It is well known that the Kolmogorov complexity is uncomputable in Turing-complete programming languages. However, what about total programming languages?
For example, is the Kolmogorov complexity of natural numbers computable if we work in Gödel's System T? (I suppose not, but I don't have a proof.) Furthermore, is it known what is the strongest computational model in which the Kolmogorov complexity is computable?