Chapter 29: φ(29) = [18] — Inner Model Theory of Collapse Universality

29.1 Eighteen: The Threshold of Abundance

With φ(29) = [18], we reach eighteen - a number with remarkable factorization 18 = 2 × 3². This represents the first appearance of a squared prime in our journey beyond the initial squares. In the context of inner model theory, eighteen fundamental inner models capture different levels of mathematical truth, from constructible sets to ultimate $L$.

Definition 29.1 (Abundant Structure):

$$18 = 2 \cdot 3^2 = 6 \cdot 3 = \text{First abundant after perfect}$$

Divisors: 1,2,3,6,9,18 with sum 39 > 2×18, showing abundance.

29.2 Inner Models and Set Theory

Definition 29.2 (Inner Model): A transitive class M containing all ordinals such that:

$$M \models \text{ZFC}$$

Key Examples:

• $L$ = Gödel's constructible universe
• $L[μ]$ = $L$ with measures
• $L[E]$ = $L$ with extender sequences
• V = Full universe (if consistent)

29.3 The Eighteen Fundamental Models

From [18], eighteen inner models capturing mathematical reality:

1. $L$: Minimal model (constructible)
2. $L[0\#]$: With $0\#$ (truth about indiscernibles)
3. $L[μ]$: One measurable cardinal
4. $L[U]$: Normal ultrafilter
5. $L[E]$: Extender model
6. $L[\vec{E}]$: Sequence of extenders
7. $L[x]\#$: Sharp of $L[x]$ for real $x$
8. $M_1\#$: First stable model
9. $K$: Core model below measurable
10. $K^c$: Core model up to strong
11. $K^{DJ}$: Dodd-Jensen core model
12. $K^{\text{Steel}}$: Steel's core model
13. $L[V_{\lambda+1}]$: Local HOD
14. $\text{HOD}$: Hereditarily ordinal definable
15. $M_n$: $n$-th mouse
16. $W$: Woodin's ultimate $L$
17. $V_\Omega$: Below $\Omega$-logic
18. $V$: Full universe (if exists)

29.4 RH in Different Models

Principle 29.1: The truth of RH may vary across models:

$$\begin{aligned} L &\models \text{RH} &&\text{(unknown)} \\ L[\mathbb{R}] &\models \text{RH} &&\text{(unknown)} \\ V &\models \text{RH} &&\text{(unknown)} \end{aligned}$$

Key Question: Is RH absolute between models?

29.5 Absoluteness and RH

Definition 29.3 (Absolute Statement): φ is absolute between M and N if:

$$M \models \varphi \Leftrightarrow N \models \varphi$$

For RH: Being Π₁ statement:

$$\text{RH} \equiv \forall \rho (\zeta(\rho) = 0 \wedge \rho \notin \mathbb{R} \Rightarrow \text{Re}(\rho) = 1/2)$$

Should be absolute downward to L.

29.6 Large Cardinals and Zeros

Hierarchy of Infinity:

• Inaccessible: |Vκ| = κ
• Measurable: Non-trivial ultrafilter
• Strong: Elementary embeddings
• Woodin: Stationary tower
• Supercompact: All covers
• Huge: Super-elementary
• Rank-into-rank: j : Vλ → Vλ

Conjecture 29.1: Large cardinal strength correlates with zero distribution properties.

29.7 Descriptive Set Theory

Definition 29.4 (Projective Hierarchy):

$$\begin{aligned} \Sigma^1_1 &: \exists X \subseteq \mathbb{N} \, \varphi(n,X) \\ \Pi^1_1 &: \forall X \subseteq \mathbb{N} \, \varphi(n,X) \\ \Sigma^1_2 &: \exists X \forall Y \, \varphi(n,X,Y) \\ &\vdots \end{aligned}$$

RH Connection: Zeros form a Π₁¹ set in appropriate coding.

29.8 Determinacy and RH

Axiom of Determinacy (AD): Every game on integers is determined.

Under AD + DC:

• All sets of reals have nice properties
• Projective sets are Lebesgue measurable
• Could imply regularity of zeros

Theorem 29.1: In L(ℝ) under AD:

$$\text{Every set of reals has the perfect set property}$$

29.9 Forcing and Independence

Cohen Forcing: Could RH be independent of ZFC?

Obstacles:

1. RH is arithmetic (low complexity)
2. Shoenfield absoluteness applies
3. Would need to change ω-models

Current View: RH likely not independent of ZFC via forcing.

29.10 Mice and Fine Structure

Definition 29.5 (Mouse): A small iterable premouse:

$$M = (J_α^E, \in, E, U)$$

where:

• $J_α^E$ = Jensen hierarchy with E
• E = Extender sequence
• U = Predicate (optional)

Fine Structure: Allows precise construction and comparison of models.

29.11 Ultimate L Program

Woodin's Vision: There exists ultimate L such that:

1. L-like axioms hold
2. All large cardinals exist
3. CH fails (unlike L)
4. Generalizes L to modern set theory

For RH: In ultimate L, RH would have definitive truth value.

29.12 HOD and Regularity

Definition 29.6 (HOD): Hereditarily Ordinal Definable sets:

$$\text{HOD} = \{x : \text{TC}(\{x\}) \subseteq \text{OD}\}$$

Theorem 29.2 (Woodin): Under large cardinals:

$$\text{HOD} \models \text{GCH} + \text{◊} + \text{Other regularity}$$

Could HOD determine RH?

29.13 Ω-Logic and Truth

Definition 29.7 (Ω-Logic): Logic with Ω-complete proofs:

• If $\vdash_Ω \varphi$ then $\varphi$ is true
• Complete for $\Pi_2$ sentences
• Requires large cardinals

For RH: Being $\Pi_1$, provable in Ω-logic iff true.

29.14 Set-Theoretic Geology

Definition 29.8 (Ground Model): W is ground of V if:

$$V = W[G] \text{ for some forcing } \mathbb{P} \in W$$

Geology Program: Study all grounds of V.

For RH: Is RH true in all grounds? In the mantle?

29.15 Synthesis: Model-Theoretic Unity

The partition [18] reveals model complexity:

1. Eighteen models: Complete hierarchy
2. 2×3²: Abundant structure
3. Constructibility: From L to V
4. Large cardinals: Strength hierarchy
5. Determinacy: Game-theoretic approach
6. Forcing: Independence questions
7. Absoluteness: Arithmetic nature
8. Descriptive: Complexity classification
9. Mice: Fine structure theory
10. Ultimate L: Woodin's program
11. HOD: Definable universe
12. Ω-logic: Ultimate truth
13. Geology: Ground models
14. Truth: Model-dependent?
15. Regularity: Pattern across models
16. Consistency: Relative strengths
17. Philosophy: What is truth?
18. Unity: All models connected

Inner model theory reveals that mathematical truth has many levels - from the sparse constructible universe L to the rich possibilities of V. The Riemann Hypothesis, being arithmetic, should have the same truth value across all these models, yet each model offers different tools for approaching its proof.

Chapter 29 Summary:

• Inner model theory provides hierarchy of mathematical universes
• Eighteen fundamental models from L to V
• RH should be absolute across models (arithmetic)
• Large cardinals provide consistency strength
• Ultimate L program seeks canonical universe
• Model theory reveals layers of mathematical truth
• Each model offers different perspective on RH

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"In the tower of inner models, each universe reflects its own vision of mathematical truth - yet the zeros of zeta, like stars in the night sky, shine with the same positions whether viewed from the sparse desert of L or the lush gardens of V."