Gödel's third incompleteness theorem (Q2834879)

To state that a formal system is incapable of proving its own consistency, as Gödel's second incompleteness theorem does, one needs to know what it means for a sentence of formal number theory, which is a statement about numbers, to `express' the consistency of that formal system. A set of conditions a provability predicate needs to satisfy so that Gödel's second incompleteness theorem holds for it can be found in in Volume 2 of the \textit{Grundlagen der Mathematik} by Hilbert and Bernays. In the 1960s examples of provability predicates were constructed for which the corresponding consistency statement are provable for strong theories of elementary number theory.NEWLINENEWLINEIn a note appended to the translation of \textit{On consistency and completeness} (1967), in which Gödel revisited the problem of the unprovability of consistency, the focus is on alternative means of expressing the consistency of a formal system, in terms nowadays referred to as `reflection principles' (``A form al system \(S\) for number theory is said to satisfy a \textit{reflection principle} for a class \(K\) of formulas if a formula of \(K\) is derivable in \(S\) only if it is true (in the standard model); \(K\)-reflection is the property of satisfying the reflection principle for \(K\)''). \(S\) is \textit{computationally sound} if it satisfies the reflection principle for the class of quantifier-free sentences of arithmetic. When S is an extension of elementary number theory, computational soundness, consistency, and the \(\Pi_1\) reflection principle for \(S\) are equivalent. ``Gödel argued, in effect, that it is the alternative expression of the notion of consistency in terms of \(\Pi_1\) reflection that we should be interested in from a foundational point of view; and he states a result that shows certain instances of the \(\Pi_1\) reflection principle for a theory meeting the conditions of the second incompleteness theorem to be underivable in the theory under assumptions about the provability predicate significantly weaker than the Hilbert-Bernays conditions.''NEWLINENEWLINEThe author of this paper discusses the background of Gödel's result and its philosophical significance, and presents a generalization of Gödel's result.