To answer the question based on the comments already posted:
Are the Godel's incompleteness theorems valid for both classical and
In a sense YES.
Goedel's incompleteness theorem(s) apply first to classical logic
Goedel's incompleteness theorem and its proof is constructive but not intuitionisticaly constructive (Goedel's paper)
Goedel himself stated in his paper that the above procedure is "constructively non-objectionable", however
a) Goedel's reference to contructivism (intuitionism), is rather formal than actual (more detailed below)
b) variations of LEM (law of excluded middle) are used throughout Godel's proof
c) combined with the use of a diagonalisation procedure
(see also Gödel’s Proof and Intuitionism for another analysis)
Does this apply the same to intuitionistic logic?
In a sense YES.
- Goedel's negative translation of classical logic into intuitionistic logic is only formal (Goedel's paper)
a) negative translation of classical logic to intuitionistic logic is not intuitionism, rather a formal analogy, because the semantics of what constitutes a construction, a proof, implication and of course the definition/construction of new entities based only on previously constructed entities is totally different, being classical than intuitionistic (and same holds for the original incompleteness proof, where these conditions are neither formalised nor met) (see also Kolmogorov's Interpretation of Intuitionistic Logic as Problems)
b) intuitionism has, in a sense, already embedded the incompletenmess theorems as it accepts statetements which can neither be proved nor refuted (at a certain given time)
c) Brouwer himself foresaw Goedel's results by a decade at least (note: Goedel himself had attended Brouwer's lectures on the foundations of mathematics)
Quoting from Artemov's Understanding Constructive Sematics (Spinoza Lecture)
And also from here
Intuitionistic vs. Classical Perspective
Intuitionists normally base their formal systems on intuition of constructive, e.g., BHK-style informal semantics, rather then on
Classical mathematicians (such as Gödel, Kolmogorov, Kleene, Novikov, and others) seek a rigorous
classical definition of the constructive semantics.
And from A. Heyting, Intuitionism An Introduction
In the light of the above Goedel's incompleteness results do indeed hold for intuitionistic logic in a formal way (with classical semantics) but not for intuitionism (which in any case does not need any incompleteness result as they are already embedded in the practice and semantics)