# Matteo Viale: Absolute model companionship, forcibility, and the Continuum Problem

13/05/21 15:11

Tuesday, May 18, 2021 17:00

For a theory \(T\) , \(T_{\exists\lor\forall}\) 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 \(T_{\exists\lor\forall}=T^*_{\exists\lor\forall}\).

The \(\{+, ·, 0, 1\}\)-theory ACF of algebraically closed field is the model companion of the theory of Fields but not its AMC as \(\exists x (x^2+1=0)\in ACF_{\exists\lor\forall}\setminus Fields_{\exists\lor\forall}\).

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

ZFC+there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently stronger notion of consistency for any mathematical problem \(\varphi\) expressible as a \(\Pi_2\) -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 \(\varphi\)).

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

*Location:***Zoom.us**: if you want to participate please contact organizers*Matteo Viale (University of Torino)*

Speaker:Speaker:

*Title*: Absolute model companionship, forcibility, and the Continuum Problem*Abstract*: Absolute model companionship (AMC) is a strengthening of model companionship defined as follows:For a theory \(T\) , \(T_{\exists\lor\forall}\) 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 \(T_{\exists\lor\forall}=T^*_{\exists\lor\forall}\).

The \(\{+, ·, 0, 1\}\)-theory ACF of algebraically closed field is the model companion of the theory of Fields but not its AMC as \(\exists x (x^2+1=0)\in ACF_{\exists\lor\forall}\setminus Fields_{\exists\lor\forall}\).

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

ZFC+there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently stronger notion of consistency for any mathematical problem \(\varphi\) expressible as a \(\Pi_2\) -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 \(\varphi\)).

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