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In Bishop's constructive mathematics, is there any literature on a possible version of the weak König's lemma, or of the compactness theorem for countable models? There is some related information here but not enough to resolve the issue.

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A detailed analysis of König's lemma in Bishop-style constructivism was carried out by Hajime Ishihara, Josef Berger and Helmut Schwichtenberg. Some references:

  1. Hajime Ishihara, Weak König’s lemma implies Brouwer’s fan theorem: a direct proof, Notre Dame J. Formal Logic 47 (2006), no. 2, 249--252.

  2. H. Schwichtenberg: A direct proof of the equivalence between Brouwer’s fan theorem and König’s lemma with a uniqueness hypothesis. J. UCS 11(12): 2086-2095 (2005)

  3. Josef Berger, Hajime Ishihara, Peter Schuster: The Weak Koenig Lemma, Brouwer's Fan Theorem, De Morgan's Law, and Dependent Choice. Reports on Mathematical Logic 47: 63-86 (2012)

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