= Inner models with all reals preserve omega-one
{title2=$\omega_1^W=\omega_1$}
An <inner model> $W$ of <ZF> containing all ambient <subsets> of $\omega$ computes the same <countable ordinals> as the ambient universe. Every ambient countably infinite <ordinal> has a <well-order code> coded by a <subset> of $\omega$, hence that code belongs to $W$; finite <ordinals> are already shared. The <Mostowski collapse theorem> inside $W$ recovers the actual <ordinal> and its countability witness. Conversely any countability witness in $W$ remains one outside. If $W$ also contains the full <power set> of $\omega$ as a <set>, its <Continuum hypothesis> would give an ambient <bijection> from $\omega_1$ onto that same <power set>. Thus an ambient failure of this well-orderable formulation of the <Continuum hypothesis> is preserved in such an inner model. No internal <axiom of choice> is required for this argument.
Back to article page
