C₂(omega): A Countable Second-Order Inner Model

• C₂(omega) is an inner model defined using a fragment of second-order logic with quantifiers restricted to countable subsets, enhancing definability beyond that of L.
• It employs a truth-folded construction in L²_ω that ensures the model satisfies ZF and AC while embedding complex large-cardinal features like inner models with finitely many Woodin cardinals.
• Under strong large-cardinal hypotheses, C₂(omega) exhibits generic absoluteness, club determinacy, and bounded power set captures, positioning it between traditional L-hierarchies and full HOD.

Searching arXiv for the exact notation and primary paper. arxiv_search(query="(Magidor et al., 25 Aug 2025)", max_results=5, sort_by="submittedDate")

$C_2(\omega)$ is the inner model $C(\mathcal{L}^2_\omega)$ based on a fragment of second-order logic in which second-order variables range over countable subsets of the domain. In this construction, definability is strengthened beyond first-order $L$-style definability but remains weaker than full second-order definability, because the quantifiers range only over countable sets in the ambient universe $V$. The resulting model is transitive, contains all ordinals, is contained in the Chang model $C_{\omega_1\omega}$, and, under strong large-cardinal hypotheses, has generic absoluteness and substantial inner large-cardinal content (Magidor et al., 25 Aug 2025).

1. Logical basis

The logic underlying $C_2(\omega)$ is $\mathcal{L}^2_\omega$. Its first-order variables $x,y,z$ range over the domain $D$ of a structure $M$, while its second-order variables $C(\mathcal{L}^2_\omega)$0 range over countable subsets of $C(\mathcal{L}^2_\omega)$1 or countable subsets of $C(\mathcal{L}^2_\omega)$2. The crucial semantic feature is externality: in a structure $C(\mathcal{L}^2_\omega)$3, the quantifiers $C(\mathcal{L}^2_\omega)$4 and $C(\mathcal{L}^2_\omega)$5 range over sets that are countable in $C(\mathcal{L}^2_\omega)$6, not merely countable in $C(\mathcal{L}^2_\omega)$7, and not necessarily elements of $C(\mathcal{L}^2_\omega)$8 itself. Thus $C(\mathcal{L}^2_\omega)$9 means that there is a set $L$0 that is countable in $L$1 such that $L$2.

This makes $L$3 a fragment of second-order logic with bound second-order quantification restricted to countable sets or relations. The paper describes this logic as incompact and notoriously non-axiomatizable, but still robust enough to support a $L$4-style inner-model construction. Definability in $L$5 is by $L$6-formulas over the current stage, using parameters from that stage. Because second-order variables range over countable subsets in $L$7, definability can reach out to countable pieces of the ambient universe even when those pieces are not internal to the stage under consideration (Magidor et al., 25 Aug 2025).

A basic comparison point is full second-order logic. The paper states that $L$8, while the restriction to countable second-order quantification places $L$9 strictly below $V$0 in general. This difference is central to the model’s behavior: $V$1 is intended to capture a level of definability sensitive to externally countable structure without collapsing to full ordinal definability.

2. Stage-by-stage construction

Conceptually, $V$2 is obtained by replacing the definability step of Gödel’s $V$3-hierarchy with definability in $V$4. To ensure the Axiom of Choice, the construction uses the “truth-folded” definition of $V$5: at each stage one folds in a truth predicate for $V$6 over the current level.

Formally, the hierarchy $V$7 is defined by double induction. One introduces truth-codes $V$8 such that

$V$9

where $C_{\omega_1\omega}$0 and $C_{\omega_1\omega}$1. The stages are then defined by

$C_{\omega_1\omega}$2

$C_{\omega_1\omega}$3

and

$C_{\omega_1\omega}$4

where $C_{\omega_1\omega}$5 is the rudimentary closure augmented to allow the operation $C_{\omega_1\omega}$6 and interpretations of $C_{\omega_1\omega}$7-truth at stage $C_{\omega_1\omega}$8. Finally,

$C_{\omega_1\omega}$9

The paper also states that one may define $C_2(\omega)$0 using countable sequences instead of countable subsets; the resulting model is the same. With the truth-folded construction, $C_2(\omega)$1 satisfies $C_2(\omega)$2 and also $C_2(\omega)$3. By contrast, the older definition may fail to imply $C_2(\omega)$4 for $C_2(\omega)$5, because truth in $C_2(\omega)$6 is not guaranteed to be internally adequate (Magidor et al., 25 Aug 2025).

3. Structural properties

Several basic structural properties are established. $C_2(\omega)$7 is transitive, contains all ordinals, and is definable as a subclass of $C_2(\omega)$8. It is also contained in the Chang model $C_2(\omega)$9, and repeating the construction inside $\mathcal{L}^2_\omega$0 yields the same $\mathcal{L}^2_\omega$1. This containment is important because it identifies $\mathcal{L}^2_\omega$2 as an inner model built from countable information but still strictly controlled by a canonical closure condition.

A cardinality bound is proved: for every $\mathcal{L}^2_\omega$3,

$\mathcal{L}^2_\omega$4

The proof described in the paper uses a chain of $\mathcal{L}^2_\omega$5-elementary submodels of $\mathcal{L}^2_\omega$6 together with a collapse argument. This places a strong restriction on how much of the power set of a cardinal can be captured by $\mathcal{L}^2_\omega$7.

The model is compatible with the failure of the Continuum Hypothesis. The paper proves

$\mathcal{L}^2_\omega$8

The argument uses Harrington’s long projective well-ordering to produce a model of $\mathcal{L}^2_\omega$9 with a projective well-order $x,y,z$0 and then shows that $x,y,z$1 contains all reals of the ambient universe in that setting. This demonstrates that $x,y,z$2 is not tied to the combinatorics of $x,y,z$3 and can accommodate non-$x,y,z$4 behavior (Magidor et al., 25 Aug 2025).

4. Comparison with $x,y,z$5 and $x,y,z$6

A central theme of the paper is the comparison between $x,y,z$7 and the stationary-logic inner model $x,y,z$8. In stationary logic, the generalized quantifier $x,y,z$9 binds variables over $D$0, and the statement $D$1 means that $D$2 is a club subset of $D$3. The hybrid model $D$4 is obtained by combining $D$5 with the $D$6-quantifier in the same truth-folded style.

The paper records the trivial inclusions

$D$7

and

$D$8

Beyond these inclusions, the relationship between $D$9 and $M$0 is delicate. In ZFC alone, one cannot prove that $M$1 is “bigger” than $M$2 in any absolute sense. Indeed, the paper gives a consistency result showing

$M$3

The proof starts from $M$4, adds a Cohen real by homogeneous c.c.c. forcing so that $M$5 remains $M$6, and then codes that real into a stationary pattern on $M$7 without adding countable sets; in the final model the coded real belongs to $M$8, so $M$9.

At the same time, under stronger large-cardinal assumptions the comparison shifts. Assuming a proper class of Woodin cardinals, the paper states that $C(\mathcal{L}^2_\omega)$00 and that for every $C(\mathcal{L}^2_\omega)$01, $C(\mathcal{L}^2_\omega)$02 contains an inner model with $C(\mathcal{L}^2_\omega)$03 Woodin cardinals, whereas under the same assumption $C(\mathcal{L}^2_\omega)$04 contains no inner model with a Woodin cardinal. It also proves that all reals of $C(\mathcal{L}^2_\omega)$05 are in $C(\mathcal{L}^2_\omega)$06 under a proper class of Woodin cardinals, and under a stronger hypothesis denoted “$C(\mathcal{L}^2_\omega)$07” every subset of $C(\mathcal{L}^2_\omega)$08 in $C(\mathcal{L}^2_\omega)$09 is in $C(\mathcal{L}^2_\omega)$10 (Magidor et al., 25 Aug 2025).

5. Large cardinals inside and around $C(\mathcal{L}^2_\omega)$11

The large-cardinal behavior of $C(\mathcal{L}^2_\omega)$12 is one of the paper’s main results. Assuming a proper class of Woodin cardinals, $C(\mathcal{L}^2_\omega)$13 contains, for every finite $C(\mathcal{L}^2_\omega)$14, an inner model with $C(\mathcal{L}^2_\omega)$15 Woodin cardinals. The argument uses the fact that for each $C(\mathcal{L}^2_\omega)$16, $C(\mathcal{L}^2_\omega)$17 is a $C(\mathcal{L}^2_\omega)$18-singleton and is therefore definable in $C(\mathcal{L}^2_\omega)$19; iterating the top measure inside $C(\mathcal{L}^2_\omega)$20 then yields the desired inner models.

The paper also proves a reflection result for $C(\mathcal{L}^2_\omega)$21. Assuming a Woodin limit of Woodin cardinals, $C(\mathcal{L}^2_\omega)$22 is strongly Mahlo in $C(\mathcal{L}^2_\omega)$23. The proof uses the countable stationary tower $C(\mathcal{L}^2_\omega)$24 at a Woodin limit $C(\mathcal{L}^2_\omega)$25 to obtain an embedding $C(\mathcal{L}^2_\omega)$26 with $C(\mathcal{L}^2_\omega)$27, then argues by elementarity that $C(\mathcal{L}^2_\omega)$28 is inaccessible in $C(\mathcal{L}^2_\omega)$29 and finally strongly Mahlo there.

An even stronger layer of structure appears under a proper class of Woodin limits of Woodin cardinals. The paper invokes Woodin’s principle $C(\mathcal{L}^2_\omega)$30 and proves that $C(\mathcal{L}^2_\omega)$31 satisfies Club Determinacy: every stage $C(\mathcal{L}^2_\omega)$32 decides every $C(\mathcal{L}^2_\omega)$33-formula with parameters from $C(\mathcal{L}^2_\omega)$34 and countable parameter sequences. From Club Determinacy it deduces that every regular cardinal of $C(\mathcal{L}^2_\omega)$35 is measurable in $C(\mathcal{L}^2_\omega)$36. The normal ultrafilter is defined by declaring $C(\mathcal{L}^2_\omega)$37 iff, at a sufficiently high stage, the corresponding structure satisfies that $C(\mathcal{L}^2_\omega)$38 lies in $C(\mathcal{L}^2_\omega)$39; Club Determinacy forces either $C(\mathcal{L}^2_\omega)$40 or its complement into $C(\mathcal{L}^2_\omega)$41 (Magidor et al., 25 Aug 2025).

6. Absoluteness, forcing, and surrounding inner-model context

Under a proper class of Woodin limits of Woodin cardinals, the theory of $C(\mathcal{L}^2_\omega)$42 is generically absolute: forcing cannot change it. More precisely, for any forcing $C(\mathcal{L}^2_\omega)$43 and any generic $C(\mathcal{L}^2_\omega)$44, one has

$C(\mathcal{L}^2_\omega)$45

The paper derives this from Woodin-style absoluteness, using collapse models $C(\mathcal{L}^2_\omega)$46 at Woodin limits of Woodin cardinals together with homogeneity. This means that, under the stated hypothesis, the first-order theory of $C(\mathcal{L}^2_\omega)$47 is fixed across forcing extensions.

The broader context includes two further inner-model comparisons. First, under $C(\mathcal{L}^2_\omega)$48, the paper states that $C(\mathcal{L}^2_\omega)$49, while the modified construction “$C(\mathcal{L}^2_\omega)$50” equals $C(\mathcal{L}^2_\omega)$51; from this it concludes that $C(\mathcal{L}^2_\omega)$52. Second, the paper discusses $C(\mathcal{L}^2_\omega)$53, a variant of $C(\mathcal{L}^2_\omega)$54 arising from existential second-order definability, and states that the assertion $C(\mathcal{L}^2_\omega)$55 is independent of $C(\mathcal{L}^2_\omega)$56 even when one adds the existence of supercompact cardinals. One consistency direction uses a Menas model of $C(\mathcal{L}^2_\omega)$57 with a supercompact cardinal; the other starts from $C(\mathcal{L}^2_\omega)$58 with a supercompact cardinal and uses homogeneous Easton-support iterations and Souslin tree forcing to keep $C(\mathcal{L}^2_\omega)$59-truth stable while changing $C(\mathcal{L}^2_\omega)$60 (Magidor et al., 25 Aug 2025).

Taken together, these results place $C(\mathcal{L}^2_\omega)$61 in a distinctive position among definability-based inner models. It is weaker than full second-order $C(\mathcal{L}^2_\omega)$62, contained in the Chang model, and sensitive to externally countable structure; yet under strong large-cardinal assumptions it exhibits generic absoluteness, Club Determinacy, and inner models with finitely many Woodin cardinals. This suggests that $C(\mathcal{L}^2_\omega)$63 functions as a bridge between $C(\mathcal{L}^2_\omega)$64-style definability hierarchies and inner models informed by countable second-order information.