Forcing and Family Contentions: Who wins the disputes?

[Tompson LEe]

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

Alice wants σσ 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 VV, leading the family to the possible world V[G]V[G]. Then Bob forces over V[G]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˙0Q˙0, ℚ˙1Q˙1, ℚ˙2Q˙2, ⋯⋯, in a forcing iteration of length ωω, ℙ=〈〈ℙα:α≤ω〉,〈ℚ˙α:α<ω〉〉P=〈〈Pα:α≤ω〉,〈Q˙α:α<ω〉〉, where the ultimate ℙP is made of the direct/inverse limit of its predecessors (depending on the version of the game). Alice wins if σσ holds in VℙVP, the ultimate future. Otherwise, Bob is the winner.

Question 1. Is there any characterization of the statements σσ 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 VV?

Clearly, Alice has a winning strategy if σσ is a consequence of ZFCZFC, a rule of nature which Bob can't change no matter how tirelessly he tries and what the initial world, VV, 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 ZFCZFC.

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 σσ, for which Alice and Bob have winning strategies in the direct and inverse limit versions of the described game respectively?

[wtf]

5 hours ago, Tompson LEe said:

The famous game-theoretic couple, Alice & Bob, live in the set-theoretic universe, VV, a model of ZFCZFC.

I'm curious as to why someone who knows forcing would ask this question on this site rather than, say, math.stackexchange where it will get knowledgable responses.

And secondly, surely you know that V is not a model of ZFC. V is a proper class, hence not a model of anything. And if we knew ZFC had a model then we'd have a consistency proof for ZFC, which we haven't got.

A strange post.

[studiot]

13 hours ago, wtf said:

A strange post.

Yes. +1

[taeto]

What happened to the math typesetting in that strange post though? Whatever it was, it is something I would want to avoid