# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 151: Henkin Witness (Logic)

Let $T$ be a theory and $\exists x \phi (x)$ be a sentence in $T$. A witness of this sentence is a constant $c$ such that $T \vdash \phi (c)$.

Similarly, a theory $T$ is said to have the witness property if wherever $\phi(x)$ is a formula with 1 free variable $x$ there is a constant $c$ such that

$T \vdash ((\exists x \phi(x)) \rightarrow \phi(c))$,

so that if $T \vdash (\exists x \phi(x))$ then $T \vdash \phi(c)$.

Witnesses and the witness property play an essential role in certain proofs of the Completeness and Compactness theorems.