 Following the logicism of Frege and Russell, the Circle considered logic and mathematics to be analytic in nature. Extending Wittgenstein’s insight about logical truths to mathematical ones as well, the Circle considered both to be tautological. Like true statements of logic, mathematical statements did not express factual truths: devoid of empirical content they only concerned ways of representing the world, spelling out implication relations between statements. The knowledge claims of logic and mathematics gained their justification on purely formal grounds, by proof of their derivability by stated rules from stated axioms and premises. (Depending on the standing of these axioms and premises, justification was conditional or unconditional.) Thus defanged of appeals to rational intuition, the contribution of pure reason to human knowledge (in the form of logic and mathematics) was thought easily integrated into the empiricist framework. (Carnap sought to accommodate Gödel’s incompleteness results by separating analyticity from effective provability and by postulating that arithmetic consisted of an infinite series of ever richer arithmetical languages; see the discussion and references in section 3.2 below.)