Iterated club shooting and the stationary-logic constructible model

This paper investigates the iteration of the stationary-logic constructible model $C(\mathtt{aa})$ by proving distributivity and stationary-set preservation for countable iterations of club-shooting forcings using mutually stationary sets, and introducing mutually fat sets to achieve stronger results for uncountable iterations, thereby demonstrating the ability to force models where $V=C(\mathtt{aa})$ and where iterated $C(\mathtt{aa})$ sequences decrease with arbitrarily large order types.

The Gist

Imagine the universe of mathematics as a giant, infinite library called The Library of All Sets. Inside this library, there are different ways to organize the books (sets). One very strict librarian, let's call him Logic, only allows books that can be built up step-by-step using a very specific set of rules.

In this paper, the author, Uri Ya'ar, is investigating a special kind of librarian named Stationary Logic (denoted as $C(aa)$). This librarian is stricter than the standard one (who builds the universe $L$) but slightly more flexible. The librarian uses a special rule: "A set exists if it appears in 'many' places," where "many" is defined by a mathematical concept called stationarity (roughly, being present in every large, unbroken section of the library).

The Core Problem: The "Self-Checking" Loop

The big question Uri asks is: If we build a universe using this strict Stationary Logic, does that new universe also think it is the final, complete universe?

• The Ideal Scenario: You build a universe $U$. You look inside $U$, and you say, "Everything in here was built by my rules." So, $U$ is its own limit.
• The Reality: Sometimes, when you build this universe, you realize, "Wait, I missed a few things! I can build more things using my own rules."
  • So, you build a second layer, $U_2$.
  • Then you look at $U_2$ and realize, "I can build even more!" So you build $U_3$.
  • This creates a descending sequence: $U \supset U_2 \supset U_3 \supset \dots$

Uri wants to know: How long can this chain of "I can build more" go on? Can it go on forever? Can it go on for a specific number of steps (like 5, 100, or a million)?

The Tools: "Club Shooting" and "Holes in the Floor"

To answer this, Uri uses a mathematical tool called Club Shooting.

The Analogy:
Imagine the library floor is covered in a sticky, permanent carpet (a Stationary Set). This carpet represents a set of numbers that is "everywhere" in a mathematical sense.

• The Goal: Uri wants to "destroy" the carpet in specific spots to change the rules of the library.
• The Method: He uses a Club Shooter. This is a machine that lays down a new, solid path (a Club Set) through the empty spaces between the sticky spots.
• The Result: Once the path is laid, the sticky spots are no longer "everywhere" (they are no longer stationary). The librarian, looking at the new floor, realizes, "Oh, those spots I thought were permanent? They're gone. I have to rebuild my universe without them."

By carefully choosing which spots to destroy, Uri can "code" information into the universe. He can make the universe "forget" a specific set of numbers, effectively hiding it from the librarian's view.

The Challenge: The "Domino Effect"

If Uri tries to destroy too many carpets at once, the machine might break the whole floor, destroying everything he built so far.

• The Problem: If you shoot a club through one set, it might accidentally destroy the "holes" you needed for the next step.
• The Solution (Mutual Stationarity): Uri introduces a concept called Mutually Stationary Sets. Imagine a group of carpets that are arranged so perfectly that if you lay a path through one, it doesn't ruin the others. They are "mutually friendly."
• The Upgrade (Mutually Fat Sets): For very long chains, he invents a new concept: Mutually Fat Sets. These are like super-carpets that are so thick and robust that even if you shoot a path through them, they remain "fat" enough to support the next round of construction. This allows him to build much longer chains than before.

The Results: How Long Can the Chain Go?

Uri proves two main things:

1. You can make the chain as long as you want (up to a point).
   Using his "Mutually Fat" tools, he shows that you can force the universe to have a sequence of "I can build more" steps of any specific length you choose (e.g., 100 steps, 1,000 steps).

   • Analogy: It's like a game of "I can build a bigger tower." Uri shows you can keep adding floors indefinitely, as long as you have the right blueprint (the mutually fat sets).

2. You can make the universe "perfect" ($V = C(aa)$).
   He also shows how to force the universe so that the librarian never realizes there's more to build. The universe becomes its own limit immediately.

   • Analogy: He builds a house so perfectly that the architect looks at it and says, "This is it. Nothing else can be added."

Why This Matters

This paper is a deep dive into the limits of mathematical construction.

• It compares two different types of "logic" (Stationary Logic vs. a simpler logic called $C^*$).
• It shows that Stationary Logic is much more powerful. In the simpler logic, you need massive "super-magnets" (large cardinals) to build long chains. But with Stationary Logic, Uri can build these long chains starting from a very basic, simple universe (the constructible universe $L$) without needing those super-magnets.

Summary in One Sentence

Uri Ya'ar invented a new mathematical "shooting" technique to carefully remove specific parts of the mathematical universe, proving that we can force the universe to have a long, descending chain of "incomplete" versions before finally settling into a perfect, self-contained reality, all without needing the most powerful mathematical tools usually required for such feats.

Technical Summary

Here is a detailed technical summary of the paper "Iterated Club Shooting and the Stationary Logic Constructible Model" by Ur Ya'ar.

1. Problem Statement and Motivation

The paper investigates the constructible model using stationary logic, denoted as $C(aa)$. This model is constructed similarly to Gödel's $L$, but using the logic $L(aa)$, which extends first-order logic with quantifiers $\text{aa}$ ("for almost all," i.e., on a club set) and $\text{stat}$ ("for stationarily many") over countable subsets.

The central questions addressed are:

1. Self-Constructibility: Does the model satisfy $V = C(aa)$? While true in $L$, it is not guaranteed in general. The paper explores the consistency of $V = C(aa)$ with $V \neq L$.
2. Iterated $C(aa)$ Sequences: If $V \neq C(aa)$, one can define a sequence of iterated models: $C(aa)^0 = V$, $C(aa)^{\alpha+1} = C(aa)^{C(aa)^\alpha}$, and intersections at limit stages. The paper asks how long a strictly decreasing sequence of such models can be.
   • Context: Previous work on $HOD$ (by McAloon, Harrington, Jech, and Zadrożny) showed that strictly decreasing sequences of arbitrary length (even $Ord$) are consistent. However, $HOD$ is constructed via second-order logic. $C(aa)$ is constructed via stationary logic, which has different expressive power.
   • Contrast: The author notes that for the model $C^*$ (constructed via cofinality-$\omega$ logic), obtaining long decreasing sequences requires large cardinals (measurable cardinals). The paper aims to show that for $C(aa)$, such sequences can be obtained over $L$ without large cardinals, highlighting the greater expressive power of stationary logic.

2. Methodology

The core methodology involves iterated forcing using club shooting techniques to "code" sets into the $C(aa)$ hierarchy.

A. Coding Mechanism

The author utilizes the fact that in $L(aa)$, one can define whether a set of ordinals is stationary. By destroying the stationarity of specific sets via forcing, one can encode information into the model.

• Club Shooting: Given a co-fat stationary set $S$, the forcing $\text{Des}(S)$ adds a club disjoint from $S$, rendering $S$ non-stationary.
• Coding: If a sequence of disjoint stationary sets $\vec{S} = \langle S_\alpha \mid \alpha < \kappa \rangle$ partitions a co-fat set, one can force a club through the complement of $\bigcup \{S_\alpha \mid \alpha \in X\}$. In the extension, $X = \{ \alpha \mid S_\alpha \text{ is not stationary} \}$. Since stationarity is definable in $C(aa)$, the set $X$ becomes an element of $C(aa)$.

B. Iteration Challenges

To create long decreasing sequences, one must iterate this coding process. The main technical challenges are:

1. Preservation: Later stages of iteration must not destroy the stationarity of sets coded in earlier stages.
2. Distributivity: To ensure the intersection of generic extensions behaves well (specifically, that the intersection of the models is a model of ZFC and that the $C(aa)$ construction stabilizes correctly), the forcing iteration must be highly distributive.

C. Key Technical Tools

To overcome these challenges, the paper introduces and utilizes two main concepts:

1. Mutually Stationary Sets (Countable Iterations):

   • Based on Foreman and Magidor, a collection of stationary sets is mutually stationary if they can be "witnessed" simultaneously by a single countable elementary substructure.
   • Theorem 2.10: Shows that countable iterations of club shooting forcings using mutually stationary sets are $\omega$-distributive and preserve the stationarity of sets not targeted for destruction. This allows for constructing models with $V = C(aa) \neq L$ and decreasing sequences of length $\omega$.

2. Mutually Fat Sets (Uncountable Iterations):

   • To achieve longer sequences (beyond $\omega$), mutual stationarity is insufficient. The author introduces Mutually Fat Sets.
   • Definition: A sequence $\{S_\kappa\}$ is mutually $\nu$-fat if for any algebra, there exists a continuous, internally approachable chain of models $\langle N_\alpha \mid \alpha \le \nu \rangle$ witnessing the fatness of the sets simultaneously.
   • Theorem 2.13: Proves that iterations of club shooting using mutually fat sets are $<\kappa_0$-distributive (where $\kappa_0$ is the first cardinal in the sequence). This strong distributivity is crucial for handling uncountable iterations.
   • Construction: The paper provides two methods to obtain mutually fat sequences:
     • Using $\square$-sequences (Global Square principle) on singular cardinals.
     • Forcing non-reflecting stationary sets.

3. Key Contributions and Results

A. Consistency of $V = C(aa) \neq L$

• Theorem 3.3: It is consistent relative to ZFC that $V = C(aa)$ and $V \neq L$. This is achieved by forcing over $L$ with a countable iteration of club shooting forcings.
• Theorem 3.6: If the complements of the coding sets form a strongly mutually fat sequence, the forcing is $<\kappa_0$-distributive. This allows the construction of a model where $V = C(aa)$ and $H(\kappa_0)^V = H(\kappa_0)^L$, preserving local structure.

B. Decreasing Iterated $C(aa)$ Sequences

• Countable Length ($\omega$): Theorem 3.9 constructs a model with a strictly decreasing sequence of iterated $C(aa)$s of length $\omega$, where $C(aa)^\omega \models \text{ZFC}$ and $C(aa)^{\omega+1} = C(aa)^V$.
  • Mechanism: The author codes a set $A$ "infinitely many times" in a triangular arrangement. Taking $C(aa)$ removes the last coding at each level, but because there are infinitely many levels, the process never terminates until the limit $\omega$.
• Arbitrary Ordinal Length: Theorem 3.10 is the main result. For any ordinal $\delta$, there is a forcing extension where the sequence $C(aa)^\alpha$ is strictly decreasing for all $\alpha \le \delta$, and $C(aa)^{\delta+1} = L[A]$.
  • This is achieved by using the mutually fat sets to perform iterations of length $\delta$ while maintaining the necessary distributivity to control the intersection of the models.

C. Comparison with $C^*$

The paper highlights a significant divergence between stationary logic ($C(aa)$) and cofinality-$\omega$ logic ($C^*$).

• For $C^*$, long decreasing sequences require large cardinals (measurable).
• For $C(aa)$, such sequences can be constructed over $L$ using only $\square$ principles. This demonstrates that stationary logic is strictly more powerful in terms of coding capabilities and flexibility regarding the structure of the constructible universe.

4. Significance

1. Advancement of Inner Model Theory: The paper extends the study of "constructible models from logics" beyond $HOD$ and $C^*$, providing a robust framework for $C(aa)$.
2. New Forcing Techniques: The introduction of mutually fat sets is a novel contribution to the theory of iterated forcing. It generalizes mutual stationarity to handle uncountable iterations with high distributivity, a tool that may be applicable to other problems in set theory involving club shooting and stationary sets.
3. Resolution of Open Questions: It answers the question of whether $V = C(aa)$ can hold with $V \neq L$ and establishes the consistency of arbitrarily long decreasing sequences of iterated $C(aa)$ models without large cardinal assumptions.
4. Distinction of Logics: It clarifies the hierarchy of expressive power between different logics used to define inner models, showing that stationary logic allows for more complex structural manipulations of the universe than cofinality-$\omega$ logic.

5. Open Questions

The paper concludes by listing several open problems:

• Can an Ord-length decreasing sequence of $C(aa)$s be constructed? (Currently limited by the need for larger coding cardinals for longer iterations).
• Do large cardinals or the failure of $\square$ restrict the length of these sequences?
• Is the length of the $C(aa)$ sequence determined by the existence of a proper class of Woodin cardinals?
• Can the results be adapted to models constructed using only ordinal stationarity (without the full $P_{\omega_1}$ context)?

In summary, Ur Ya'ar's paper successfully adapts Zadrożny's coding framework to the context of stationary logic, introducing the powerful concept of mutually fat sets to overcome distributivity barriers, thereby proving that the $C(aa)$ hierarchy can be manipulated to have arbitrary decreasing lengths over $L$.