I am investigating to what extent extensions of the librationist property or set theory £ may support relative inaccessible sets; see Librationist Closures of the Paradoxes and Elements of Librationism for details on £. The language is as LAST minus identity plus set abstracts as extensionality fails radically; the language of £ also has its own truth set or property $\mathrm{T}$ and an enumerator € which given the semantical set up gives a bijection from $\omega$ (see below) to the full universe.
£ is geared to deal with semantical and set theoretical paradoxes in a novel and fully type free manner, and it is besides that able to recover impredicative classical mathematics in the sense of $ID_{<\omega}+BI$ without further assumptions and it is accounted for by a Herzbergerian style semantics which can be carried out in $\textbf{L}_{\varsigma}$ where $\varsigma$ is the least $\Sigma_{3}$-admissible ordinal. With further postulations concerning the closure of manifestation points (see below) under capture (see below), I have e.g. shown that £ + the Skolem-Frankel Postulation interprets ZFC (see Theorem 21.32 2).
Thinking in £ requires quite some adjustment from the ordinary as we have total liberty in the use of set terms. But if we want the isolated set or property to do mathematical work we need to show that the isolated set term corresponds with a kind set, or sort or property, which is to say that the set is not paradoxical. It e.g. requires some work to show that the Leibnizian-Russellian definition of identity $a=b\triangleq\forall u(a\in u\rightarrow b\in u)$ works and that the set $\{x|x=a\}$ is kind for all $a$. It is also a theorem of £ that $\omega=\{x|\forall y(\emptyset\in y\wedge\forall z(z\in y\rightarrow z'\in y)\rightarrow x\in y)\}$ is kind, where $\emptyset =\{x|x\neq x\}$ and $y'=\{x|x\in y\vee x=y\}$.
We do not straightforwardly have standard transfinite recursion in £, but we have manifestation points; however, we show below in Theorem 2 that versions of transfinite recursion can be captured with manifestation point, and it may be that this can facilitate answers to my main query.
In Theorem 1 here we rehearse the definition of manifestation points in section 6 of 1 for convenience and to show that we can make use of instances with parameters; $\mathbf{T}B$ abbreviates $\exists y(y\in\{y|B\})$ where $y$ does not occur in $B$, $\Vdash_{M}A$ signifies that $A$ is a maxim and thence non-paradoxical and $ACM$ is the alethic comprehension principle (see e.g. 2, p. 13):
Theorem 1:
If $A(x,y,z)$ is a formula with the variable indicated we
can find a term $h(z)$ such that $\Vdash_{M}\forall w,z(w\in h(z)\leftrightarrow\mathbf{TT}%
A(w,h(z),z))$.
Proof:
Let $d=\{\langle x,g \rangle|A(x,\{u|\langle u,g \rangle\in g\},z)\}$ and $h(z)=\{y|\langle y,d \rangle\in d\}$. If we spell out, we have: $h(z)=\{y|\langle y,\{\langle x,g \rangle|A(x,\{u|\langle u,g \rangle\in
g\},z)\}\rangle\in\{\langle x,g \rangle|A(x,\{u|\langle u,g \rangle\in g\},z)\}\}$. By $ACM $
we have that $\Vdash_{M}\forall w(w\in h(z)\leftrightarrow\mathbf{T}
\langle w,\{\langle x,g \rangle|A(x,\{u|\langle u,g \rangle\in g\},z)\}\rangle\in\{\langle x,g \rangle|A(x,\{u|\langle u,g \rangle\in g\},z)\})$,
and so by $ACM $ again it follows that $\Vdash_{M}\forall w(w\in
h(z)\leftrightarrow\mathbf{TT}A(w,\{u|\langle u,\{\langle x,g \rangle|A(x,\{u|\langle u,g \rangle\in g\},z)\}\rangle\in
\{\langle x,g \rangle|A(x,\{u|\langle u,g \rangle\in g\},z)\}\},z)$
But this is $\Vdash_{M}\forall w(w\in h(z)\leftrightarrow\mathbf{TT}A(w,h(z),z))$, as wanted. As we only invoked generalizable prescriptions also $\Vdash_{M}\forall w,z(w\in
h(z)\leftrightarrow\mathbf{TT}A(w,h(z),z))$. Obviously, we can have more than one parameter or a vector of parameters in $A$.
I send the reader to sections 17-19 and 21 of 2 for the definitions of domination ($\mathcal{D}(z,u)$) and capture ($\mathcal{C}(z,u,w)$) used below and just remark here that $u$ indicates the set of parameters allowed and that domination serves to give the set of definable subsets of a set and that librationist capture entails pairing and collection and replacement and specification and choice in the context.
$y \ is \ a \ Skolem \ from \ u:$
$$S(y,u)\triangleq(\omega\in y\wedge\forall z(z\in y\rightarrow\mathcal{D}(z,u)\in y)\wedge \\ \forall w, z(w\in y \wedge \mathcal{C}(z,u,w)\rightarrow z\in y))\wedge\ \\ \forall z(z\in y\rightarrow\cup z\in y))$$
$u \ is \ a \ Fraenkel \ of \ x:$
$$F(x,u) \triangleq \forall y(S(y,u)\rightarrow x\in y)$$
$The \ definable \ echelon \ by \ manifestation \ point$:
$$\Vdash_{M}\forall x(x\in\dot{D}\leftrightarrow\mathbf{TT}F(x,\dot{D}))$$
With the Skolem-Fraenkel Postulation mentioned above and Postulate 20.1 of 2 we are able to show that $\dot{D}$ is kind and hereditarily kind. We are thence able to show that £ via $\dot{D}$ interprets ZFC by a consideration extending an interpretation of ZF in a system S=ZF with collection and with a weak power set minus extensionality by Harvey Friedman in The Consistency of Classical Set Theory Relative to a Set Theory with Intuitionistic Logic, J. of Symbolic Logic, Vol. 38, No. 2, (1973), pp. 315-319.
With these manifestation points we can rather straightforwardly ascend inaccessible cardinals to the point that all sets are members of a Grothendieck universe and to the level of the first hyper inaccessible cardinals when providing further postulations as the Skolem-Fraenkel postulation. I have some hunch that we reach some limit to such a process of reaching extensions of £ somewhere around or below Mahlo, and that this could be of some interest.
To illustrate on how we may start an ascent, let $B(v,w)$ abbreviate $\exists p,q(v=<p,q>\wedge q=\{z|\forall y(S(y,u)\wedge\forall w(w\in p\rightarrow \exists r(<w,r>\in v\wedge r\in y)\rightarrow z\in y)\}$. Let $IN$ be the manifestation point of $B(v,w)$ as per above so that $\forall x,y(<x,y>\in IN \leftrightarrow \mathbf{TT}B(<x,y>,IN))$. As identity is kind-preserving, we have $\vDash_{M}\forall x,y(<x,y>\in IN \leftrightarrow B(<x,y>,IN))$. We let $IN(p)$ abbreviate the operator $\{z|\forall y(S(y,u)\wedge\forall w(w\in p\rightarrow \exists r(<w,r>\in IN\wedge r\in y)\rightarrow z\in y)\}$. We say that u is a hyper of x just if $H(u,x)=\forall y(S(y,u)\wedge\forall z(z\in y\rightarrow IN(z)\in y)\rightarrow x\in y)$. We finally take $\dot{H}$ as the manifestation point of $H(u,x)$ and with some work we show that $\vDash_{M}\forall x(x\in \dot{H}\leftrightarrow H(\dot{H},x)\leftrightarrow \forall y(S(y,\dot{H})\wedge\forall z(z\in y\rightarrow IN(z)\in y)\rightarrow x\in y))$
Adapting 1 section 9 (p. 352) let $KIND(x)\triangleq \forall y(\mathbf{T}y\in x\wedge\mathbf{T}y\notin x)$ and $\mathbb{H}$ be the manifestation point of the formula $KIND(x)\wedge x\subset y$ so that we have $\vDash_{M}\forall x(x\in\mathbb{H}\leftrightarrow\mathbf{TT} (KIND(x)\wedge x\subset\mathbb{H})).$ One should notice that $\nvDash_{M}KIND(\mathbb{H})$. However, if $\vDash_{M}a\in\mathbb{H}$ then $a$ is KIND and iterative and hereditarily KIND and iterative. 1 establishes that $\vDash_{M}\omega\in\mathbb{H}$ and that $\mathbb{H}$ is closed under the Jensen rudimentary functions. Importantly, the ordinary power set operation is notoriously paradoxical in £ for sets or properties $b$ such that $\nvDash_{M}\{x|x=x\}\subset b$ (see section 7 of 1). Moreover, $\mathbb{H}$ is highly non-extensional. It is essential that $\mathbb{H}$ is closed under the notions of domination and librationist capture pointed to above, and it is these features that help us isolate sets in $\mathbb{H}$ which under supplementary postulations interpret ZFC and beyond; Skolem-relative uncountability is induced for domination in the context of $\mathbb{H}$.
Theorem 2: If $F:\mathbb{H}\to \mathbb{H}$ and $A$ is a definable subset of $\mathbb{H}$ then there is a function $f$ such that $\vDash_{M}\forall x,y(<x,y>\in f\leftrightarrow \mathbf{TT}(x\in A\wedge y=F(\{<u,v>|<u,v>\in f\wedge u\in x\}))$.
The proof is by taking the appropriate manifestation point.
Theorem 3: $\vDash_{M}<a,b>\in f\Leftrightarrow \ \vDash_{M}a\in A\wedge b=F(f\upharpoonright a)$.
We leave out the proof of this.
We now attempt to define Mahlo-cardinals. Let $Tr^{2}(x)$ signify that $x$ is a transitive set and that all members of x are transitive and let $ORD=\{x|x\in \mathbb{H}\wedge Tr^{2}(x)\}$. Let $\mathbf{IN}=\{x|\exists y(y\in ORD\wedge x=\{z|z\in IN(y)\wedge Tr^{2}(z)\}$ Let $Func(f)$ express that $f$ is a function, and let $Dom(f)$ denote the domain of $f$. $NFD(x,f)$ stands for `f is a normal function on the ordinals in x', i.e more precisely: $$(f\in\mathcal{D}(x^{2})\wedge Func(f)\wedge Dom(f)=x\wedge(\delta\prec\eta\prec x\rightarrow f(\delta)\prec f(\eta)\prec x)\wedge (Lim(\gamma)\rightarrow f(\gamma)=\bigcup_{\xi\prec\gamma}f(\xi)).$$
Theorem 4: With manifestation point we can define the following function $C$:
$\vDash_{M}<\alpha,\beta, \Xi>\in C\leftrightarrow\mathbf{TT}(<\alpha,\beta>\in Ord^{2}\wedge(\lnot\exists x(x\in\alpha)\wedge\lnot\exists x(x\in\beta)\rightarrow\Xi=\mathbf{IN})\wedge
\forall \kappa(\beta=\kappa +1\rightarrow \\ \Xi=\{\lambda|\exists\Psi(<\alpha,\kappa,\Psi>\in C\wedge\lambda\in\Psi\wedge\forall f(NDF(\lambda,f)\rightarrow \exists\gamma(\gamma\in\Psi\wedge \gamma=f(\gamma))))\})\wedge \\ \forall\kappa(\alpha=\kappa+1\rightarrow\Xi=\{\lambda|\exists\Psi(<\kappa,\beta,\Psi>\in C\wedge\lambda\in\Psi\wedge\forall\delta(\delta\prec\lambda\rightarrow \\ \exists\Upsilon(<\kappa,\beta+\delta,\Upsilon>\in C\wedge\lambda\in\Upsilon)))\})\wedge \\ (Lim(\alpha)\rightarrow\Xi=\{\lambda|\exists\delta\exists\Delta(\delta\prec\alpha\wedge<\delta,\beta,\Delta>\in C\wedge\lambda\in\Delta)\}) \wedge \\ (Lim(\beta)\rightarrow\Xi=\{\lambda|\exists\delta\exists\Delta(\delta\prec\beta\wedge<\alpha,\delta,\Delta>\in C\wedge\lambda\in\Delta)\}))$
Theorem 5:
$\vDash_{M}<\alpha,\beta, \Xi>\in C\Leftrightarrow \ \vDash_{M}<\alpha,\beta>\in Ord^{2}\wedge(\lnot\exists x(x\in\alpha)\wedge\lnot\exists x(x\in\beta)\rightarrow\Xi=\mathbf{IN})\wedge
\forall \kappa(\beta=\kappa +1\rightarrow \\ \Xi=\{\lambda|\exists\Psi(<\alpha,\kappa,\Psi>\in C\wedge\lambda\in\Psi\wedge\forall f(NDF(\lambda,f)\rightarrow \exists\gamma(\gamma\in\Psi\wedge \gamma=f(\gamma))))\})\wedge \\ \forall\kappa(\alpha=\kappa+1\rightarrow\Xi=\{\lambda|\exists\Psi(<\kappa,\beta,\Psi>\in C\wedge\lambda\in\Psi\wedge\forall\delta(\delta\prec\lambda\rightarrow \\ \exists\Upsilon(<\kappa,\beta+\delta,\Upsilon>\in C\wedge\lambda\in\Upsilon)))\})\wedge \\ (Lim(\alpha)\rightarrow\Xi=\{\lambda|\exists\delta\exists\Delta(\delta\prec\alpha\wedge<\delta,\beta,\Delta>\in C\wedge\lambda\in\Delta)\}) \wedge \\ (Lim(\beta)\rightarrow\Xi=\{\lambda|\exists\delta\exists\Delta(\delta\prec\beta\wedge<\alpha,\delta,\Delta>\in C\wedge\lambda\in\Delta)\})$
We can now state the Mahlo-Regulation:
$$\vDash_{M}<\alpha,\beta>\in Ord^{2}\Rightarrow \ \vDash_{M}\exists x\exists\Psi(<\alpha,\beta,\Psi>\in C\wedge x\in\Psi)$$
With £ plus the Mahlo-Regulation We can define a manifestation point analogously with $\dot{D}$ above, and with appropriate postulations we achieve that the manifestation point is KIND and hereditarily KIND.
I stress that £ itself of course does not commit to the consistency of ZFC.
Now as to my questions: Some places in the literature it is suggested that somewhere around Mahlo is the limit for how far we can build up inaccessible cardinals from below. How far further can we press on with manifestation points such as here? Suggestions are welcome.