 But there is an important point: Assuming large cardinal axioms (ADL(ℝ) suffices), although one can produce outer models in which δ̰13 &gt; ℵ2 it is not currently known how to produce outer models in which δ̰13 &gt; ℵ3 or even ΘL(ℝ) &gt; ℵ3. Thus it is an open possibility that from ZFC +ADL(ℝ) one can prove ΘL(ℝ) ≤ ℵ3. Were this to be the case, it would follow that although large cardinals cannot rule out the definable failure of CH they can rule out the definable failure of 2ℵ0 = ℵ2. This could provide some insight into the size of the continuum, underscoring the centrality of ℵ2. 