The compactness theorem for countable (Tarski?) models is equivalent to the weak König's lemma by a result of H. Friedman and others as noted [here](https://mathoverflow.net/questions/228389), in the context of classical logic. The weak König's lemma in constructive mathematics has been extensively studied as noted [here](https://mathoverflow.net/questions/228468). What is the status of the *compactness theorem for countable models* itself in constructive mathematics.


One relevant paper seems to be *Moerdijk, Ieke; Palmgren, Erik. Minimal models of Heyting arithmetic. J. Symbolic Logic 62 (1997), no. 4, 1448–1460.*


One relevant issue is the mutual relation of the following three items:


(1) compactness theorem for countable models;


(2) weak Koenig's lemma;


(3) lesser limited principle of omniscience (LLPO).


In what constructive setting are these principles equivalent? More specifically: how much of the proof of equivalence sketched in the answer to [this question](https://mathoverflow.net/questions/228389) can be retained in a suitable constructive setting, using references mentioned in the answer to [this question](https://mathoverflow.net/questions/228468)?



Constructive compactness for countable models?
I share Vladimir Kanovei's view that the syntactical approach might be a good oppportunity to constructive NSA in much the same way as it was for Nelson's Internal Set Theory.


