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IIn the context of a weak formal theory called Basic IntuitionisticMathematics $\mathsf{BIM}$, we study Brouwer's Fan Theorem and a strongnegation of the Fan Theorem, Kleene's Alternative (to the Fan Theorem). Weprove that the Fan Theorem is equivalent to contrapositions of a number ofintuitionistically accepted axioms of countable choice and that Kleene'sAlternative is equivalent to strong negations of these statements. We alsodiscuss finite and infinite games and introduce a constructively useful notionof determinacy. We prove that the Fan Theorem is equivalent to theIntuitionistic Determinacy Theorem, saying that every subset of Cantor spaceis, in our constructively meaningful sense, determinate, and show that Kleene'sAlternative is equivalent to a strong negation of a special case of thistheorem. We then consider a uniform intermediate value theorem and acompactness theorem for classical propositional logic, and prove that the FanTheorem is equivalent to each of these theorems and that Kleene's Alternativeis equivalent to strong negations of them. We end with a note on a possiblyimportant statement, provable from principles accepted by Brouwer, that onemight call a Strong Fan Theorem.

The Fan Theorem, its strong negation, and the determinacy of games