What Does Godel's Second Incompleteness Theorem Show?

Abstract

NO CONSISTENT formal system that contains a certain amount of finitary number theory can prove its own consistency. This statement of Gddel's second Incompleteness Theorem, which is more or less in G6del's own words,' is imprecise but the imprecision is remediable. What counts as a consistent formal system, what constitutes its containing the requisite amount of finitary number theory and what it is for a system not to be able to prove its own consistency are all susceptible of exact specification. Nevertheless, in textbook treatments of it, the second Incompleteness Theorem (unlike the first) does not very often receive a fully general statement which is any more precise than this. Often it is applied to some particular formal system, to yield the result that such and such a sentence is not a theorem of that system; and if it is then characterized as a generalization of this result, and the generalization is spelt out, then the statement above is typical of what one finds.2 One explanation for this is that precision here is particularly hard to come by and, once attained, very much lacking in intuitive impact. But I shall argue that there are also non-heuristic reasons why it can be more appropriate to invoke this imprecise statement than a corresponding precise version. First, I need to broach the question of what a corresponding precise version would look like. This in turn requires some declaration of the canons of precision that I am adopting. I shall proceed to lay these down stipulatively. Given any mathematical claim, there will be some theory in which it is embedded (be it set theory, number theory, analysis or