NEVEREND

NEVEREND has been redefined on November 27th, 2023 by FG googologist Wiftyx3.

The definition was strenghtened to approach the smallest inaccessible cardinal θ, which is bigger than any non-logically defined ordinal. He stated, that for every domain of $A$, which denoted the non-logically defined ordinals, there would be a bigger domain $B$, that included NEVEREND, along with every other logically defined ordinal that approaches θ, and $A \subseteq B \iff \beta \notin A$ where $\beta$ denotes NEVEREND.

Keep in mind that $\alpha < {\beta} < \theta : \alpha \in A \land \beta \in B \land {\theta} \in C$. We can still define bigger variants of NEVEREND using a formula for all fixed points, but first, let's define this ordinal using logical formulas, so we can construct our fixed point formula.

Let NEVEREND denote $\forall A \exists B \exists C(\alpha \in A \iff\beta\notin A \land \beta \in B \iff \theta \notin B \land\theta \in C \land A \subseteq B \rightarrow B \subseteq {C} \implies A \subseteq {C}:\alpha < \beta < \theta)$

To define bigger variants, we are going to use $[\psi ]$ formulas. Let's define every fixed point of NEVEREND.

The fixed point of NEVEREND is $f : [\psi]_n \mapsto [\psi]_{[\psi]_\alpha}$. This tells us, that if a formula of type $n$ would get applied to the same formula, with a type of some formula that has the type of an ordinal $\alpha$ - which will always be the current ordinal or the ordinal's $n$-th fixed point - , we get every fixed point of NEVEREND.

More on this number later...