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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