Math 557 Sep 19
Henkin Theories
Key Concepts
An \(\mathcal{L}\)-theory \(T\) is a Henkin theory if for every sentence of the form \(\exists x \psi\) there exists a constant \(\mathcal{L}\)-term \(t\) such that \[ T \vdash_{\mathcal{L}} \exists x \psi \; \to \; \psi_{t/x} \]
Questions:
- What are Henkin theories needed for?
- Do they exist?
- Can we extend a given consistent theory to a Henkin theory that is still consistent?
Problems
The basic construction step for extending an \(\mathcal{L}\)-theory \(T\) to a Henkin theory is for every sentence of the form \(\sigma \equiv \exists x \psi\),
- add a new constant symbol \(c_\sigma\) to \(\mathcal{L}\)
- add the formula \(\exists x \psi \; \to \; \psi_{c_\sigma/x}\) to \(T\).
The Henkin extension \(T_H\) of \(T\) is obtained by an iterative process:
Single iteration step:
- \(\mathcal{L}' = \mathcal{L} \cup \{c_\sigma \colon \sigma \: \mathcal{L}\text{-sentence of the form } \exists x \psi \}\)
- \(\Gamma = \{ \exists x \psi(x) \to \psi_{c_\sigma/x} \colon \sigma \: \mathcal{L}\text{-sentence of the form } \exists x \psi \}\)
- \(T' = T \cup \Gamma\) (an \(\mathcal{L}'\)-theory)
Iteration process:
- \(\mathcal{L}_0 = \mathcal{L}, \; T_0 = T\)
- \(\mathcal{L}_{n+1} = \mathcal{L}'_n, \; T_{n+1} = T_n'\)
- \(\textstyle \mathcal{L}_H = \bigcup_{n \in \mathbb{N}} \mathcal{L}_n, \; T_H = \bigcup_{n \in \mathbb{N}} T_n\)
\(T_H\) is no longer a purely symbolic extension of \(T\) (in the sense that we simply extend the language), since we add new sentences to \(T\) (the sentences in \(\Gamma\) for each iteration step).