  Brouwer’s example shows that there is a case where the double negation principle in the form of \(\forall x\in \mathbb{R}(\neg \neg Px \rightarrow Px)\), leads to a contradiction (“The Non-equivalence of the Constructive and the Negative Order Relation on the Continuum”, in Dutch). The first publication of a strong counterexample in English had to wait till 1954, in “An Example of Contradictority in Classical Theory of Functions” (1954F). This polemical title should be understood as follows: if one keeps to the letter of the classical theory but in its interpretation substitutes intuitionistic notions for their classical counterparts, one arrives at a contradiction. So it is not a counterexample in the strict sense of the word, but rather a non-interpretability result. As intuitionistic logic is, formally speaking, part of classical logic, and intuitionistic arithmetic is part of classical arithmetic, the existence of strong counterexamples must depend on an essentially non-classical ingredient, and this is of course the choice sequences.