The cheese numbers[1] are a hierarchy of numbers coined by googology wiki user Mumuji.

The system is ill-defined as we will explain later, and hence the numbers are ill-defined.


Definition

We copy the definition from the original source:[1]

It starts as follows:
c_1 = f_{22222222222}(22222222222222222222222)
c_{n +1} = f_{gamma_{gamma_{c_n}}(c_{n})
Where gamma is the gamma in the gamma nought ordinal.
And _ is subscript
And f is FGH.
n-cheese numbers
1-c_m = c_m
n-c_m = c_{((n-1)-c_m)}
After the equals sign, the first - is minus and second is for the nth n-cheese number.

We will explain problems of the original definition in the next section.


Issues

Similar to other articles based on personal websites which are not peer-reviewed, the system has many errors. For example, the use of variables without quantification makes the definition ambiguous. The lack of a fixed definition of fundamental sequences makes the fast-growing hierarchy ill-defined.

Due to the lack of the precise quantification of a variable and the precise definition of the domain, there are two ways to compute 1-c_1:

  1. It can be evaluated as c_1 by the first rule.
  2. It can be evaluated as c_{0-c_1} by the second rule.

However, 0-c_1 is ill-defined because the rule yields an infinite loop.

It is a common mistake for googologists to only specify rules to evaluate without specifying the domain. Since the notion of a function is defined as a pair of the domain and the assignment, the lack of the domain makes the function ill-defined in the way above.


Intended behaviour

Here, we copy explanation by the creator:

According to the definition, it means:
\(c_0 = f_{22222222222}(22222222222222222222222)\)
\(c_{n+1} = f_{\Gamma_{\Gamma_{c_n}}}(c_{n})\)
It can be extended as the following:
\(1-c_m = c_m\)
now for the nth cheese number:
\(n-c_m = c_{((n-1)-c_m)}\)

So, this means that for something like \(3-c_3\), we compute it as follows: \(3-c_3 = c_{(2-c_3)}\) and so on. Unfortunately, the explanation is incorrect. For example, the value of c_1 deffers from the original one, and the value of c_0 was not defined in the original definition.


References

  1. 1.0 1.1 Mumuji, Cheese Numbers. Retreived 00:20, 31 May 2021 (UTC).
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