 Like others in the Vienna Circle, Carnap saw the Tractatus as a basis for solving the perennial problem of accommodating the truths of logic and mathematics to empiricism. Wittgenstein was no empiricist, but his characterization of logical truth as a pure artifact of representation, empty of empirical significance, gave the Vienna Circle what they needed. Their logicism extended Wittgenstein’s truth-functional conception of logic to the whole of mathematics, and their empiricism interpreted Wittgenstein’s atomic sentences as sense perceptions, along the lines of the Elementarerlebnisse at the basis of Carnap’s Aufbau. This version of Wittgenstein made mathematics consistent with empiricism. However, its weaknesses were obvious even to the Circle: First, from the empiricist perspective of the Vienna Circle, Wittgenstein’s truth-functional conception of logic seemed to require finitism; even the most abstract theories of physics could then employ only restricted, finitary quantification rather than unrestricted quantification over a potentially infinite number of observations ( Aufbau: §180; Carnap 1936–37: §23). And second, Wittgenstein’s exclusion of “elucidations” (meta-sentences about the object sentences framed in the language of science) was not only paradoxical (it made all statement and discussion of logical principles meaningless) but seemed at odds with recent developments in the foundations of mathematics, especially Hilbert’s metamathematics, and the brilliant use made of this approach by Skolem, Tarski, and by one of Carnap’s own students, Kurt Gödel. (In turn, Gödel’s arithmetization of syntax and his incompleteness theorems would have an enormous impact on Carnap’s Logical Syntax of Language just a few years later—see the supplement on  Logical Syntax of Language  for more details.)