3 |
Department of Mathematics, National University of Singapore, Singapore 119076 |

4 |
The Downward Löwenheim–Skolem Theorem states that every model of a countable theory has an elementary countable submodel. |

5 |
Now known as the Sacks Splitting Theorem. |

6 |
The Jump Theorem states that every degree that is r.e. in and above 0′ is the jump of an r.e. degree. |

7 |
An object of type 0 is a nonnegative integer. An object of type n + 1 is a set of objects of type n. n+1E is the characteristic function of the set of objects of type n. In Kleene’s theory of recursion in objects of finite type, n+1E is recursive if and only if n = 0. He defined the 1-section of n+1E to be the set of type 1 objects recursive in n+1E. He proved that the 1-section of 2E is HYP. |

8 |
Kleene developed a computation theory of higher type objects generated by n-th iterates of the power set operation on ℕ. A unified approach from the set-theoretic point of view, E-recursion, was introduced by Dag Normann in 1967. |

9 |
Yiannis Moschovakis. |

10 |
The problem asks if there is a minimal α-degree for every admissible ordinal α. |