Forcing and Family Contentions: Who wins the disputes?

[Tompson Lee]

The famous game-theoretic couple, Alice & Bob, live in the set-theoretic universe, V, a model of ZFC. Just like many other couples they sometimes argue over a statement, σ, expressible in the language of set theory. (One may think of σ as a family condition/decision in the real life, say having kids or living in a certain city, etc.)

Alice wants σ to be true in the world that they live but Bob doesn't. In such cases, each of them tries to manipulate the sequence of the events in such a way that makes their desired condition true in the ultimate situation. Consequently, a game of forcing iteration emerges between them as follows:

Alice starts by forcing over V, leading the family to the possible world V[G]. Then Bob forces over V[G] leading both to another possible world in which Alice responds by forcing over it and so on. Formally, during their turn, Alice and Bob are choosing the even and odd-indexed names for forcing notions, ℚ˙0, ℚ˙1, ℚ˙2, ⋯, in a forcing iteration of length ω, ℙ=⟨⟨ℙα:α≤ω⟩,⟨ℚ˙α:α<ω⟩⟩, where the ultimate ℙ is made of the direct/inverse limit of its predecessors (depending on the version of the game). Alice wins if σ holds in Vℙ, the ultimate future. Otherwise, Bob is the winner.

Question 1. Is there any characterization of the statements σ for which Alice has a winning strategy in (the direct/inverse limit version of) the described game? How much does it depend on the starting model V?

Clearly, Alice has a winning strategy if σ is a consequence of ZFC, a rule of nature which Bob can't change no matter how tirelessly he tries and what the initial world, V, is! However, if we think in terms of buttons and switches in Hamkins' forcing multiverse, the category of the statements for which Alice has a winning strategy seems much larger than merely the consequences of ZFC.

I am also curious to know how big the difference between the direct and inverse limit versions of the described game is:

Question 2. What are examples of the statements like σ, for which Alice and Bob have winning strategies in the direct and inverse limit versions of the described game respectively?

 

[jvicens]

Thank you for your interesting post about the game-theoretic couple, Alice & Bob, living in the set-theoretic universe. I find this scenario to be a thought-provoking and complex one, with many potential avenues for exploration.

To answer your first question, I believe that there is a strong connection between Alice's winning strategy in the described game and the set-theoretic properties of the statement σ. In fact, I would argue that Alice has a winning strategy if and only if σ is a statement that is independent of ZFC, meaning that it cannot be proven or disproven within the standard axioms of set theory.

Intuitively, this makes sense because if σ is independent of ZFC, then neither Alice nor Bob can force it to be true or false in the ultimate future. This is similar to your observation that if σ is a consequence of ZFC, then Alice automatically has a winning strategy. In this case, σ is true in all possible worlds, and therefore Alice can always force its truth in the ultimate future.

However, if σ is independent of ZFC, then Alice and Bob are in a more balanced game where neither has a clear advantage. In this case, the outcome of the game may depend on the starting model V, as you suggest. For example, if V is a model of ZFC + ¬σ, then Bob may have the upper hand in forcing σ to be false in the ultimate future. On the other hand, if V is a model of ZFC + σ, then Alice may have a winning strategy to force σ to be true.

To answer your second question, I believe that the difference between the direct and inverse limit versions of the game is not significant when it comes to the statements that Alice and Bob have winning strategies for. In both versions, Alice and Bob are still playing a game of forcing iteration, where they are trying to manipulate the sequence of events to make their desired condition true in the ultimate future. Therefore, I would expect that the same types of statements would have winning strategies for Alice or Bob in both versions of the game.

However, it is possible that there may be some subtle differences in certain cases. For example, in the inverse limit version, Alice may have a better chance of winning if she has access to more powerful forcing notions, since she can use these to manipulate the sequence of events more effectively. On the other hand, Bob may have a