Carnap...

Carnap, Schlick, and other logical empiricists applied the methods of structural analysis to produce a wide-ranging critical and reformative project, conceived by at least some of its adherents as having radical and utopian social consequences as well.[^141] Especially in its pejorative application against “metaphysics,” the project involved, as recent scholarship has demonstrated, significant and central misunderstandings of Wittgenstein’s original project.[^142] Nevertheless it demonstrated the relevance of the specific methods of logical analysis to broader questions of philosophy of science, politics, and culture, and consolidated the legacy of these methods for the logically based styles of philosophical analysis and reflection that became more and more popular, especially in the USA and Britain, following World War II.

Around the same time, the continuation, by philosophers associated more or less directly with the Circle’s central project, of Frege’s original attempt to display the logical foundations of mathematics, produced a set of radical results, of a mostly negative character, that demonstrated in a fundamental way the inherent instabilities involved in the attempt to analyze their structure.

Kurt Gödel’s 1931 “On Formally Undecidable Propositions of Principia Mathematica ” reported what would become the best-known and most historically decisive of these results, the two famous “incompleteness” theorems showing that any consistent axiomatic system powerful enough to describe the arithmetic of the natural numbers will formulate truths that cannot be proven within that system.

The result was widely perceived as demonstrating the failure of the logicist program of reducing mathematics to logic that had been begun by Frege and continued by Russell and Hilbert. It turned on the possibility of constructing, in any sufficiently strong system, a sentence asserting its own unprovability within that system. The resulting sentence is true but, since it is true, cannot be proven.

In reaching it, Gödel used the metalogical technique of “arithmetization” to represent the syntax of a formal system, including the notions of proof and consequence, within that system itself. Working independently with a similar metalogical technique, Alfred Tarski showed in 1933 the indefinability of arithmetical truth within a formal system of arithmetic[^143] .