## Archive for September 2nd, 2010

### Theories and Reduction

September 2, 2010

A theory $\Gamma$  is a set whose members are just the sentences for a language $\textup{L}$ that follow from the set.  In other words, $\Gamma$ is a theory in the language $\textup{L}$ if, and only if, $\Gamma$ is closed under logical consequence -i.e.,

$\Gamma \models \psi$ and $\psi$ is a sentence $\Longrightarrow \psi \in \Gamma$.

The elements of $\Gamma$ are the theorems of $\Gamma$.

Hellman notes that a theory, construed syntactically in this way, is essentially connected to the notion of reducibility, but is not so connected to the notion of determination. I’ll talk about reduction now, and then in the next update I’ll deal with determination.

Reduction applies in the realm of definitions, which are syntactic entities, that facilitate the elimination of definable terms.  He elaborates on this point in a footnote.  If in $\alpha$-structures, $\phi$ reduces $\psi$, there’s a not-necessarily-recursively-enumerable theory where every definition enabling the reduction of $\psi$ to $\phi$ is provable.  And its provability doesn’t depend on whether or not $\alpha$ is the set of models that theory.  Once you have $\alpha$, you can set out the theory,

$\bigcap \{\gamma : (\exists m) (m \in \alpha \wedge m$ is a model of $\gamma )\}$

This theory is the intersection of the theories of each of the models in $\alpha$ and contains every definition needed to reduce $\psi$ to $\phi$.    Physical Reduction, PR, is equivalent to,

In $\{ m: m$ models $\bigcap \{\gamma : (\exists m') (m' \in \alpha \wedge m'$ is a model of $\gamma )\}\}$, $\phi$ reduces $\psi$.

This means that if reducibility holds for a collection of structures, then, and only then, it holds for the set of models for all sentences that are true in each member of that set of structures (even if the set of structures is a proper subset of the set of all model of sentences true in each of those structures).

This won’t work for determination, as it is not so connected to the syntactic nature of theories.