The Σ 2 theory of D h ( ⩽ h O ) as an uppersemilattice with least and greatest element is decidable

Computability ◽  
2021 ◽  
pp. 1-21
Author(s):  
James Barnes
Keyword(s):  
Initial Segment ◽  
Quantifier Theory

The decidability of the two quantifier theory of the hyperarithmetic degrees below Kleene’s O in the language of uppersemilattices with least and greatest element is established. This requires a new kind of initial segment result and a new extension of embeddings result both in the hyperarithmetic setting.

scholarly journals Lattice initial segments of the hyperdegrees

2010 ◽  
Vol 75 (1) ◽  
pp. 103-130 ◽  
Author(s):  
Richard A. Shore ◽  
Bjørn Kjos-Hanssen
Keyword(s):  
Distributive Lattice ◽  
Representation Theorem ◽  
Initial Segment ◽  
Finite Lattice ◽  
The Other ◽  
Locally Finite ◽  
Other Hand ◽  
Lattice Representation ◽  
Quantifier Theory

AbstractWe affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, . In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable, locally finite lattice) is isomorphic to an initial segment of . Corollaries include the decidability of the two quantifier theory of , and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of ω1ck. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ω1. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of .

scholarly journals Imaging of the Axon Initial Segment

2019 ◽  
Vol 89 (1) ◽  
Author(s):  
Jessica Di Re ◽  
Cihan Kayasandik ◽  
Gonzalo Botello‐Lins ◽  
Demetrio Labate ◽  
Fernanda Laezza
Keyword(s):  
Initial Segment ◽  
Axon Initial Segment
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