Forcing as a computational process

Por • 20 jul, 2020 • Sección: Educacion

Joel David Hamkins, Russell Miller, Kameryn J. Williams

We investigate how set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for information about a model of set theory ⟨M,∈M⟩, we explain senses in which one may compute M-generic filters G⊆P∈M and the corresponding forcing extensions M[G]. Specifically, from the atomic diagram one may compute G, from the Δ0-diagram one may compute M[G] and its Δ0-diagram, and from the elementary diagram one may compute the elementary diagram of M[G]. We also examine the information necessary to make the process functorial, and conclude that in the general case, no such computational process will be functorial. For any such process, it will always be possible to have different isomorphic presentations of a model of set theory M that lead to different non-isomorphic forcing extensions M[G]. Indeed, there is no Borel function providing generic filters that is functorial in this sense.

arXiv:2007.00418v1 [math.LO]

Logic (math.LO)

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