The Tractatus Logico-Philosophicus consists of seven sets of numbered propositions or statements, which are believed to be about the connection between language and the world. It seems to present an account of the essence of language as expressive of thought. Thought, according to Wittgenstein, is limited to what is factual so that the propositions of language are representations of the world. The propositions of logic, on the other hand, convey no factual information—logic consists of tautologies. Logic is very useful, but all of its conclusions are true by definition.

Wittgenstein believed that a meaningful sentence must have a precise structure that is made up of simple (in Russell’s language, “atomic”) sentences or simple names. Atomic sentences are pictures of states of affairs. Working backwards from this “picture theory of meaning” it would follow that, given the ideal logical language, the world itself has a logical structure.

Wittgenstein was to later abandon this view in favor of philosophical activity that consisted of descriptive analysis of ordinary language. But before he did that, the Tractatus had enormous influence on the new twentieth century school of thought known as logical positivism.

Kurt Gödel (1906–1978) is famous for his theorem about mathematical systems, which appeared in a 1931 article titled “On Formally Undecidable Propositions in Principia Mathematica and Related Systems,” originally published in German in the 1931 volume of the journal Monatshefte für Mathematik (Monthly Journal of Mathematics). According to Gödel’s Theorem, every formal (mathematical or logical) system is incomplete because there can always be a sentence expressing a truth that can’t be proved in the system. To prove his theorem, Gödel invented a method for correlating formulas in logic with positive integers.