ABSTRACT
As well as models of how we compute, there are models of how we accumulate knowledge. In this chapter I am going to briefly review the basic logic needed for the most familiar of these models — the axiomatic theory. We will also begin to look at the close links between the axiomatic model
and our model of computability. When we know more about these two kinds of model, we will see that the parallels between them will help our understanding of both. For instance, in Chapter 8, we will see that one of the most important mathematical results of the last century — Go¨del’s Incompleteness Theorem — is reducible to a simple fact from computability theory. Remember — it was this theorem which had such negative implications for Hilbert’s Programme, in particular in relation to the Provability Theme I mentioned in Chapter 1. At this point I should mention that Chapters 3 and 8 dealing with logic can
be safely missed out, if you are only interested in the computability theory.
