First-order logic and computer science

In the previous article, we discussed the halting problem and its relation to Godel’s incompleteness theorem. The essence is as follows: suppose I am programmed to do something I don’t predict I will (this is allowed by the definition of a computer). Fundamentally, this means that where \(T\) is my theory of my mind and \(G\) is the thing I will do, \(T\) entails that \(G\iff \lnot (T\vdash G)\). Our belief in \(G\) then stems from our belief in the soundness of \(T\), that \((T\vdash G)\implies G\) – thus if \(T\) were sound, it wouldn’t believe in its own soundness.