Gap ordinal

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A gap ordinal is an ordinal \(\alpha\) such that \((L_{\alpha+1}\setminus L_\alpha)\cap\mathcal P(\omega)=\varnothing\).^p.364 An ordinal \(\alpha\) is said to start a gap if \(\alpha\) is a gap ordinal but for all \(\beta<\alpha\), \((L_\alpha\setminus L_\beta)\cap\mathcal P(\omega)\neq\varnothing\).^p.368

Gap ordinals are very large. This is because, if \( \alpha \) is a gap ordinal, then \( L_\alpha \cap \mathcal{P}(\omega) \) satisfies second-order arithmetic, despite not containing all subsets of \( \omega \). Therefore, if \( \alpha \) is a gap ordinal, it is admissible, recursively inaccessible, recursively Mahlo, nonprojectible, and more. However, there can still be countable gap ordinals.

There is a nice analogy between gap ordinals and cardinals. Note that \( \alpha \) is a cardinal if, for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \). If \( \alpha \) is infinite, we have \( \pi \subseteq \gamma \times \alpha \subseteq \alpha \times \alpha \subseteq V_\alpha^2 \subseteq V_\alpha \) and thus \( \pi \in V_{\alpha + 1} \). Thus, "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in V_{\alpha + 1} \)" is equivalent to being a cardinal. Meanwhile, the least ordinal satisfying the similar but weaker condition "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in L_{\alpha + 1} \)" is equal to the least gap ordinal, since it's equivalent to \( L_\alpha \) satisfying separation.

Harvey Friedman proved that \(L_{\beta_0}\cap\mathcal P(\omega)\) does not satisfy \(\Sigma^0_5\) determinacy.

Longer gaps

Given any ordinal \(\gamma<\omega_1^L\), it is possible to find an \(\alpha<\omega_1^L\) such that \((L_{\alpha+\gamma}\setminus L_\alpha) \cap \mathcal{P}(\omega) = \emptyset\).^p.365 If such an \( \alpha \) starts a gap, then it is said to start a gap of length \( \gamma \). It is possible for \( \alpha \) to start a gap of length \( > \alpha \): for example, the least \( \alpha \) so that \( \alpha \) starts a gap of length \( \alpha^+ \) is equal to the least admissible which is not locally countable.

There can also be second-order gap, and more. An ordinal \( \alpha \) is said to start an \( \eta \)th-order gap of length \( \gamma \) if \( (L_{\alpha+\gamma} \setminus L_\alpha) \cap \mathcal{P}^\eta(\omega) = \emptyset \) and, for all \( \beta < \alpha \), \((L_\alpha \setminus L_\beta) \cap \mathcal {P}^\eta(\omega) \neq \emptyset\). The least ordinal which starts a second-order gap is greater than the least \( \alpha \) which starts a first-order gap of length \( \alpha \), and more.

If \( 0^\sharp \) exists, then, for any countable \( \eta\) and any \( \gamma \) at all, there is a countable \( \delta \) which starts an \( \eta \)th-order gap of length \( \gamma \). Meanwhile, if \( V = L \), then there is no countable ordinal starting a first-order gap of length \( \omega_1 \).

Citations