Cyklus' Number
Cyklus' Number is a proposed large number that surpasses all known large numbers, including Rayo's Number. It is designed to be not only mathematically larger than any previously defined number but also fundamentally beyond human and computational comprehension. The number is intended to be defined through multiple independent frameworks, ensuring that it surpasses all conceivable limits in logic, computation, physics, and novel mathematical constructions.
Rayo's Number is the largest number definable in first-order set theory with a limited number of symbols. To exceed this, Cyklus' Number must be defined within a formal system more expressive than that used to define Rayo's Number. This could involve second-order set theory, large cardinal axioms, or an entirely new meta-system designed to encapsulate and extend all known formal mathematical expressions.
Definition: Cyklus' Number is the output of the fastest-growing function definable in a system stronger than that used to define Rayo's Number.
In computability theory, some functions grow too quickly for any Turing machine to compute, such as the Busy Beaver function BB(n). Cyklus' Number is intended to be uncomputable even by machines that transcend traditional computational limits, including oracle machines that can decide the Halting Problem.
Definition: Cyklus' Number is the largest number that is uncomputable by any physically or theoretically realizable computational model, including hypercomputers with access to undecidable problems.
From a physical standpoint, any number expressible in our universe is bounded by fundamental physical limits such as the Bekenstein Bound, which sets a limit on the amount of information that can be stored in a given region of space. The number of possible quantum states in the observable universe is estimated to be around 10^120. Cyklus' Number must therefore exceed all conceivable numbers derivable from known physical laws and even hypothetical multiverse models.
Definition: Cyklus' Number is a number that exceeds the total number of possible quantum states of the largest possible physically realizable multiverse.
A novel approach to defining Cyklus' Number is through a self-referential recursive function that outgrows all known functions in an iterative process. Suppose we define a function CYK(n) where each step produces a function that grows faster than any previously defined function in any system known up to that step. If we let CYK(∞) be the limit of this process, we obtain a number that is recursively beyond all describable mathematical constructs.
Definition: Let CYK(n) be a function where CYK(n+1) grows faster than any function definable within the system that produced CYK(n). Cyklus' Number is CYK(∞), meaning it transcends all known formal systems recursively.
To emphasize its extreme magnitude, Cyklus' Number is sometimes described in terms of data storage. If stored as raw data, it would require 3.333... (decimal) CYKbytes, where 1 CYKbyte = 1024 geopbytes, and 1 geopbyte = 1024 brontobytes. The unit hierarchy follows the standard binary progression, making the CYKbyte an astronomical unit of data that exceeds any current or foreseeable data storage capacities, reinforcing that Cyklus' Number is incomprehensible in conventional terms.
Cyklus' Number is a conceptual extension of large number theory that aims to push beyond all known mathematical, computational, and physical limits. By defining it through multiple independent approaches, it establishes itself as a number that truly surpasses everything in this universe and beyond. While its precise formalization is still in development, its role as a theoretical upper bound on expressible numbers makes it a unique and compelling concept in googology.