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](https://mathoverflow.net/questions/136371) but not enough to resolve the issue.



Compactness in Bishop's constructive mathematics
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**](http://dx.doi.org/10.1305/ndjfl/1153858649), *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](http://www.jucs.org/jucs_11_12/a_direct_proof_of). 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](http://rml.tcs.uj.edu.pl/rml-47/3-Schuster.pdf). Reports on Mathematical Logic 47: 63-86 (2012)


