# Absolute model companionship, the continuum problem, and forcibility

06.05.2021 15:00 - 16:30

Matteo Viale (Università degli Studi di Torino, Italy)

Absolute model companionship (AMC) is a strengthening of model companionship defined as follows:

For a theory $$T$$, $$T_{\exists\vee\forall}$$ 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 $$T_{\exists\vee\forall}=T^*_{\exists\vee\forall}$$.

The theory $$\mathsf{ACF}$$ of algebraically closed field is the model companion of the theory $$\mathsf{Fields}$$ of fields but not its AMC as $$\exists x(x^2+1=0)\in \mathsf{ACF}_{\exists\vee\forall}\setminus \mathsf{Fields}_{\exists\vee\forall}$$. Any model complete theory $$T$$ is the AMC of $$T_{\exists\vee\forall}$$.

We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) $$2^{\aleph_0}=\aleph_2$$ is the unique solution to the continuum problem which can be in the AMC of a partial Morleyization of the $$\in$$-theory $$\mathsf{ZFC}$$ enriched with large cardinal axioms.

We also show that (assuming large cardinals) forcibility overlaps with the apparently stronger notion of consistency for any mathematical problem $$\psi$$ expressible as a $$\Pi_2$$-sentence of a (very large fragment of) third order arithmetic ($$\mathsf{CH}$$, the Suslin hypothesis, the Whitehead conjecture for free groups, are a small sample of such problems $$\psi$$).

Partial Morleyizations can be described as follows: let $$F_{\tau}$$ be the set of first order $$\tau$$-formulae; for $$A\subseteq F_\tau$$, $$\tau_A$$ is the expansion of $$\tau$$ adding atomic relation symbols $$R_\phi$$ for all formulae $$\phi$$ in $$A$$ and $$T_{\tau,A}$$ is the $$\tau_A$$-theory asserting that each $$\tau$$-formula $$\phi(\vec{x})\in A$$ is logically equivalent to the corresponding atomic formula $$R_\phi(\vec{x})$$. For a $$\tau$$-theory $$T$$, $$T+T_{\tau,A}$$ is the partial Morleyization of $$T$$ induced by $$A\subseteq F_\tau$$.

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