Let $V$ be the class of all sets, where sets are defined like in $MK$ as elements of classes.

Properties of $V$ whose negations are unbounded (by element-hood & subset-hood) in $V$ would be called "top" properties of $V$, formally:

$\mathcal{Define: }\ \varphi \text { is a top-property of } V$, if and only if:
$$\varphi(V) \wedge \forall x \in V \ \exists y \in V \ (x \in y \wedge x \subset y \wedge \neg \varphi(y))$$

for example if $V$ is the class of all sets of $MK$ then properties of "proper class", "has a subclass that is a proper class", "larger than any set", "of upper bound size on classes", etc.. all those are top-properties, those are not shared with any set, but there can be top properties that are shared with some sets like "is transitive", "is super-transitive", "is not a union of a strictly smaller set in size of strictly smaller sets in size", "is not reachable by power and union from below", "is bigger in size than all of its elements" etc...

We can also coin a weaker notion of top property of $V$, denote it by top* property, as follows:

$\mathcal{Define: }\ \varphi \text { is a top*-property of } V$, if and only if:

$$\varphi(V) \wedge \forall x \in V \ \exists y \ (x \in y \wedge x \subset y \wedge \neg \varphi(y))$$

This notion uese bounding in classes instead of sets.

Now, lets take the theory $K_2^+(W)$, which is but a slight modification of Harvey Friedman's $K_2(W)$ theory (p:7), remove axiom 3 of $K_2^+(W)$ and replace it with an axiom of pairing on sets, and replace the last axiom in $K_2^+(W)$ with the following scheme:

  1. Top*-Reflection: if $\varphi$ is a unary predicate definable in $L(\in)$, then:

$$\varphi \text{ is a top* property of }V \to \\ \exists x \in W \ \forall y \subseteq W [\varphi(y) \to y \subseteq x]$$

To paraphrase Harvey here, we'd say: the top* pure set properties of the universe of today are intensely reflected in some elements of yesterday's world! Call this theory $``K_2^{++}(W)"$.

Another approach which more suites top reflection, is to replace top*-reflection in $K_2^{++}(W)$ by top-reflection scheme, and also add union, power, on sets (as in $MK$), and a limitation of size axiom on sets, more specifically: a class that is equinumerous to a subclass of a set, is a set. Call the resulting theory $``K_2^{3+}(W)"$.

I tend to think that both of these approaches are very risky and mostly inconsistent. However if not inconsistent, then clearly they would prove the last axiom of $K_2(W)$ easily because "is trasnitive" is a top property of $V$ in any of these approaches, and I'd conjecture that both these theories would be stronger than $K_2(W)$.

Question: is there a clear inconsistency with $K_2^{++}(W), K_2^{3+}(W)$ class theories?

The motivation behind that kind of theories is written in full in Harvey's document(p.1), in brief it is to investigate a kind of reflection principle into a sub-world of the universe, in relation to producing large cardinal axioms. In Harvey's you don't see the universe of sets explicitly formalized, here I do that, and let it actively participate in this kind of reflection.


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