Introduction[]
Infinite-Class Googology, otherwise known as ICG, is a formation of Googology invented by the user AbsolutelyArsenal in order to attempt to solve a problem he set the day beforehand: ‘The Class Number, C, is defined as the first number larger than 1, for which C belongs in Class C of Fictional Googology’. His solution was to remove the limit on the number of classes entirely.
Definitions[]
- A concept is defined as an entity which has been imagined, and therefore made to exist under the Existential Axiom (EA).
- A number is a concept which has a numerical value, to which it can be compared with other numbers.
- A formulation is defined as a method from which numbers can be generated.
- A class is defined as a set which is defined uniquely by a numerical formulation, with which new numbers can be formulated from the formulation of the class, in addition to all formulations which define a class below said class.
- An operation is defined as a set function which operates on a given input or set of inputs, and provides an output which is directly dependent on the input.
Fundamental Axioms[]
- The Axiom of Limitless Value, also known as the ALV, assures that there exists no smallest or biggest number. Mathematically, for every number A, there exists two numbers B and C, for which B<A<C.
- The Axiom of Limitless Class, also known as the ALC, assures there exists not a largest class, to which every possible number can be generated by formulations defining classes up to that class.
- The Axiom of Numerical Formulation, also known as the ANF, assures that, for every number, there necessarily exists at least one formulation which can create that number.
- The Axiom of Classification, also known as the AC, assures that every number can be put into one and only one class.
- The Axiom of Inclusivity of Formulation and Class, known as the AIFC, assures that there is not a formulation to which not a number can be created from. By extension, there exists not an empty class.
- The First Axiom of Incompleteness of Formulation, also known as the AIF1, assures that there is not a formulation to which there exists not a number which cannot be created from the formulation. For every formulation, there necessarily exists a number which the formulation is unable to generate.
- The Second Axiom of Incompleteness of Formulation, also known as the AIF2, assures that there is not a formulation to which there does not exist a number which cannot be created, given said formulation, and the formulations of every class under that which the formulation defines. In essence, there is no class which can contain every number.
- The Axiom of Formulation of Incremental Class, also known as the AFIC, assures that for every class, there exists a higher class defined by a higher formulation, for which there exists a number which was previously not able to be formulated by the previous class, but can be formulated given the higher class.
- The Axiom of Incompleteness of Infinite-Class Googology, also known as the AIICG, assures that there exists at least one number which cannot exist given the premises of ICG.
- The Axiom of Uniqueness of Infinite-Class Googology, also known as the AUICG, assures that there exists at least one number which can only exist by the premises of ICG, and not in any other possible mathematical formulation.
- The Axiom of Intermediate Value, also known as the AIV, assures that for any two numbers A and B, where A<B, there necessarily exists a value C which satisfies A<C<B.
- The Axiom of Exclusivity of Identity, also known as the AEI, assures that there is not a single operational identity which applies to every number. There is no additive or multiplicative identity common to every class and formulation in ICG.
- The Axiom of Exclusivity of Operation, also known as the AEO, assures that for every operation with infinite possible outputs, there necessarily exists at least one input or set of input values, for which the operation cannot be performed.
- The Axiom of Description, also known as the AD, assures that there exists not a number which cannot be at least partially representable by a lexical or numerical description defining what it can or cannot be.
- The Axiom of Eternal Conception, also known as the AEC, assures that once a concept has been created, it cannot be removed or deleted.
- The Axiom of Operations in Class, also known as the AOC, assures that there exists not an operation which can give an output of a different class to the input.
Universal Operations[]
A universal operation is defined as an operation which can operate on any set of inputs. This does not break the AEO because there are not infinite possible outputs. There are only four basic universal operations.
- The equality function, =: This takes in two arguments, A and B. The result of A=B is TRUE if the mathematical value of A is equal to that of B; FALSE if otherwise. As an example, if P is defined as ‘0+1’ and Q is defined as ‘2-1’, then P=Q returns TRUE.
- The equivalence function, ==: This takes in two arguments, A and B. The result of A==B is TRUE if the mathematical value of A is equal to that of B and their definition is equivalent, and FALSE otherwise. As an example, if P and Q are defined as above, then P==Q returns FALSE because one is defined by addition and one is defined by subtraction.
- The lesser value comparison function, <: This takes in two arguments, A and B. The result of this operation is TRUE if the mathematical value of A is less than that of B; and FALSE otherwise.
- The greater value comparison function, >: This takes in two arguments, A and B. The result of this operation is TRUE if the mathematical value of A is greater than that of B; and FALSE otherwise.