm{ZF}-\mathrm{Foundation})+\mathrm{DC}+\mathrm{PCMeas}.
\end{gathered}}
\]

There is one important modification: we should **not immediately descend to \(L(\mathcal M)\)**, since that could lose DC and the measurable cardinals. Instead, retain the full hierarchy over an atom copy, perform our construction there, and take the pure kernel at the end.

### 1. Atomize the second-order structure

Suppose
\[
V\models(\mathrm{ZF}-\mathrm{Foundation})+\mathrm{DC}+\mathrm{PCMeas}
\]
contains a full second-order model \(\mathcal M\models\theta\).

Use your two-sorted presentation, with \(S_1\) representing all subsets of \(S_0\), and replace its underlying objects by atoms. This is the structural setup of [your post](https://mathoverflow.net/a/506933).

Let \(A\) be the resulting set of atoms, and work in the full cumulative hierarchy \(\mathscr H(A)\) formed using the original universe’s power sets. Then
\[
\mathscr H(A)\models\mathrm{ZFA}+\mathrm{DC}+\mathrm{PCMeas}.
\]

Indeed:

- DC transfers because a sequence of elements of a set has bounded \(A\)-rank.
- The pure kernel has exactly the original subsets of every ordinal. Measures on ordinals are pure, so the ambient measurable cardinals remain measurable.
- The atomized structure is still full for second-order quantification.

Choose a sufficiently large
\[
B=V_\lambda(A),\qquad
\operatorname{cf}^{\mathscr H(A)}(\lambda)>\omega,
\]
containing the structure and its relevant power sets.

### 2. Run the preparation and labelling construction with atoms

The core-model preparation has the following ZFA version.

Collapse \(B\) and code its membership structure, including the distinguished atoms and equality of indices, by a real \(r\). Apply the core-model package in the pure kernel of the collapse extension, obtaining \(N_r\).

Now form the **ZFA companion of \(N_r\)**: internally adjoin the coded atom-index set, construct the hierarchy over those formal atoms, and relabel them by the actual atoms using the collapse enumeration. Its pure kernel is \(N_r\), it satisfies choice, and it contains the original \(B\).

Finite recoding gives the same companion: the pure model \(N_r\) is unchanged, and the induced bijection between formal atom indices belongs to \(N_r\). If the enumeration has repetitions, take the quotient by the coded equality relation.

Consequently,
\[
W=\mathrm{HOD}_{B\cup\{B\}}^{\text{companion}}
\]
descends to \(\mathscr H(A)\), contains \(B\), and satisfies
\[
\mathrm{ZFA}+\mathrm{DC}+\mathrm{PCMeas}+\mathrm{SVC}(B).
\]

The arguments are the same as before: ground closure of \(B\) supplies DC, while the pure kernel and its HOD supply the canonically lifted measures.

Our DC-preserving Prikry labelling construction also works over ZFA. Names are well-founded above the atoms; the target labels may be atoms just as well as sets. The bounded-presentation argument still gives surjections onto **all** sets, including atomful ones. Thus we obtain
\[
U\models
\mathrm{ZFA}+\mathrm{DC}+\mathrm{PCMeas}
+\mathrm{SVC}+\mathrm{KW}_1^{*,\omega},
\]
while preserving the protected power sets. In particular, \(\mathcal M\) remains a full second-order model of \(\theta\) in \(U\).

### 3. Purify the structure by taking fibers

This is the useful simplification.

Let \(M\) be the combined domain of the two-sorted structure. In \(U\), choose
\[
f:\kappa^\omega\twoheadrightarrow M
\]
and define
\[
g(m)=f^{-1}\{m\}.
\]

The map \(g\) is injective, and every \(g(m)\) is pure: it is a set of sequences of ordinals. Hence \(g[M]\), together with all transported relations, belongs to
\[
P=\operatorname{Ker}(U).
\]
Thus \(P\) contains a copy \(\mathcal M^\flat\) of the original relational structure.

Crucially, this copy is **full**. If
\[
Y\in P,\qquad Y\subseteq g[S_0],
\]
then \(g^{-1}[Y]\) exists in \(U\). By preservation of the original power set, it is represented by some \(p\in S_1\). The transported incidence relation therefore represents \(Y\) in \(\mathcal M^\flat\).

Conversely, every represented subset of \(g[S_0]\) is pure and belongs to \(P\). Therefore
\[
P\models\mathcal M^\flat\models\theta
\]
with full second-order semantics.

Notice that \(g\) itself need not belong to \(P\); its range and the transported relations do.

### 4. The pure kernel retains the entire target theory

We have \(P\models\mathrm{ZF}\), and:

- **DC:** a DC sequence through a pure set is itself pure.
- **PCMeas:** \(P\) and \(U\) have the same ordinals, subsets of ordinals, and measures on ordinals.
- **\(\mathrm{KW}_1^{*,\omega}\):** they have the same \(\kappa^\omega\), and a surjection from \(\kappa^\omega\) onto a pure set has pure graph.
- **SVC:** if \(C\) is an SVC seed in \(U\), fiber-code it by a pure set \(C^\flat\). Surjections
  \[
  \alpha\times C^\flat\twoheadrightarrow X
  \]
  onto pure \(X\) have pure graphs, so \(P\models\mathrm{SVC}(C^\flat)\).

Thus \(P\) satisfies the advertised target theory and contains a full second-order model of \(\theta\). This proves the conservativity statement.

So Foundation is removed from the base by your structural setup, with **fiber purification replacing the final atom-removal step**. No separate Jech–Sochor or DC-transfer theorem is needed.

