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. 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:

This question is inspired by one question, which was about representations that can be realised homologically by an action on a graph (i.e., a 1-dimensional complex).Many interesting integral representations of groups arise via homology from a group acting on a simplicial complex that is homotopy equivalent to a wedge of spheres. A classical example is the action of groups of Lie type on spherical buildings. On homology this gives an integral form of the Steinberg representation.One may ask if there exists a complex of lower dimension than the Tits building that realises the (integral) Steinberg representation in this way. I am guessing that the answer is No, but how to prove it?More generally, given an integral G-representation that can be realised as the homology of a spherical complex with an action of G, is there an effective lower bound on the dimension of such a complex? One obvious lower bound is given by the minimal length of a resolution by permutation representations. Is this something that has been studied?

What does (x-a)(x-b)(x-c)...(x-z) equal to? Solution :