**In Collaboration with Adam Cooper-Terán, René López, Regina García, Stefano Gaudiano, Juan Arras, Paloma Rincón, y Ese Chuy**

This project is about generating a space to tell the story of two of the most important mathematical events in the 20th century. The first, is Kurt Gödel’s Incompleteness proof and the second is Alan Turing’s Halting problem. These two proofs are relevant because they show the limits of reason–therefore the limits of formal systems (e.g., the computer), and allow to sense the immeasurability of the infinite. The present project is not only an attempt to explain and visualize the proofs but seeks to present them within a historical and social context.

It took me more than 10 years to arrive to this result. After many conversations with friends and collaborators. I’m so happy to finally arrive to this, a full infinite circle.

### PRELUDE

We begin with the best explanation there is about the nature of numbers, that is set theory explained by Sesame Street.

*Sesame Street: Fish to Infinity*

Understanding numbers as sets soon takes us into problems, problems dealing with infinity. Can we have sets of infinities? Can we have larger sets with infinities? This can create paradoxes as it is shown in Russell’s Paradox.

*Russell’s Librarian’s Catalog–Set Theory Paradox*

Then, to understand the context and importance of Gödel and Turing’s discoveries, we can watch this documentary, clearly explaining how everything began with Cantor and his idea that there were infinities larger than others infinities.

*Dangerous Knowledge BBC documentary*

### INCOMPLETO

The exhibition continues with the original work below by Adam Cooper-Terán. It is an animation summarizing different characters and elements surrounding the historical period when the Incompleteness proof by Kurt Gödel and the Halting problem by Alan Turing were developed.

*Perfect Forms by Adam Cooper-Terán*

### KURT GÖDEL

*Mirror-Kurt Gödel’s Incompleteness Proof by Regina García*

In this animation the essence of Gödel’s incompleteness proof is represented. A set of numbers represent two states at the same time. Each set represent the value of the numbers and specific phrases. With this proof, he showed that a logical system could never have all the rules required to develop arithmetic. Such system no matter what, will always be incomplete.

*Adele: Gödel’s wife by Adam Cooper-Terán*

This film shows Gödel’s personal life. His wife playing two roles simultaneously, as a burlesque dancer and as a housewife. The idea was to make her represent two states of mind at the same time just like the Incompleteness Proof. Gödel’s problems with food are alluded too.

We close this section with a pencil drawing summarizing Gödel’s life.

*Kurt Gödel by René López*

### ALAN TURING

*Turing’s Halting Problem by Regina García*

The central idea of Turing’s Halting Problem is represented in this video. That is, Turing continues Gödel’s attempt to show the limits of reason and formal systems. Back in the early developments of modern science, a dream was born. To construct a mechanical machine that could compute for very long periods of time searching for answers to complex problems. Turing showed that it is impossible to have that. The reason is that we cannot know ahead of time whether the machine is really solving a problem or whether it is in a loop. Also, we cannot know systematically ahead of time which problems can be solved by a computer and which ones cannot.

*Manuscript Found in a Pocket by Adam Cooper-Terán*

In this animated video we present an adaptation of the story Manuscrito Encontrado en un Bolsillo (Maunscript Foound in a Pocket) from Julio Cortazar. We played with the idea that Alain Turing used a logical system to find romantic love within the London subway. So he will use random number cards to predict a head of time where his potential lovers will change stations, if they changed stations according to the cards, he will follow and approach them, if not, he would have to let them go.

We close this section with a pencil drawing summarizing Turing’s life.

*Alan Turing by René López*

## EXHIBITION

Incompleto was presented as an art show at Cub0 Gallery in Ciudad Juárez. It was important to present it in November 2019 to make it a homage to the movie Blade Runner as it is set in this date. It connects with Incompleto because after Turing wrote the paper (On Computable Numbers with Applications to the Entscheidung-Problem, 1936) where he creates the Turing machine and the Halting Problem (the basis for artificial intelligence and its limits), he writes a paper in which he explores whether a machine can be as smart and capable as a human–making it impossible to distinguish between a machine and a computer (Computing Machinery and Intelligence, 1950). He explored this idea with his Turing test. Such test is emulated in Blade Runner as a test to identify androids (The novel Blade Runner: Do Androids Dream of Electric Sheep? was published in 1968, years after Turing’s paper). At some point he was worried that computers will feel different and thus bad about themselves for not being like humans.

*Megan Fox in Blade Runner*

It was also important to present the exhibition in a white small cube. We designed the space so as to feel on one hand very rational, straight lines, white. But also on the other hand, we emulated the idea of a padded room from a mental asylum. Cantor and Gödel, were secluded in such places a few times. As it is stated in the documentary Dangerous Knowledge (2007), they were put there so that the rational architecture would make them come back to their senses–the very two people testing and finding the limits of reason.

*Incompleto exhibition at Cub0 Gallery*

The poster and program were designed to emulate the idea of an insane asylum but also dedicated to the androids who died in November 2019.

*Incompleto poster and program by Juan Arras*

We decided to include the idea of a revelation. Revelation as something that can surpass reason and can be more powerful. The idea was created when I told my friend Stefano Gaudiano that once upon a time when I was dancing, I had the revelation of two giants appearing and revealing a deep message. Such message can not be put into words and thus has to remain secret. This is how he interpreted this. This work was outside the gallery, as a revelation escaping reason. The drawings also lead to Incompleto as as it is itself a revelation about the limits of reason.

*Revelation-Incompleto by Stefano Gaudiano*

Below is some of the documentation of visitors to the event:

*Visitors inside the exhibition*

I would like to thank Paloma Rincón and Ese Chuy for helping me with the design of the exhibition space. And to Neida Molina y Octavio Castrejón for being space custodians.

*Gabriela Durán, Paloma Rincón y Ese Chuy*