The *-variation of the Banach-Mazur game and forcing axioms

This paper introduces a new poset property defined via a variation of the Banach-Mazur game that strengthens $(\omega_1+1)$-strategic closedness, proves that the Proper Forcing Axiom (PFA) is preserved under forcing with such posets, and applies this result to reproduce Magidor's theorem on the consistency of PFA with weak square principles while distinguishing the property from $(\omega_1+1)$-operational closedness.

The Gist

Imagine the universe of mathematics as a giant, infinite library. Inside this library, there are millions of books (mathematical truths), but some shelves are missing, and some books are torn. Mathematicians use a tool called Forcing to build new shelves and repair torn books, essentially creating "parallel universes" of mathematics where different rules might apply.

However, there's a catch. When you build a new universe, you don't want to accidentally break the fundamental laws that make the library work. One of the most important laws is the Proper Forcing Axiom (PFA). Think of PFA as the "Golden Rule" of this library: it ensures that the library is well-organized, that you can always find a book you're looking for, and that the structure remains stable.

The paper you asked about is about finding a new, safer way to build these parallel universes without breaking the Golden Rule.

The Game of "Banach-Mazur"

To understand the author's solution, we need to play a game. Imagine a game played between two people, Player I (the Challenger) and Player II (the Defender), on a giant chessboard made of mathematical conditions.

• The Goal: Player II wants to keep playing forever without getting stuck. Player I wants to trap Player II so they have no legal move left.
• The Rules:
  • In the old version of the game, Player I picks one move at a time.
  • In this paper's new version (the *-variation), Player I gets to pick a whole handful of moves (a countable set) at once.

If Player II can still win this harder game (where the opponent throws a handful of obstacles at them instead of just one), the mathematical structure is called $*$-tactically closed.

The Analogy: The Bridge Builder

Let's use a metaphor to make this concrete.

Imagine you are building a bridge across a river (the mathematical universe).

• The River: Represents the gap between what we know and what we want to prove.
• The Planks: Represent the mathematical conditions.
• The Old Strategy (Strategic Closedness): You are allowed to lay down one plank at a time. You have a plan (a strategy) to keep the bridge going. But, if someone throws a whole pile of rocks (a set of conditions) at you all at once, your single-plank plan might fail, and the bridge collapses.
• The New Strategy ($*$-Tactical Closedness): The author introduces a new type of bridge builder. This builder is so skilled that even if the opponent throws a pile of rocks (a countable set of conditions) at them, the builder can still lay down a plank that supports the whole pile.

The paper proves that if you use this "Super Builder" (a poset that is $*$-tactically closed) to build your new universe, the Golden Rule (PFA) will survive. It won't break.

Why Does This Matter?

Before this paper, mathematicians knew two main ways to build safe universes:

1. The "Super Strong" Way: Very rigid rules that guarantee safety but are hard to use.
2. The "Operational" Way: A slightly looser set of rules (introduced by the author in a previous paper) that also kept the Golden Rule safe.

The author asks: "Are these two ways actually different, or are they just the same thing in disguise?"

To answer this, the author plays a game of "What if?"

• They show that there are some mathematical truths that the "Super Strong" way protects, but the "Operational" way destroys.
• They also show that there are other truths that the "Operational" way protects, but the new "$*$-Tactical" way destroys.

The Conclusion: These two methods are different. They are like two different types of safety harnesses. One might save you from falling off a cliff, while the other saves you from a fire. You need to know which one to use depending on the situation.

The Big Result: Magidor's Theorem

The paper ends with a practical application. It uses this new "Super Builder" technique to prove a famous theorem by a mathematician named Magidor.

Magidor wanted to know: "Can we have the Golden Rule (PFA) AND a specific, slightly weird pattern of numbers (called the Square Principle) existing at the same time?"

For a long time, people thought the answer was "No." But using this new game-theory approach, the author shows that Yes, you can have both. It's like proving you can have a perfectly organized library (PFA) even if the books are arranged in a slightly unusual spiral pattern (the Square Principle).

Summary in One Sentence

This paper introduces a new, tougher version of a mathematical game to define a "super-safe" way of building new mathematical universes, proving that this method preserves the most important laws of the universe while showing it is distinct from other known safe methods.

Technical Summary

Here is a detailed technical summary of the paper **"The -Variation of the Banach-Mazur Game and Forcing Axioms"* by Yasuo Yoshinobu.

1. Problem Statement and Motivation

The central problem addressed in this paper is identifying broader classes of forcing posets that preserve the Proper Forcing Axiom (PFA).

• Background: It is known that PFA is preserved by $\omega_2$-closed posets and, more generally, by operationally closed posets (introduced by the author in a previous work). However, $(\omega_1 + 1)$-strategic closedness is insufficient to preserve PFA (e.g., the poset adding a $\square_{\omega_1}$-sequence is strategically closed but destroys PFA).
• The Gap: Researchers seek a strengthening of strategic closedness that is strong enough to preserve PFA but broad enough to include posets that do not preserve other axioms (like Chang's Conjecture) or that force specific combinatorial principles (like weak square principles).
• Goal: To introduce a new game-theoretic property of posets, prove that it preserves PFA, and demonstrate that this property is distinct from (and incomparable with) the previously defined operational closedness.

2. Methodology

The author employs Banach-Mazur games on posets as the primary tool for defining closure properties.

• Standard Game ($G_{\alpha}(P)$): Two players, I and II, take turns choosing conditions. Player I chooses a single condition, and Player II must choose a stronger condition. Player II wins if they can continue for $\alpha$ turns.
• *The -Variation ($G^*(P)$): The author introduces a variation where Player I chooses a countable set of conditions ($A_\gamma \in [P]^{\leq \aleph_0}$) at each turn, rather than a single condition. Player II must choose a condition $b_\gamma$ that is a common extension of the set $A_\gamma$ (or its boolean infimum).
• New Definitions:
  • $\ast$-Tactical Closedness: Player II has a winning strategy in $G^*(P)$ that depends only on the set of conditions chosen by Player I so far (the "current position"), not on the history of the game or the turn number.
  • $\ast$-Operational Closedness: Player II has a winning strategy that depends on the set of conditions chosen by Player I and the ordinal number of the turn.
• Auxiliary Posets: The author constructs auxiliary posets ($R$ and $S$) derived from a given poset $P$ and its winning strategy. These auxiliary posets are used to analyze the interaction between the forcing and the ground model's combinatorial principles (like Chang's Conjecture and Square principles).
• Preservation Proofs: The preservation of PFA is proven using a density argument involving elementary submodels and the specific structure of the winning strategies in the $G^*$ game.

3. Key Contributions and Results

A. Preservation of PFA

• Theorem 3.1: PFA is preserved under forcing with any $\ast$-operationally closed poset.
  • Significance: This generalizes the author's previous result on operational closedness. Since $\ast$-tactical closedness is a special case of $\ast$-operational closedness, PFA is also preserved by $\ast$-tactically closed forcings.

B. Application: Magidor's Theorem

• Theorem 3.3 (Magidor): The consistency of PFA with the statement that $\square_{\kappa, \omega_2}$ holds for all $\kappa \geq \omega_2$.
  • Proof: The author shows that the natural poset $P_{\square_{\kappa, \lambda}}$ (forcing the weak square principle) is $\ast$-tactically closed (Theorem 2.6). By combining this with the preservation theorem, the consistency result is reproduced.

C. Distinction Between Closure Notions

The paper establishes that $\ast$-tactical closedness and operational closedness are incomparable properties under the assumption of $MA^+(\omega_1\text{-closed})$.

• Result 1 (Section 5): Under $MA^+(\omega_1\text{-closed})$, every $\ast$-tactically closed forcing preserves Chang's Conjecture (CC).
  • Implication: Since there exist operationally closed posets that force the failure of CC, Operational Closedness $\nRightarrow$ $\ast$-Tactical Closedness.
• Result 2 (Section 6): Under $MA^+(\omega_1\text{-closed})$, every operationally closed forcing forces the failure of $SCP^-$ (Setwise Climbability Property minus the function $f$).
  • Implication: Since $SCP^-$ can be forced by a $\ast$-tactically closed poset (Theorem 4.6), $\ast$-Tactical Closedness $\nRightarrow$ Operational Closedness.

D. Combinatorial Equivalences

The paper introduces the Setwise Climbability Property ($SCP$) and its variant $SCP^-$.

• Theorem 4.6:
  • $SCP$ is equivalent to $MA_{\omega_2}$ restricted to the class of $\ast$-operationally closed posets.
  • $SCP^-$ is equivalent to $MA_{\omega_2}$ restricted to the class of $\ast$-tactically closed posets.
  • This characterizes these combinatorial principles as Martin-type axioms for specific classes of posets, analogous to how $\square_{\omega_1}$ characterizes strategic closedness.

4. Significance

1. Refinement of Forcing Preservation: The paper provides a finer classification of posets that preserve PFA. It shows that PFA is robust enough to survive forcings that are "weaker" than full $\omega_2$-closure but "stronger" in a specific game-theoretic sense ($\ast$-tactical/operational).
2. Resolution of Magidor's Consistency: It offers a streamlined proof of Magidor's theorem regarding the consistency of PFA with weak square principles, utilizing the new $\ast$-tactical closedness property.
3. Separation of Axioms: By demonstrating the incomparability of operational and $\ast$-tactical closedness, the paper clarifies the landscape of forcing axioms. It shows that different strengthenings of strategic closedness preserve different fragments of set-theoretic consequences (e.g., one preserves Chang's Conjecture while the other destroys it).
4. Game-Theoretic Characterization: The work deepens the connection between Banach-Mazur games and forcing axioms, showing that modifying the rules of the game (allowing Player I to pick countable sets) yields new, useful closure properties that are distinct from standard strategic or operational closure.

In summary, Yoshinobu introduces a novel game-theoretic framework ($\ast$-variation) that successfully captures a class of forcings preserving PFA, distinct from previous classes, and uses this framework to resolve consistency questions regarding square principles and to separate different forcing axioms based on their preservation properties.