Consider the language of groups. Show that the theory $T= \{\sigma\mid A\vDash\sigma\}$ is a Henkin theory, where $A$ is a fixed nontrivial group.

Question

This is Van Dalen's exercise 3.1.1. We wish to show $T= \{\sigma\mid A\vDash\sigma\}$ is a Henkin theory, where $A$ is a fixed non-trivial group.

I'm familiar with the fact that a Henkin theory is a theory where all sentences of the form $\exists x\varphi(x)$ exhibit a witness. That being said, my guess on the best way to do this is as follows:

Let $\varphi$ be an arbitrary formula, and consider the sentence $\exists x\varphi (x)$.

With this specific $\varphi$, construct a constant symbol $c$ that is a witness for this specific sentence $\exists x\varphi (x)$.

I'm not too sure how to go about this, though.... If someone could help, that would be great.

Answer 1

See page 110: "Show that $T$ is not a Henkin theory.

Hint

Consider the language of groups and their axioms (page 80).

We have one constant $e$ for the neutral element.

What is a non-trivial group? A group that has more than one element, i.e. a group where there is at least one element $x \ne e$.