_{1}is the Church-Kleene ordinal? Intuitively, it seems so. If I understand correctly, ω

_{1}

^{CK}is the first ordinal without a *constructible* bijection to the natural numbers, and the axiom of constructibility implies that if there is no constructible bijection, then there's no bijection at all, and thus ω

_{1}

^{CK}is the first uncountable ordinal in such a set theory, which is the definition of ω

_{1}.

Is this right, or am I missing something? And if I'm missing something, would there be some similar situation that would imply this result? An "axiom of computability" for example?