In the preceding chapters, we have been guided by the following, seemingly harmless extrapolation from our experience with finite sets: infinite universes can be surveyed in their totality. In particular can we in a global manner determine whether A ⊨ ∃xϕ(x) holds, or not. To adapt Hermann Weyl’s phrasing: we are used to think of infinite sets not merely as defined by a property, but as a set whose elements are so to speak spread out in front of us, so that we can run through them just as an officer in the police office goes through his file. This view of the mathematical universe is an attractive but rather unrealistic idealisation.
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