27 - Reflections

Summary

After the glory days of the 1930s, Gödel's comments on the details of his incompleteness theorems were few and far between. However, he did add a brief footnote to the 1967 translation of a much earlier piece on ‘Completeness and consistency’. And Gödel thought that his brisk remarks in that footnote were sufficiently important to repeat them in a short paper in 1972, in a section entitled ‘The best and most general version of the unprovability of consistency in the same system’.

Gödel makes two main points. We explain the first of them in Section 27.2. We then go on to prove some results about reflection principles which hopefully throw light on his second point. And we'll return to develop that second point further in the next chapter, where we touch on Hilbert's Programme.

It will do no harm at all, however, to begin with a summary of …

The Second Theorem: the story so far

(a) Start with the p.r. relation Prf_T (m, n), which obtains when m is the super g.n. of a T-proof of the wff with g.n. n. Assuming T is p.r. adequate, this relation can be canonically captured in T by an open wff Prf_T (x, y) whose components recapitulate step by step the natural p.r. definition of Prf_T (along the lines we gave for the case of PA back in Section 15.9).