proving the completeness of first-order logic,

Gödel began by devising a method that would transform statements about the process of proof into statements about the properties of numbers.

be 2ℵ0, which will thus equal one of those larger alephs.

6 Stability was the personal virtue that stood above all others, and that faith was rewarded by a society that could be counted on to return it in full measure.

advanced economic liberalism and expanding democracy—both of which now unleashed forces that turned against them. Only 6 percent of Austrians had the right to vote in the 1860s. But the subsequent expansion of the franchise empowered populist and nationalist movements whose competing demands could not be met.

he plainly disagreed with the reverence for Wittgenstein’s idea that mathematics, like language, was merely a tool, a set of rules or a syntax that had no inherent meaning in itself.

collection of working parts consisting of neurons and their interconnections. If, however, the mind is nothing but a calculating machine, then it is subject to the limitations of the Incompleteness Theorem, which leads to the thorny fact that numbers possess at least some properties that are beyond the power of the human mind to establish: “So this

every even number greater than 2 is the sum of two primes.

completeness and decidability.