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lo.logic – Can this Ackermann love clique idea formulated with out including a raw of set-hood achieve the consistency of Ord is Mahlo?

A pleasant evolution of the idea introduced on the prior posting to Mathoverflow, is that one can dole with the understanding of smallness altogether! And can labor solely within the language of clique idea, so no necessity so as to add any raw understanding over the predicates of equality and membership. To re-iterate the axioms:

Define: $clique(X) iff exists Y ( X in Y)$

**Extensionality:** $forall A forall B: forall Z (Z in A Leftrightarrow Z in B) implies A=B$

**Comprehension:** $exists X forall Y (Y in X iff clique(Y) land phi)$; the place $X$ not free in $phi$.

**Limitation of Size:** $X in V iff not exists f:V hookrightarrow X $; $“hookrightarrow”$ for “*injections*“; $V$ is the category of all units.

**Foundation:** $X neq emptyset implies exists Y in X (Y cap X = emptyset)$

**Reflection:** if $phi_1,..,phi_n$ are *absolute clique theoretic formulation* by which $Y$ is free, and their parameters amongst symbols $A,B$; and if $pi_i$ is the system $“forall A,B are H_{<W} [ forall Y(phi_i Rightarrow H_{<W}(Y)) to exists X < W forall Y(Y in X Leftrightarrowphi_i) ]”$

then: $$exists W: pi_1 land … land pi_n $$

Where $H_{<W}$ is the predicate “hereditarily strictly subnumerous to $W$” outlined by the predicated courses being strictly subnumerous to $W$ and each component of their transitive closures being too strictly subnumerous to $W$; and $<$ stands for strict subnumerousity outlined within the habitual sense. A absolute clique theoretic system means a bounded $“in V”$ formula whose predicates are $in,=$ and its terms are variables, except $V$ which solely seems in bounding all quantified variables in it.

I pretense that this idea can construe Ackermann and Morse-Kelley clique idea [with modified size limitation axiom of every class equinumerous to a set is a set], and too Muller’s extension of Ackermann’s. Also, I cerebrate it interprets Tarski–Grothendieck clique idea (TG). I’m probably not positive if it reaches as much as the consistency of $ORD$ is Mahlo. It does all of that within the habitual language of clique idea!

Is ORD is Mahlo, the require consistency power of this idea?

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