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Forcing and Family Contentions: Who wins the disputes?

Tompson Lee

New member
Jul 6, 2018
4
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?