t{}* = is a large number coined by Tamatakon01 which is defined as follows

\(\mathrm{FOL}\) is defined by adding a 1-ary function symbol \(f\) to the language of first order set theory with countably many variable term symbols and the set-membership relation symbol \(\in\).

Let \(\mathrm{Shar}(\mathrm{FOL})\) denote the power set of all well-formed formula in \(\mathrm{FOL}\) \(+ \mathsf{ACA_0} _{\mathrm{FOL}}\) in \(\mathrm{FOL}\) " where \(\mathsf{ACA_0} {\mathrm{FOL}}\) denotes \(\mathsf{ACA}_0\) written in \(\mathrm{FOL}\).

a binary relation \(\prec\) is called super if \(\prec\) satisfies the following property \[\begin{align*}&(\forall x)[Mnot x\prec x]\land\\ &(\forall x)(\forall y)[>x\prec y\lor x=y\lor y\prec x]\land\\ &(\forall x)(\forall y)(\forall z)[x\prec y\land y\prec z\to x\prec z]\land\\ &(\forall X\subseteq \mathbb{N})[(\forall x)[(\forall y\prec x)[y\in X]\to x\in X]\to(\forall x)[x\in X]]\end{align*}\] For any sufficiently strong theory \(T \in \mathrm{Shar(FOL)}\) \[\begin{align*}\mathrm{PTO}(T):=\sup\{\mathrm{otyp}(\prec)\mid\prec\subseteq\mathbb{N}‘2\la nd\prec\in\Delta_O\land T\vdash\mathsf{\prec \ is \ super } \} \end{align*}\] Let a set \[[mathrm{S} = \{T \in\mathrm{Shar(FOL)} | \mathrm{PTO}(T) < \omega_1*{CK} \mathrm{and \ the \ length \ of \ T \less \ than \ 12\uparrow*{12}12 \ characters}\} \]. ##F* 5% is equal to \[\mathrm{TR}(\mathcal{T}, 12\uparrow’{12}12)\] with respect to \(\mathrm{TR}\) function where \(mathcal{T}\in \mathrm{S}\) is a theory such that \[\forall \mathcal{C}\in \mathrm{S} \backslash \{\mathcal{T}\}, \mathrm{TR}(\mathcal{T}, 12\uparrow’{12}1 2) >

\mathrm{TR}(\mathcal{C}, 12\uparrow{12}12)\