25 - The derivability conditions

Summary

In the last chapter, we gave some headline news about the Second Theorem for PA, and about how it rests on the Formalized First Theorem. In this actionpacked chapter, we'll say something about how the Formalized First Theorem is proved. More exactly, we state the so-called Hilbert-Bernays-Löb derivability conditions on the provability predicate for theory T and show that these suffice for proving the Formalized First Theorem for T, and hence the Second Theorem. We'll also prove a very nice theorem due to Löb.

More notation

To improve readability, let's introduce some neat notation. We will henceforth abbreviate Prov_T(⌜ϕ⌝) simply by ▪_Tϕ. So Con_T can now alternatively be abbreviated as ¬▪_T⊢.

Two comments. First, note that our box symbol actually does a double job: it further abbreviates the long predicative expression already abbreviated by Prov_T, and it absorbs the corner quotes that turn a wff into the standard numeral for that wff's Gödel number. If you are logically pernickety, then you might be rather upset about introducing a notation which in this way rather disguises the complex logical character of what is going on. But my line is that abbreviatory convenience here trumps notational perfectionism.

Second, we will very often drop the explicit subscript from the box symbol, and let context supply it. We'll also drop other subscripts in obvious ways. For example, here's the Formalized First Theorem for theory T in our new notation:

The turnstile signifies the existence of a proof in T's deductive system.