He would have been interested in Ed Lorenz' work (from the butterfly effect fame) I suppose. According to Tim Palmer:
"Despite being known for his pioneering work on chaotic unpredictability, the key discovery at the core of meteorologist Ed Lorenz's work is the link between space-time calculus and state-space fractal geometry. Indeed, properties of Lorenz's fractal invariant set relate space-time calculus to deep areas of mathematics such as Gödel's Incompleteness Theorem."
"Consider a point p in the three-dimensional Lorenz state space. Is there an algorithm for
determining whether p belongs to IL? There are certainly large parts of state space which don’t
contain any part of IL. However, suppose p was a point which ‘looked’ as if it might belong to
IL. How would one establish whether this really is the case or not? If we could initialise the
Lorenz equations at some point which was known to lie on IL, we could then run (1) forward to
see if the trajectory passes through p. If the integration is terminated after any finite time and
the trajectory still hasn’t passed through p, we can’t really deduce anything. We can’t be sure
that if the integration was continued, it would pass through p at some future stage.
The Lorenz attractor provides a geometric illustration of the Gödel/Turing incompleteness
theorems: not all problems in mathematics are solvable by algorithm. This linkage has been
made rigorous by the following theorem [7]: so-called Halting Sets must have integral Hausdorff
dimension. IL has fractional Hausdorff dimension - this is why it is called a fractal. Hence we can
say that IL is formally non-computational. To be a bit more concrete, consider one of the classic
undecidable problems of computing theory: the Post Correspondence Problem [46]. Dube [16]
has shown that this problem is equivalent to asking whether a given line intersects the fractal
invariant set of an iterated function system [4]. In general, non-computational problems can all
be posed in this fractal geometric way."
I wonder what is meant by this question. In a very naive sense of the question, the type of mathematics Gödel worked on (foundational?) wasn't really computation oriented or even formulaic, so the ability to perform lots of calculations very quickly probably wasn't of much use to him. Indeed, computers where available to him in later years.
A more interesting interpretation of the question (to me) is what if he had access to computers used as mind amplification devices. For example; such as by using Mathmatica or Maple to explore and visualize theorems and things. I'd imagine the benefit of this activity for someone like Gödel would be "not much". Computations and simulation have inherit limitations; precision and rounding errors for scientific computation, and the fact they can only model what we can imagine for another. These people such as Gödel, Neumann, and their ilk, new this, they begat the era of computation we have today, and all the limitations that involved. Neumann in particular was famous for, when presented with your problem, would tell how to solve it.
What's new today that they may not have foreseen is the vast level of internetworking and human communication that arose from ubiquitous presence of computers and networks.
Something that strikes me about the present article is the fact that Gödel, keen to see Rucker before he knew him, was not so keen to converse with him after their first encounter. One might think Gödel was not too impressed with Rucker, maybe found him boring and dull for example.
"What's new today that they may not have foreseen is the vast level of internetworking and human communication that arose from ubiquitous presence of computers and networks."
In an alternate universe, Godel published his proofs in a paper on arXiv in 2002. However it was largely overlooked or dismissed as the work of a crank, so he went back to his day job at MSR.
A casual reference to the paper in a comment on on Math Overflow five years later led to wider discussion and eventually to Scott Aaronson publicising it on his blog. Within a year the two theorems were accepted by the global mathematics community.
Since then, numerous blogs, subreddits, and Facebook posts have challenged the legitimacy of the proofs or claimed prior credit. Speculation persists on some parts of the internet that GCHQ or the NSA knew of the theorems as early as the 1950s.
Perhaps the idea of computers would have been more useful than the computers themselves. Several of the ideas in his proof are simpler if you can say "computable" rather than "recursive" or whatever. So perhaps if he had been thinking in these terms he could have done more work sooner.
The way things turned out the work from this era defined what computers can and can't do, not the other way around, which would likely not satisfy many mathematicians.
He was right there at the Institute for Advanced Study (IAS) when they built the seminal IAS machine whose plans were distributed widely to help germinate computing around the world.
