One of his incompleteness theorems is similar to the liar paradox.
Liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). - Wikipedia
The paradox of Godel's 1st incompleteness theorem makes a similar assertion to that of the liar paradox.
What does this have to do with Welles?
A movie from 1949: The Third Man.
In a previous post I mentioned Charlie's observation that: "if the mathematicians can't get the paradox out of their system when they're creating it themselves, the poor economists are never going to get rid of paradoxes, nor are the rest of us."
Which brings me to a classic scene* from that movie. Here's a setup of the monologue from that scene by Filmsite.org...
With murderous fluency, he [Harry Lime played by Welles] contemplates the greater productivity of a warring, strife-ridden culture and civilization that is plagued by warfare and violence, versus a peaceful one. The corruptible Lime cynically justifies his black market criminal activities by recognizing that despite appearances, good and evil (black and white, peace and war, up and down, etc.) are complementary concepts.
...and the monologue as delivered by Welles (with a smug grin):
"In Italy for thirty years under the Borgias, they had warfare, terror, murder, bloodshed - but they produced Michelangelo, Leonardo Da Vinci, and the Renaissance.
In Switzerland they had brotherly love, 500 years of democracy and peace, and what did they produce? The Cuckoo Clock."
Unlike Citizen Kane (1941), the movie was not written by Orson Welles but he did come up with that monologue.
*The actors in this scene [Joseph Cotton and Orson Welles] both first appeared together in Citizen Kane.