What is Gödel's second incompleteness theorem?

Great mathematical theorems often have cloud-like identities, being more like a cloud of non-equivalent formulations than being one sharp result. Gödel’s incompleteness theorems are no exception to this rule. In the case of the Second Incompleteness Theorem, the situation is even more dramatic. There seems to be no precise mathematical formulation that covers our intuitive ‘coordinate-free’ understanding of the theorem. How to formulate (a reasonable version of) the theorem? In my talk I address this question and provide some further information about the theorem. To be specific:

• I provide a reasonably general preliminary formulation of the Second Incompleteness Theorem.
• I discuss the matter of intensionality in metamathematics. An inportant example of intentionality is the fact that whether one has the second incompleteness theorem or not may depend on the choice of the representation of the axiom set.
• I consider two ways to address the problem of giving a coordinate-free formulation of the theorem.
• I provide some examples of less well-known applications of the theorem.