Types of Theories

A Theory $\Gamma$ is a nonempty set of $\mathcal{L}(\tau )$-sentences (where $\mathcal{L}(\tau )$ is a logic of vocabulary $\tau$), the sentences $\phi \in \Gamma$ are called ‘axioms.‘

Some notes on the nature of theories

Consistent Theories: A theory $T$ is consistent if $T⊢\neg (\varphi \wedge \neg \varphi )$, or equivalently, $T⊢(\varphi \vee \neg \varphi )$. That is, for a formula $\varphi \in \mathcal{L}[\tau ]$, $T$ can only prove either $p$ or $\neg p$, not both. $\to$ Inconsistent theories (i.e., T is inconsistent if T ($⊢\varphi \wedge \neg \varphi$)) are complete as a consequence of the principle of explosion, the theory T becomes trivial, proving all statements w/in the language (including all $\varphi$ and $\neg \varphi$).

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Complete Theories: A theory $T$ is complete if $T⊢\varphi \vee \neg \varphi$ for all $\varphi \in \mathcal{L}[\tau ]$.

E.g., the simplest example of a complete theory is to consider the set of all sentences true in the structure $\mathcal{M}$, denoted Th($\mathcal{M}$).

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Deductively closed systems: A theory T is deductively closed if every logical consequence (i.e., a theorem of T) is an axiom, that is, T is deductively closed if $T⊢\varphi$ and $\varphi \in T$. This $⟹$ that we effectively cannot prove any ‘new’ statements which are not axioms. $\to$ Recall: $\varphi \in T$ is defined as ”$\varphi$ is an axiom of T.”

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Maximal Theories: A maximal theory T is a theory T where either $\varphi \in T$ or $\neg \varphi \in T$ for any $\varphi \in \mathcal{L}[\tau ],$ this $⟹$ that either $\varphi$ or $\neg \varphi$ is an axiom. A maximal theory can also be defined as having the properties of being both complete and deductively closed. $\to$ This is related to the maximal consistent extension of T

Note: Maximal theories are taken to be consistent

• Inconsistent theories are not necessarily maximal or deductively closed, as only the addition of a single contradictory statement is needed in order to prove any statement via explosion.
• $\to$ i.e., Consider the inconsistent theory $PA+\text{"0=1"}$

Lemma: Suppose $T$ is a maximal and finitely-satisfiable $\mathcal{L}$-theory. If $T_0\subseteq T$ is finite and $T_0\models \phi$, then $\phi \in T$. Proof: If $\phi \notin T$, then, since $T$ is maximal, $\neg \phi \in T$. But this $⟹T\cup \{\neg \phi \}$ is a finite unsatisfiable subset of $T$. Thus, a contradiction.

A maximal consistent extension of $T$ is an expansion of $T$ to a maximal theory $T^{\prime }$ where formulas of the language are added as axioms to $T$ without there being inconsistencies/contradictions. An incomplete theory (that is, there is some $\phi$ such that $\phi$ is undecidable) will have multiple possible maximal consistent extensions as $T\cup \{\phi \}$ and $T\cup \{\neg \phi \}$ are both consistent.

References

Logics, Structures & Models Proof Theory Model Theory D.Marker.pdf The Compactness Theorem