In this theory $K_2(W)$ (page 7) Harvey Friedman argue for weakening Sub-world Separation into SS-. He did that in order to evade making $W$ transitive. But he could have done that by simply reverting to the more natural way, which is the origin of the Reducibility axiom (which is a kind of reflection principle), that of changing Reducibility to the following:

if $\varphi$ is a predicate definable by a formula in $L(\in)$ from parameters $x_1,..,x_n$, then all closures of:

$$x_1,..,x_n \in W \wedge \varphi(W) \to \exists x_{n+1} \in W (\varphi(x_{n+1}))$$; are axioms.

I mean if we keep sub-world separation as it is in $K(W)$, (i.e.; don't weaken it to SS-), and replace Reducibility with the above, and keep the other axioms of $K_2(W)$ as they are, call the resulting system $K^* _2(W)$, would that be weaker than $K_2(W)$ or otherwise be inconsistent?


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