More on setwise climbability properties

This paper introduces two variations of setwise climbability properties, demonstrating that the first type is equivalent to known principles and consistent with the Proper Forcing Axiom (PFA), while the second type characterizes Martin-type axioms via generalized Banach-Mazur games and is inconsistent with PFA, alongside an analysis of their compatibility with large fragments of PFA.

The Gist

Here is an explanation of the paper "More on Setwise Climbability Properties" using simple language and creative analogies.

The Big Picture: Building Towers and Playing Games

Imagine you are an architect trying to build a massive tower that reaches into the sky (representing infinite mathematical structures). In set theory, mathematicians often study "square principles" (like $\square$), which are like blueprints that allow you to build these towers in a very specific, rigid way.

However, some mathematicians (like the authors of this paper) are interested in "fragments" of these blueprints. These are like smaller, partial blueprints that don't build the whole tower but still have interesting properties.

The paper introduces two new ways to tweak these blueprints, which they call "Full" and "End-Extension" variations. To understand if these new blueprints work, the authors use a metaphorical game called the Banach-Mazur Game.

The Game: Climbing the Ladder

Imagine a game played by two people, Player I and Player II, on a giant ladder.

• Player I tries to set traps or block the path.
• Player II tries to climb the ladder forever without falling.

If Player II has a winning strategy (a guaranteed way to keep climbing no matter what Player I does), the "ladder" (or the mathematical structure) has a special property called Climbability.

The paper looks at how different rules for this game change the outcome:

1. The "Full" Variation: This is like saying, "Player II wins if they can climb the ladder and reach the very top, filling every single rung."

   • The Result: The authors found that this version isn't actually new. It's just a combination of two old, known rules. It's like discovering that a "Super Sandwich" is just a regular sandwich plus a slice of cheese. They proved these are consistent with a famous rule called PFA (Proper Forcing Axiom), which is like a "Golden Rule" of mathematics that keeps the universe stable.

2. The "End-Extension" Variation: This is the tricky one. Here, the rule is stricter. Player II must not only climb the ladder but also ensure that every step they take extends the previous steps perfectly, like adding a new floor to a building that fits exactly on top of the one below.

   • The Result: This is a brand-new rule. It turns out this version is much more dangerous to the "Golden Rule" (PFA). If you try to build a structure using these rules, you might break the Golden Rule.

The Twist: Two Types of "Good" Builders

The paper introduces a fascinating distinction between two types of "good" mathematical structures (posets):

• Indestructibly Proper: These are like Steel Buildings. No matter what storm (mathematical forcing) hits them, they stand firm.
• Absolutely Proper: These are like Reinforced Concrete. They are even stronger; they remain standing even if the ground itself changes (if the universe of math changes).

The authors discovered a surprising gap between these two:

• The "End-Extension" rules (the dangerous ones) are compatible with Absolutely Proper buildings. You can have a universe where these new rules exist, and the "Reinforced Concrete" buildings still stand.
• However, they are incompatible with Indestructibly Proper buildings. If you try to mix the "End-Extension" rules with the "Steel Buildings," the whole structure collapses.

The "Magic Mirror" (Mapping Reflection Principle)

To prove that the "End-Extension" rules break the "Steel Buildings," the authors used a tool called the Mapping Reflection Principle (MRP).

• Analogy: Imagine you are looking in a mirror that reflects your image back to you, but the reflection is slightly distorted. MRP is a rule that says, "If you keep looking in this mirror, eventually you will see a pattern that reveals a contradiction."
• The authors used this "magic mirror" to show that if you try to have both the "End-Extension" rules and the "Steel Buildings" at the same time, the mirror reflects a logical contradiction. Therefore, they cannot coexist.

Summary of Findings

1. The "Full" Rules: These are safe. They are just old rules in disguise and play nicely with the Golden Rule (PFA).
2. The "End-Extension" Rules: These are new and powerful. They act like a "Martin-type axiom" (a rule that says "if you have a certain type of game, you can find a solution").
3. The Conflict: The "End-Extension" rules are strong enough to break some versions of the Golden Rule (specifically, they break the "Indestructibly Proper" version) but are weak enough to coexist with the strongest version ("Absolutely Proper").

Why Does This Matter?

In the world of mathematics, we want to know how far we can push the rules before the whole system breaks. This paper draws a very precise line in the sand. It tells us exactly how much "climbability" we can add to our mathematical universe before we destroy the stability provided by the Proper Forcing Axiom.

It's like finding out exactly how much weight you can put on a bridge before it collapses, but realizing that if you use a specific type of steel (Absolutely Proper), you can actually carry even more weight than you thought possible.

Technical Summary

Here is a detailed technical summary of the paper "More on Setwise Climbability Properties" by Bernhard König and Yasuo Yoshinobu.

1. Problem Statement and Context

The paper investigates setwise climbability properties, which are combinatorial principles introduced by the second author (Yoshinobu) as fragments of Jensen's Square Principle ($\square_{\omega_1}$). These principles are known to be equivalent to Martin-type axioms ($MA_{\omega_2}$) for specific classes of posets defined by game-theoretic closure properties (variations of generalized Banach-Mazur games).

The central problem addressed is the classification and consistency of variations of these climbability properties. Specifically, the authors aim to:

1. Determine if new variations of these principles are equivalent to known combinatorial principles or forcing axioms.
2. Analyze the consistency of these new principles with the Proper Forcing Axiom (PFA) and its fragments.
3. Investigate the relationship between different game closure properties (specifically $\ast$-variations vs. $\ast\ast$-variations) and their impact on the preservation of PFA.

2. Methodology

The authors employ a combination of combinatorial set theory and forcing techniques:

• Combinatorial Definitions: They define new variations of the setwise climbability principles ($SCL^-$ and $SCL$) by adding "fullness" or "end-extension" conditions.
• Game Theory: They introduce a new variation of the generalized Banach-Mazur game, denoted as $G^{\ast\ast}(P)$, which is a stricter version of the previously studied $G^\ast(P)$. This game defines new closure properties for posets: $\ast\ast$-operationally closed and $\ast\ast$-tactically closed.
• Forcing and Martin's Axiom: They characterize the new principles as $MA_{\omega_2}$ for the classes of posets defined by these new game closures.
• Preservation Analysis: They utilize Mapping Reflection Principle (MRP) and strengthenings of properness (absolute and indestructible properness) to separate these principles from PFA and its fragments.
• Model Construction: They construct specific forcing extensions to demonstrate consistency and inconsistency results.

3. Key Contributions and Results

A. "Full" Variations ($SCL^-_f$ and $SCL_f$)

The authors introduce "full" variations where the union of the climbing sets covers the target ordinal.

• Equivalence: They prove that $SCL^-_f$ is equivalent to $SCL^- + CL_{\omega_1}$ (where $CL_{\omega_1}$ is a specific combinatorial principle), and $SCL_f$ is equivalent to $SCL$.
• Consistency with PFA: Since $SCL$ is known to be consistent with PFA (via $\ast$-operationally closed forcing), these "full" variations are also consistent with PFA.

B. "End-Extension" Variations ($SCL^-_e$ and $SCLe$)

The authors introduce "end-extension" variations where the sequence of sets must be end-extending (a stronger condition than just $\subseteq$-increasing).

• New Characterization: These principles are characterized as $MA_{\omega_2}$ for posets that are $\ast\ast$-tactically closed or $\ast\ast$-operationally closed.
• The $\ast\ast$-Variation: The game $G^{\ast\ast}(P)$ requires Player I to add conditions that are strictly stronger than the previous move in a specific way, making it a "harder" game for Player I than $G^\ast(P)$.
• Equivalence of Principles: Surprisingly, they prove that $SCL^-_e$ implies $SCLe$ (and thus implies $SCL$), showing that the end-extension condition on the system is sufficient to generate the full pair.
• Implication of $AP_{\omega_1}$: They prove that $SCL^-_e$ implies the Approachability Property ($AP_{\omega_1}$).
• Inconsistency with PFA: Since PFA negates $AP_{\omega_1}$ (a result by Foreman and Todorcevic), $SCL^-_e$ is inconsistent with PFA. This contrasts with the original $SCL$ and $SCL^-$, which are consistent with PFA.

C. Fragments of PFA and Properness

The paper introduces two strengthenings of properness to analyze the extent of PFA fragments preserved by $\ast\ast$-closed forcing:

1. Absolutely Proper: A poset $P$ is absolutely proper if it remains proper in any $\sigma$-closed extension.
2. Indestructibly Proper: A poset $P$ is indestructibly proper if $P \times Q$ is proper for every $\sigma$-closed poset $Q$.

Key Separation Results:

• Theorem 4.3: Assuming PFA, forcing with any $\ast\ast$-tactically closed poset preserves $MA_{\omega_1}(\text{absolutely proper})$. Consequently, $SCL^-_e$ is consistent with $MA_{\omega_1}(\text{absolutely proper})$.
• Theorem 4.12: $MA_{\omega_1}(\text{indestructibly proper})$ negates $SCL^-_e$.
• Theorem 4.16: Assuming PFA, forcing with any $(\omega_1+1)$-strongly strategically closed poset preserves $MA_{\omega_1}(\text{indestructibly proper})$.

These results establish a strict hierarchy:
$$\text{PFA} \implies MA_{\omega_1}(\text{indestructibly proper}) \implies \neg SCL^-_e$$
$$\text{PFA} \implies MA_{\omega_1}(\text{absolutely proper}) \implies SCL^-_e \text{ is consistent}$$

This separates the class of $(\omega_1+1)$-strongly strategically closed posets (which preserve indestructible properness) from $\ast\ast$-tactically closed posets (which do not).

4. Significance

• Refinement of Square Principles: The paper provides a finer granularity for understanding fragments of Jensen's Square Principle, distinguishing between those compatible with PFA and those that are not, based on subtle variations in combinatorial definitions.
• Game-Theoretic Distinctions: It highlights a significant difference between the $\ast$-variation and the $\ast\ast$-variation of Banach-Mazur games. While the rule changes appear minor, they lead to drastically different consistency results regarding PFA.
• Separation of Forcing Axioms: The work successfully separates $MA_{\omega_1}(\text{absolutely proper})$ from $MA_{\omega_1}(\text{indestructibly proper})$ using the principle $SCL^-_e$. This clarifies the landscape of fragments of PFA.
• New Open Questions: The paper concludes by posing questions about the direct implications between $(\omega_1+1)$-strongly strategic closure and $\ast\ast$-tactical closure, as well as the equivalence of $CL_{\omega_1} + SCL^-$ and $SCL$.

In summary, the paper demonstrates that while "full" variations of climbability properties remain within the realm of PFA-consistent principles, "end-extension" variations introduce a level of combinatorial strength (via $AP_{\omega_1}$) that is incompatible with full PFA but consistent with significant fragments of it, thereby mapping a new territory in the hierarchy of forcing axioms.