ON THE DECIDABILITY OF THE ${{\rm{\Sigma }}_2}$ THEORIES OF THE ARITHMETIC AND HYPERARITHMETIC DEGREES AS UPPERSEMILATTICES.
We establish the decidability of the ${{\rm{\Sigma }}_2}$ theory of both the arithmetic and hyperarithmetic degrees in the language of uppersemilattices, i.e., the language with ≤, 0 , and $\sqcup$. This is achieved by using Kumabe-Slaman forcing, along with other known results, to show given finite...
| Publicado en: | Journal of Symbolic Logic Vol. 82; no. 4; pp. 1496 - 1519 |
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| Autor principal: | |
| Formato: | Artículo |
| Publicado: |
Cambridge University Press
Dec2017
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| Materias: | |
| Acceso en línea: | Ver este registro en EBSCOhost |
| Sumario: | We establish the decidability of the ${{\rm{\Sigma }}_2}$ theory of both the arithmetic and hyperarithmetic degrees in the language of uppersemilattices, i.e., the language with ≤, 0 , and $\sqcup$. This is achieved by using Kumabe-Slaman forcing, along with other known results, to show given finite uppersemilattices ${\cal M}$ and ${\cal N}$, where ${\cal M}$ is a subuppersemilattice of ${\cal N}$, that every embedding of ${\cal M}$ into either degree structure extends to one of ${\cal N}$ iff ${\cal N}$ is an end-extension of ${\cal M}$. |
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