A Preliminary Twistor Model Toward Metalogic Geometry: From Projective Geometry Perspectives

Integration science is an advancement of cognitive science. This paper opens a new topic called metalogic geometry that aims to integrate metalogic with the mathematical twistor theory. We first revisit Gödel methods used in metalogic including Gödel numbering, expressibility, definability, self-referential statement, and proof. Second, it revisits the core ideas of the twistor theory. To follow the Penrose idea: Light rays as twistors, we define the notions of Gödel ray and Penrose cone. By the expressibility and definability, a pair of Gödel numbers compose a Gödel ray (or Tarski ray). A family of Gödel rays composes a Penrose cone. The intersection point of a Penrose cone yields a Gödel point. Gödel rays are projected as twistors. Each Gödel point is projected to a Riemann celestial sphere. Twistors and Riemann spheres assemble the picture of twistor space. The meaning of this work is discussed in the concluding remarks.