 In 1908, Brouwer had shown that \(\neg \neg(A \vee \neg A)\); in 1923, when Hilbert’s program was in full swing, this result inspired Brouwer to say that “We need by no means despair of reaching this goal [of a consistency proof for formalized mathematics]”; see section 2.1 above for the  full quotation.  (At the time, Brouwer suspected that \(\neg \neg A \rightarrow A\) was weaker than PEM; Bernays quickly corrected this impression in a letter to Brouwer (Brouwer 1925E: 252n.4 [Mancosu 1998: 292n.4]).)