C²(ω, aa): Canonical Inner Model

• C²(ω, aa) is an inner model combining countable second-order quantification with the stationary aa‐quantifier, capturing nuanced definability traits.
• The construction uses a transfinite C(𝓛)-hierarchy incorporating truth predicates and rudimentary closure to analyze large-cardinal relationships.
• Under strong large-cardinal hypotheses, C²(ω, aa) exhibits enhanced determinacy properties and ensures regular cardinals are measurable within the model.

$C2(\omega,aa)$, more commonly written $C^2(\omega,aa)$, is an inner model obtained by applying the $C(\mathcal L)$-construction to the logic $\mathcal L^2_\omega(aa)$, the extension of a restricted second-order logic by the stationary-logic $aa$-quantifier. It combines two constructions studied side by side: $C^2(\omega)$, based on second-order variables ranging over countable subsets and relations, and $C(aa)$, based on the $aa$-quantifier of stationary logic. In the 2025 treatment of inner models from second-order logics, $C^2(\omega,aa)$ is analyzed as a combined model with stronger large-cardinal consequences than those established for $C^2(\omega)$ alone (Magidor et al., 25 Aug 2025).

1. Logical basis and semantic ingredients

The starting point is the logic $C^2(\omega,aa)$0, defined as second-order logic in which the bound second-order variables range over countable subsets and relations on the domain (Magidor et al., 25 Aug 2025). The paper emphasizes a crucial semantic point: these subsets and relations are chosen from the ambient universe $C^2(\omega,aa)$1 and assumed to be countable in $C^2(\omega,aa)$2, but they need not be elements of the domain, and they need not be countable in the sense of the domain. This distinguishes $C^2(\omega,aa)$3 from full second-order logic $C^2(\omega,aa)$4, whose second-order quantifiers range over all subsets and relations of the domain.

From this logic, the paper defines

$C^2(\omega,aa)$5

The model $C^2(\omega,aa)$6 is the corresponding $C^2(\omega,aa)$7-model arising from stationary logic $C^2(\omega,aa)$8, where the additional quantifier $C^2(\omega,aa)$9 is interpreted using stationarity on countable subsets. The combined model is then

$C(\mathcal L)$0

where $C(\mathcal L)$1 extends $C(\mathcal L)$2 by the $C(\mathcal L)$3-quantifier (Magidor et al., 25 Aug 2025).

This organization places $C(\mathcal L)$4 within the general program of producing canonical inner models from extended logics. In this case, the two logical enrichments are distinct in kind: one enlarges definability by allowing quantification over countable second-order objects, while the other imports the stationary-logic $C(\mathcal L)$5-quantifier.

2. Construction via the $C(\mathcal L)$6-hierarchy

The explicit hierarchy is given first for $C(\mathcal L)$7. The paper states that the model is built by a transfinite double induction on a hierarchy $C(\mathcal L)$8 together with a class $C(\mathcal L)$9 of truth predicates (Magidor et al., 25 Aug 2025). The recursion is

$\mathcal L^2_\omega(aa)$0

The rudimentary closure operation $\mathcal L^2_\omega(aa)$1 is explicitly said to include the operation $\mathcal L^2_\omega(aa)$2. The model is then

$\mathcal L^2_\omega(aa)$3

The same section remarks that one can equivalently define $\mathcal L^2_\omega(aa)$4 using countable sequences instead of countable subsets; the resulting model is the same (Magidor et al., 25 Aug 2025). For $\mathcal L^2_\omega(aa)$5, the same style of $\mathcal L^2_\omega(aa)$6-construction is used, now with countable second-order quantification and the $\mathcal L^2_\omega(aa)$7-quantifier combined. The paper also states that one must use the “new” definition of $\mathcal L^2_\omega(aa)$8 from earlier work, with truth predicates folded in, because it is not clear whether the older version is adequate to truth in itself; the same design principle is carried over to $\mathcal L^2_\omega(aa)$9.

At the level of method, this construction shows that $aa$0 is not defined by a single forcing or extender recipe. Rather, it is produced by a logical hierarchy in the $aa$1 style, with definability controlled by the semantics of countable second-order quantification together with stationary logic.

3. Relation to $aa$2, $aa$3, and neighboring inner models

The paper records the basic inclusions

$aa$4

and

$aa$5

where $aa$6 is the inner model from the cofinality quantifier $aa$7 (Magidor et al., 25 Aug 2025). These are described as trivial inclusions in ZFC.

A central comparative claim is that $aa$8 appears to be a much bigger inner model than $aa$9, although this cannot be literally true in ZFC alone (Magidor et al., 25 Aug 2025). The paper treats this as a consistency-strength comparison rather than an absolute theorem of ZFC. Its evidence is large-cardinal-theoretic: assuming a proper class of Woodin cardinals, $C^2(\omega)$0 is in $C^2(\omega)$1 but not in $C^2(\omega)$2. Under the same assumption, all reals of $C^2(\omega)$3 are in $C^2(\omega)$4; under a strong enough large-cardinal hypothesis denoted $C^2(\omega)$5, subsets of $C^2(\omega)$6 that belong to $C^2(\omega)$7 are also in $C^2(\omega)$8 (Magidor et al., 25 Aug 2025).

These comparisons locate $C^2(\omega)$9 as a genuine amalgam rather than a mere notational convenience. It contains both $C(aa)$0 and $C(aa)$1, while the relationship between the two constituent models remains partly conjectural when isolated from the combined construction.

4. Structural properties of $C(aa)$2

Several general structural facts are established for $C(aa)$3 and provide context for the combined model. One theorem states that for any $C(aa)$4,

$C(aa)$5

The paper says this is used to show a certain structural tameness, while also remarking that the bound is probably far from optimal in large-cardinal contexts (Magidor et al., 25 Aug 2025).

A further inclusion places the model inside the Chang model: $C(aa)$6 The proof strategy described in the paper is to repeat the $C(aa)$7-construction inside the Chang model and show that it gives the same result (Magidor et al., 25 Aug 2025). From this, together with large-cardinal assumptions, the paper derives a forcing-absoluteness statement: assuming a proper class of Woodin limits of Woodin cardinals, the theory of $C(aa)$8 cannot be changed by set forcing.

These properties suggest a distinctive combination of strength and restraint. On one side, $C(aa)$9 is strong enough to capture canonical sharp-like objects. On the other, it remains bounded by the Chang model and subject to explicit cardinality estimates on its power sets.

5. Large-cardinal content of $aa$0

The main theorem in this direction states: if there is a proper class of Woodin cardinals, then $aa$1. Moreover, $aa$2 contains, for every $aa$3, an inner model with $aa$4 Woodin cardinals (Magidor et al., 25 Aug 2025). The argument proceeds through the canonical sharps $aa$5. The paper notes that for every $aa$6, $aa$7 is a $aa$8-singleton, and therefore $aa$9 can be defined in, and belongs to, $C^2(\omega,aa)$0. By iterating the top measure inside $C^2(\omega,aa)$1, one obtains an inner model with $C^2(\omega,aa)$2 Woodin cardinals.

This is contrasted sharply with $C^2(\omega,aa)$3. Under a proper class of Woodin cardinals, the reals of $C^2(\omega,aa)$4 form a countable $C^2(\omega,aa)$5-set, and in particular $C^2(\omega,aa)$6 (Magidor et al., 25 Aug 2025). The comparison is one of the paper’s main reasons for treating $C^2(\omega,aa)$7 as substantially stronger than $C^2(\omega,aa)$8 in large-cardinal environments.

A second major theorem states that, assuming a Woodin limit of Woodin cardinals, $C^2(\omega,aa)$9 is strongly Mahlo in $C^2(\omega)$0 (Magidor et al., 25 Aug 2025). The proof uses a stationary tower embedding $C^2(\omega)$1 from the countable stationary tower forcing at a Woodin limit $C^2(\omega)$2, together with absoluteness properties of the relevant Chang-model-type constructions. The paper explicitly remarks that it is open whether this can be strengthened to weak compactness or measurability of $C^2(\omega)$3 in $C^2(\omega)$4.

6. Club Determinacy and measurability in $C^2(\omega)$5

The strongest theorem stated specifically for the combined model concerns Club Determinacy and measurability (Magidor et al., 25 Aug 2025). The paper introduces $C^2(\omega)$6, the club filter of $C^2(\omega)$7, and uses Woodin’s principle CM$C^2(\omega)$8: for all $C^2(\omega)$9, if $C^2(\omega,aa)$00 and $C^2(\omega,aa)$01, then either $C^2(\omega,aa)$02 or its complement is in $C^2(\omega,aa)$03.

Woodin’s theorem is quoted in the form: if there is a proper class of Woodin limits of Woodin cardinals, then CM$C^2(\omega,aa)$04 exists (Magidor et al., 25 Aug 2025). Using this, the paper defines a Club Determinacy property for the levels $C^2(\omega,aa)$05 of the $C^2(\omega,aa)$06-construction and proves:

Assuming a proper class of Woodin limits of Woodin cardinals, $C^2(\omega,aa)$07 satisfies Club Determinacy.

From this the paper derives the corollary:

Assume a proper class of Woodin limits of Woodin cardinals. Then every regular cardinal of $C^2(\omega,aa)$08 is measurable in $C^2(\omega,aa)$09.

The paper identifies this as the stronger result for the combination $C^2(\omega,aa)$10. In effect, the addition of the $C^2(\omega,aa)$11-quantifier upgrades the large-cardinal conclusions available from the countable-second-order construction by itself. A plausible implication is that the stationary-logic component is not merely auxiliary: it is the ingredient that converts the underlying logical hierarchy into one satisfying a strong internal measure-theoretic regularity principle.

7. Adjacent variants and open problems

The paper places $C^2(\omega,aa)$12 within a broader landscape of inner models from extended logics (Magidor et al., 25 Aug 2025). One adjacent object is HOD1, presented there as a variant of HOD associated with the model $C^2(\omega,aa)$13. The theorem stated is that the question whether HOD1 is the same as HOD cannot be decided on the basis of ZFC even if one adds the assumption that there are supercompact cardinals.

Several comparison problems around $C^2(\omega,aa)$14 and $C^2(\omega,aa)$15 are left open. The paper explicitly highlights whether $C^2(\omega,aa)$16 follows from large cardinals, whether $C^2(\omega,aa)$17 satisfies GCH under such hypotheses, and exactly how large $C^2(\omega,aa)$18 can be in terms of internal large cardinals (Magidor et al., 25 Aug 2025). These questions show that the combined model is not only a repository of strong consequences but also a test case for the general problem of extracting canonical inner models from logics that quantify over countable structure.

In that sense, $C^2(\omega,aa)$19 occupies a precise position in current inner-model theory. It is defined by a concrete $C^2(\omega,aa)$20-construction, related to both $C^2(\omega,aa)$21 and $C^2(\omega,aa)$22, bounded above by $C^2(\omega,aa)$23, and—under very strong large-cardinal hypotheses—rich enough to make every regular cardinal of $C^2(\omega,aa)$24 measurable inside the model.