TITLE: Kolmogorov complexity is the strongest noncomputable function
QUESTION [19 upvotes]: Yury I. Manin says that Kolmogorov complexity (in some nontrivial sense) is the strongest noncomputable function ("Колмогоровская сложность... невычислима... она во многих интересных смыслах заслуживает титул универсальной невычислимой функции... в некотором смысле слова(нетривиальном) это такая самая сильная невычислимость которая может существовать":http://youtu.be/nnZPqnwoD64?t=15m39s).
What is the exact wording of this statement?
upd: my translate of dialog on this video:
designations and definitions: 
Let $u$ - is partial recursive function between $\mathbb{Z}_+$ and $X$, where $X$ is countable set.
$K_u(x) = min \{ m \in \mathbb{Z}_+ | u(m) = x\}$ or infinity
Statement: $\exists u ($optimal Kolmogorov numeration$): \forall v($function as$ u) \exists c(u, v) > 0  \forall x \in X: K_u(x) \le K_v(x)$ 
Kolmogorov's order$(u) \mathbb{Z}_+ \to X$ - in order of order Lower_to_higher
Dialog:
Misha Verbitsky: Is $u$ bijection?
Yury Manin: No, and it is focus and big trap. $u$ is not define on many $n$, many
Misha Verbitsky: And no for any $m$ too?
Yury Manin: It get all $x$.
Michael Tsfasman: No, because sometimes
Yury Manin: Optimal, optimal get all $x$. Not all $m$ are programs only some. And its non-computable... Furthermore it (in many interesting ways) deserves the title of the Universal noncomputable function. 
Somebody: Complexity?
Yury Manin: Complexity and Kolmogorov's order too.
Somebody: Order? It is order of increasing complexity?
Yury Manin: Order of increasing complexity. They are non-computable. In a manner (nontrivial) it is the strongest noncomputable. If you have oracle that give you things in Kolmogorov's order then very mach things became computable (I'll show it for codes). I wrote somewhere that civilization is such oracle, that we do produce scientific knowledge in order of increasing its Kolmogorov complexity...

REPLY [4 votes]: .


'Now, a Turing-degree is the set of all problems that are Turing-equivalent to a given problem. What are some examples of Turing-degrees? Well, we've already seen two examples: (1) the set of computable problems, and (2) the set of problems that are Turing-equivalent to the halting problem[1]. Saying that these Turing-degrees aren't equal is just another way of saying that the halting problem isn't solvable.
Are there any Turing-degrees above these two? In other words, is there any problem even harder than the halting problem? Well, consider the following "super halting problem": given a Turing machine with an oracle for the halting problem, decide if it halts! Can we prove that this super halting problem is unsolvable, even given an oracle for the ordinary halting problem? Yes, we can! We simply take Turing's original proof that the halting problem is unsolvable, and "shift everything up a level" by giving all the machines an oracle for the halting problem. Everything in the proof goes through as before, a fact we express by saying that the proof "relativizes."'----Lecture by Scott Aaronson


[1]Finding the Kolmgorov complexity is Turing equivalent to deciding the halting problem.(For how,look at the comments by @usul below this post here.) 
So we see there is an arithmetical hierarchy of oracles ( and even intermediate ones) and as such there is no strongest Oracle or its equivalent non computable function (as an example this non computable function  can be  taken as  Chaitin's $\omega$ for its class.)
In my opinion ; all problems which are "natural" and have actually arisen  in practice
; the hardest akin to NP hard; can be solved by the halting oracle.. hence perhaps Kolmogorov complexity in the strongest noncomputable function which occurs in * practice*.
 We will have to qualify the above as it is a subjective statement eg the function AreThereInfinitelyManyTwinPrimes() can not be calculated by the Halting oracle and hence more in-computable than Kolmogorov complexity K() here on MO