2021/09/06 by Matteo Viale, Viale, Matteo
Mathematics · Medicine · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Neurological and metabolic disorders
paper · pdf · doi:10.48550/arxiv.2109.02285
Absolute model companionship (AMC) is a strict strengthening of model companionship defined as follows: For a theory T, T∃\vee∀ denotes the logical consequences of T which are boolean combinations of universal sentences. T^* is the AMC of T if it is model complete and T∃\vee∀=T^*∃\vee∀. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) 2ℵ0=ℵ2 is the unique solution to the continuum problem which can be in the AMC of a "partial Morleyization" of the ∈-theory ZFC+"there are class many supercompact cardinals". We also show that (assuming large cardinals) forcibility overlaps with the apparently weaker notion of consistency for any mathematical problem ψ expressible as a Π2-sentence of a (very large fragment of) third order arithmetic (CH, the Suslin hypothesis, the Whitehead conjecture for free groups are a small sample of such problems ψ). Partial Morleyizations can be described as follows: let Formτ be the set of first order τ-formulae; for A⊆ Formτ, τA is the expansion of τ adding atomic relation symbols Rϕ for all formulae ϕ in A and Tτ,A is the τA-theory asserting that each τ-formula ϕ(x)∈ A is logically equivalent to the corresponding atomic formula Rϕ(x). For a τ-theory T T+Tτ,A is the partial Morleyization of T induced by A⊆ Formτ. Finally we characterize a strong form of Woodin's axiom (*) as the assertion that the first order theory of Hℵ2 as formalized in a certain natural signature is model complete.