From Diagonalization to Infinity: the Utility of Negation Space

This paper attempts an experimental approach of defining a "negation space" which is diagonalized away from all identities defined by the Law of Identity. Both Godel’s G and Turing’s diagonalizing algo D are fixed points in this negation space, supporting its ontological reality. This negation space is shown to serve as a novel derivation of zero, thus allowing a unique starting point for chains of diagonalized identities. Using the diagonalization function x ↔ ¬P1(x) and this zero, we show that self-referential fixed points can be determined for each of the ordinals (and therefore the natural numbers and cardinals). In other words, once negation space is established, the same logic that produces the Burali-Forti Paradox (and Cantor’s Paradox) can be seen as a constructive basis for the foundation of infinity via self-referential fixed points. This allows for a direct derivation of arithmetic foundations that are only axiomatically assumed in Peano Arithmetic and ZFC, eliminating the need for the Axiom of Infinity. The creation of such a negation space and similar diagonalization approaches can also be used to derive the reals and complex numbers. This diagonalization approach coincides with the Neti Neti concept of consciousness or Witness, i.e., Neti Neti’s semantic definition ("I am not this, I am not that") seems semantically equivalent to our diagonalization function, x ↔ ¬P1(x). Philosophically, this experimental approach may reconceptualize the relationship between logic and diagonalization, changing the ontological basis of mathematics from being focused on object ("noun") consistency to also guaranteeing function ("verb") consistency. Whereas ZFC creates classes to avoid the unconstrained computation of the Burali-Forti Paradox and Cantor’s Paradox, the diagonalization approach gives ontological truth status to absence. If the framework holds, what it seems to introduce is a convergence between axiomatic-deductive mathematics and computational mathematics at an ontological level, allowing for new epistemologies of arithmetic and logical foundations