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This is an English summary of my Japanese blog post submitted to a Japanese googology event.


History

It is well-known that Rayo's number cannot be formalised in \(\textsf{ZFC}\) set theory, because it refers to the truth in "the universe". More precisely, Rayo originally intended to refer to the truth in this real universe, which might make sense in philosophy but does not make sense in matheatics. On the other hand, it is natural to consider the truth in the universe \(V\) of first order set theory instead. The precise meaning of the statement "Rayo's number cannot be formalised in \(\textsf{ZFC}\) set theory" is that the alternative definition referring to the truth in \(V\) is not formalisable in \(\textsf{ZFC}\) set theory as long as it is consistent.

Two years ago, I invented a new strategy to "imitate" the truth predicate in \(V\) within \(\textsf{ZFC}\) set theory, and constructed a \(\textsf{ZFC}\)-variant of Rayo's function called CoRayo function. It refers to the truth predicate of each segment of von Neumann hierarchy instead of \(V\) itself, and hence is formalised in \(\textsf{ZFC}\) set theory. Although the actual strength is unknown, I expected it to be the strongest function formalised in \(\textsf{ZFC}\) set theory due to an evidence derived from reflection principle for each finite segment of \(\textsf{ZFC}\) set theory.


Definition

Now I contructed a \(\textsf{ZFC}\) variant of Fish number 7 called CoFish number 7. The strategy is quite simple and natural. Since \(m(m,n)\) map in the definition of Fish number 6 ​can be generalised by replacing the defining formula of \(m(0,2)\) by the equality with an arbitrary functional \(\mathbb{N}^{\mathbb{N}} \to \mathbb{N}^{\mathbb{N}}\), I just replaced it by the CoRayo functional instead of the Rayo system in the definition of Fish number 7. (Since the definition of \(m(m,n)\) map is not so easy to understand, I added introductions and precise formulations for the reader who does not know Fish numbers 6 and 7, though.)

Namely, CoFish number 7 is just Fish number 7 except that I formally replaced the occurrence of the Rayo functional in its definition by the CoRayo functional. Since Fish number 7 itself is just Fish number 6 except that Fish formally replaced the defining formula of \(m(0,2)\) as the equality with the Rayo functional, CoFish number 7 can be described just as Fish number 6 except that Fish formally replaced the defining formula of \(m(0,2)\) as the equality with the CoRayo functional.


Analysis

By definition, CoFish number 7 is much geater than CoRayo number, which I expected to be the largest number formalised in \(\textsf{ZFC}\) set theory. Therefore I expect that CoFish number 7 is the largest number formalised in \(\textsf{ZFC}\) set theory.

By the way, this is not the first time when I invented a googological system inspired by Fish's works. My system for 巨大数楼閣数 is a generalisation of Fish's \(\textrm{TR}\) function using the strategy to strengthen systems using functions, which is also the main strategy of Fish numbers. 巨大数楼閣数 makes sence as long as you fix a formal theory \(T\) in which you are working, e.g. \(\textsf{ZFC}\) set theory, like \(TR\) function, and is expected to be the strongest computable large number whose well-definedness is provable in \(T\). For example, if we are working in \(\textsf{ZFC}\) set theory as usual, i.e. setting \(T = \textsf{ZFC}\), then 巨大数楼閣数 is a computable large number formalised in \(\textsf{ZFC}\) set theory and is expected to be the strongest among computable large numbers formalised in \(\textsf{ZFC}\) set theory.

If my expectations of the strength of 巨大数楼閣数 and CoFish number 7 are correct, then it is interesting that both of the currently strongest compputable large number formalised in \(\textsf{ZFC}\) set theory and the currently strongest uncomputable large number formalised in \(\textsf{ZFC}\) set theory are inspired by Fish's works.

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