I'm pretty certain that the author was making the subtle suggestion that Gödel would delight in concocting questions that would confound Wolframs' Alpha.
Fans of Stanislaw Lem might look at it this way...
("Have it compose a poem about a haircut. But lofty, noble, tragic, timeless, full of love, treachery, retribution, quiet heroism in the face of certain doom! Six lines, cleverly rhymed, and every word beginning with the letter S!")
Just a guess here, but maybe the author thinks that Kurt Gödel actually "...is widely regarded as the most important innovator in scientific and technical computing today."?
Then again, I don't even understand what the term 'technical computing' means.
I also completely miss the connection and I clearly wouldn't put those two in the same league. Godel was a very thoughtful if troubled soul who totally shook the foundations of mathematics.
No, that would require a modicum of modesty. I think Mr Wolfram compared himself somewhere in writing to Newton himself, but I cannot recall where I read that.
Some of Kurt Godel's most famous works are his 2 incompleteness theorems, which state the inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest.
Which basically means that there are things that just can't be computed but are quite obvious for us humans.
Firstly, the incompleteness theorem (there's only one) is not about computation but about proofs; specifically that there are some things within an axiomatic system which are true which cannot be proven such within the system itself.
Secondly, this does not mean that these things are unprovable, period. For instance, Gödel showed that you can prove these things perfectly fine, you just have to use different system. But this system will have its own unprovable statements.
Thirdly, computability was Turing's domain, not Gödels, and has to do with the kinds of algorithms you can use to perform proofs, not the system the proof is for. For example: second-order logic (SOL) can prove things about first-order logic (FOL) that FOL cannot prove about itself, but both systems can by utilized by a universal turing machine.
Fourthly, there is no known thing that cannot be proven, period, which is obvious for us humans.
There are two things called his incompleteness theorems, but only one actually deals with completeness. The other deals with consistency, which is not coming up with contradictory answers. Basically, it says that any system that asserts it is consistent is necessarily inconsistent. This is I suppose related to things which are true about the system which cannot be proven true in the system, but only in a roundabout way.
Computability and completeness are also definitely related, but, not quite so tightly as you implied. Computability is a property of algorithms, while completeness is a property of axiomatic systems. Algorithms operate on axiomatic systems, but are not the systems they implement, and there's no inherent connection between being able to prove statements about a system, and the algorithms used to compute proofs for that system. As I said, SOL can prove FOL to be complete and consistent, and you can do this on a universal turing machine, but FOL cannot prove itself to be complete and consistent even doing this on a universal turing machine.
Of course, if your computer _is_ an axiomatic system (and all are), then the incompleteness theorem(s) do say things about what that computer can do, and we know that all turing complete computers have precisely the same sorts of constraints in that regard.
The real thing that was irksome, tho, was the assertion that there are some things that are obvious to humans but which are uncomputable. This is just unfounded. Unless you're supposing human brains are somehow not computing things (collections of neurons are computers, after all, just not von Neumann architecture), there is no way for this statement to be true.
It's because of the self-referential, contradictory nature of Wolfram Alpha's answer. Kind of like trying to assign a truth value to the statement "this statement is false".
By putting in "how to program", and having it misinterpret it as "What is the world's most powerful computational software?", having the answer come out as "Wolfram Mathematica", and having the whole thing be powered by Wolfram Mathematica, you get a contradiction (maybe even 2)... in Gödelian style.
Not only that. Wolfram's attempt to take Godel's place in mathematics is contradicted by this answer, which is a self referential demonstration that the Wolfram axiomatic system (both the person and Alpha) cannot produce the genius of Godel, and is thus incomplete. Thus, Wolfram merely proves Godel's genius, via the original form of Godel's genius in the first place.
Basically, Wolfram is ingenious at proving his disingenuity.
Because Alpha's praise of Mathematica and Wolfie is like a rear world example of the incompleteness theorem? The limits of errr... I'm stretching this as far as I can, but it's not enough.
