HOD1: Inner Model via Existential Second-Order Definability

• HOD1 is an inner model defined via existential second-order formulas, serving as a variant to classical HOD in set theory.
• It compares ordinary HOD with a model generated by extended definability, illustrating independence phenomena under supercompact assumptions.
• Forcing and coding strategies demonstrate that the equation HOD = HOD1 remains undecidable in ZFC augmented with large cardinal axioms.

Searching arXiv for papers specifically about HOD1 as the inner model $C(\Sigma^1_1)$. $\mathrm{HOD}_1$ is an inner model of set theory defined by

$\mathrm{HOD}_1 = C(\Sigma^1_1),$

where $\Sigma^1_1$ is the existential fragment of second-order logic, consisting of formulas of the form

$\exists R_1\ldots \exists R_n\,\varphi,$

with $\varphi$ first-order. In the $C(\mathcal L)$-framework, $\mathrm{HOD}_1$ is obtained by replacing ordinary first-order definability in a constructibility-style hierarchy with existential second-order definability. It is therefore a definability-based variant of HOD rather than the usual class of hereditarily ordinal-definable sets. The central current result is that the statement $\mathrm{HOD}=\mathrm{HOD}_1$ is independent of ZFC plus “There is a supercompact cardinal” (Magidor et al., 25 Aug 2025).

1. Definition and logical setting

The paper "New inner models from second order logics" defines $\mathrm{HOD}_1$ by

$\mathrm{HOD}_1$0

with

$\mathrm{HOD}_1$1

This places $\mathrm{HOD}_1$2 inside the general $\mathrm{HOD}_1$3-construction, where one forms an inner model by closing stages under definability in an extended logic $\mathrm{HOD}_1$4 rather than only under first-order definability (Magidor et al., 25 Aug 2025).

In this setup, $\mathrm{HOD}_1$5 is not introduced as an ad hoc modification of HOD, but as one instance of a broader program studying inner models arising from stronger logical resources. The logic $\mathrm{HOD}_1$6 is stronger than ordinary first-order logic, yet much weaker than unrestricted second-order logic. This makes $\mathrm{HOD}_1$7 a natural nearby variant of HOD from the standpoint of definability theory.

The paper does not supply a further equivalent formulation beyond this syntactic one. The relevant formal content is precisely that $\mathrm{HOD}_1$8 is the $\mathrm{HOD}_1$9-model associated with existential second-order formulas (Magidor et al., 25 Aug 2025).

2. Relation to ordinary HOD

The usual $\mathrm{HOD}_1 = C(\Sigma^1_1),$0 is the class of hereditarily ordinal-definable sets, defined using ordinary first-order formulas with ordinal parameters. By contrast, $\mathrm{HOD}_1 = C(\Sigma^1_1),$1 is built from existential second-order definability. The two constructions are therefore adjacent in motivation but distinct in mechanism (Magidor et al., 25 Aug 2025).

The paper is explicit that $\mathrm{HOD}_1 = C(\Sigma^1_1),$2 is a variant of HOD rather than HOD itself. It also notes that $\mathrm{HOD}_1 = C(\Sigma^1_1),$3 “extends e.g. $\mathrm{HOD}_1 = C(\Sigma^1_1),$4,” although no structural theorem about that extension is developed there (Magidor et al., 25 Aug 2025). What is developed in detail is the comparison problem

$\mathrm{HOD}_1 = C(\Sigma^1_1),$5

A crucial point is that the paper does not prove any absolute inclusion such as

$\mathrm{HOD}_1 = C(\Sigma^1_1),$6

in ZFC. Instead, it proves that equality itself is undecidable over a strong large-cardinal background. This shifts the comparison from a straightforward inclusion question to a relative-consistency problem (Magidor et al., 25 Aug 2025).

This suggests that replacing first-order definability by existential second-order definability does not have a uniform effect on the resulting inner model across all universes of set theory. The comparison is sensitive to forcing and coding constructions rather than being fixed by the formal strength of the logic alone.

3. Independence of $\mathrm{HOD}_1 = C(\Sigma^1_1),$7

The central theorem proved in Section 3 of "New inner models from second order logics" is: $\mathrm{HOD}_1 = C(\Sigma^1_1),$8 More precisely, assuming the consistency of ZFC with a supercompact cardinal, both

$\mathrm{HOD}_1 = C(\Sigma^1_1),$9

and

$\Sigma^1_1$0

are consistent with that theory (Magidor et al., 25 Aug 2025).

This refines an earlier result cited in the paper, namely that $\Sigma^1_1$1 is already independent of ZFC. The new contribution is that adjoining large cardinal axioms of this strength does not resolve the question (Magidor et al., 25 Aug 2025).

The significance is methodological as much as structural. Large cardinals often settle or calibrate the consistency strength of canonical statements about inner models, but here even the presence of a supercompact cardinal does not determine whether existential second-order definability collapses back to ordinary HOD. That establishes $\Sigma^1_1$2 as a genuinely nontrivial variation in the landscape of definability-based inner models.

4. Model where $\Sigma^1_1$3

For the positive consistency direction, the proof uses Menas’s model of $\Sigma^1_1$4 with a supercompact cardinal. The paper states that in this model every set is coded by the class of ordinals $\Sigma^1_1$5 such that

$\Sigma^1_1$6

and that this coding pattern can be captured by $\Sigma^1_1$7 (Magidor et al., 25 Aug 2025).

The argument is then straightforward in outline. Since every set is recoverable from an ordinal coding pattern visible to $\Sigma^1_1$8, one obtains

$\Sigma^1_1$9

Because the same model also satisfies $\exists R_1\ldots \exists R_n\,\varphi,$0, it follows that

$\exists R_1\ldots \exists R_n\,\varphi,$1

there (Magidor et al., 25 Aug 2025).

The role of the coding

$\exists R_1\ldots \exists R_n\,\varphi,$2

is central. The proof does not rely on a direct structural collapse of existential second-order definability to first-order ordinal definability; rather, it exploits a universe in which every set has already been encoded into a sufficiently transparent ordinal pattern. This suggests that equality between HOD and $\exists R_1\ldots \exists R_n\,\varphi,$3 can be forced by making all sets uniformly recoverable from coding visible to the richer definability apparatus.

5. Model where $\exists R_1\ldots \exists R_n\,\varphi,$4

The negative consistency direction is more elaborate. Starting from a universe with

$\exists R_1\ldots \exists R_n\,\varphi,$5

and a supercompact cardinal $\exists R_1\ldots \exists R_n\,\varphi,$6, the proof constructs a model $\exists R_1\ldots \exists R_n\,\varphi,$7 in which a real $\exists R_1\ldots \exists R_n\,\varphi,$8 satisfies

$\exists R_1\ldots \exists R_n\,\varphi,$9

Hence

$\varphi$0

in that model (Magidor et al., 25 Aug 2025).

The construction uses a sequence $\varphi$1 of the first weakly compact cardinals above $\varphi$2, with

$\varphi$3

It first “marks” the $\varphi$4 by violating GCH exactly at each $\varphi$5, while preserving weak compactness of each $\varphi$6 and introducing no new weakly compact cardinals. Then, over the resulting universe, it performs a reverse Easton iteration adding Cohen subsets at inaccessible cardinals below $\varphi$7. Finally, for those $\varphi$8, it destroys weak compactness of $\varphi$9 by adding a $C(\mathcal L)$0-Souslin tree $C(\mathcal L)$1 (Magidor et al., 25 Aug 2025).

The crucial technical claim is

$C(\mathcal L)$2

To prove this, the paper compares $C(\mathcal L)$3-truth across four forcing-related models. For every $C(\mathcal L)$4-formula $C(\mathcal L)$5 and $C(\mathcal L)$6, the following are equivalent: $C(\mathcal L)$7 The forward implications use upward persistence of existential second-order truth under forcing extensions, while the converse direction uses homogeneity of the forcing from $C(\mathcal L)$8 to $C(\mathcal L)$9 (Magidor et al., 25 Aug 2025).

Because $\mathrm{HOD}_1$0 remains unchanged through the critical step, the real $\mathrm{HOD}_1$1 is not seen by $\mathrm{HOD}_1$2. Ordinary HOD, however, can detect both which ordinals are the marked $\mathrm{HOD}_1$3 and which of them have lost weak compactness. Thus $\mathrm{HOD}_1$4 becomes ordinal-definable in $\mathrm{HOD}_1$5, yielding

$\mathrm{HOD}_1$6

(Magidor et al., 25 Aug 2025).

6. Conceptual significance

The paper treats $\mathrm{HOD}_1$7 as a nearby but nontrivial perturbation of HOD. The question

$\mathrm{HOD}_1$8

therefore tests how robust hereditary ordinal definability is under a modest strengthening of the underlying logic from first-order to existential second-order definability (Magidor et al., 25 Aug 2025).

The independence theorem shows that even strong large cardinal assumptions do not settle whether this strengthening changes the resulting inner model. In one direction, coding can force complete agreement. In the other, carefully arranged forcing extensions allow ordinary ordinal definability to recover information that remains invisible to the $\mathrm{HOD}_1$9-construction (Magidor et al., 25 Aug 2025).

A plausible implication is that the decisive issue is not merely the formal expressive power of $\mathrm{HOD}=\mathrm{HOD}_1$0, but the interaction between that expressive power and the ambient coding architecture of the universe. In particular, homogeneity and persistence of $\mathrm{HOD}=\mathrm{HOD}_1$1-truth can prevent $\mathrm{HOD}=\mathrm{HOD}_1$2 from seeing distinctions that HOD can recover through ordinally definable global patterns.

The paper does not establish substantive comparisons between $\mathrm{HOD}=\mathrm{HOD}_1$3 and the other inner models studied there, such as $\mathrm{HOD}=\mathrm{HOD}_1$4, $\mathrm{HOD}=\mathrm{HOD}_1$5, or $\mathrm{HOD}=\mathrm{HOD}_1$6. Its treatment of $\mathrm{HOD}=\mathrm{HOD}_1$7 is separate and focused almost entirely on the comparison with ordinary HOD (Magidor et al., 25 Aug 2025).

7. Position within the study of inner models from extended logics

Within the broader $\mathrm{HOD}=\mathrm{HOD}_1$8-program, $\mathrm{HOD}=\mathrm{HOD}_1$9 occupies a natural intermediate position: stronger than purely first-order definability, but still governed by a sharply delimited logical fragment. The paper’s Section 3 does not develop a fine structure for $\mathrm{HOD}_1$0, nor does it derive large-cardinal content internal to $\mathrm{HOD}_1$1. Its sole substantive theorem about the model is the undecidability of equality with HOD under a supercompact cardinal assumption (Magidor et al., 25 Aug 2025).

That narrow focus is itself informative. It indicates that $\mathrm{HOD}_1$2 is being used as a test case for a general phenomenon: changing the logic in a constructibility-style hierarchy can produce inner models whose relationship to classical definability-based models is not settled by ZFC, and not even by strong large cardinal hypotheses.

In that sense, $\mathrm{HOD}_1$3 functions as an especially clear example of the sensitivity of inner-model formation to the choice of definability notion. The paper’s final conclusion is accordingly not that $\mathrm{HOD}_1$4 has a fixed structural relation to HOD, but that the equation

$\mathrm{HOD}_1$5

remains open to forcing variation even in universes with supercompact cardinals (Magidor et al., 25 Aug 2025).