Monotone versus non‐monotone projective operators
For a class of operators Γ$\Gamma$, let |Γ|$|\Gamma |$ denote the closure ordinal of Γ$\Gamma$‐inductive definitions. We give upper bounds on the values of |Σ2n+11,mon|$|\Sigma ^{1,mon}_{2n+1}|$ and |Π2n+21,mon|$|\Pi ^{1,mon}_{2n+2}|$ under the assumption that all projective sets of reals are determ...
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| Veröffentlicht in: | The Bulletin of the London Mathematical Society 2024-12, Vol.57 (1), p.256-264 |
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| container_title | The Bulletin of the London Mathematical Society |
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| creator | Aguilera, J. P. Welch, P. D. |
| description | For a class of operators Γ$\Gamma$, let |Γ|$|\Gamma |$ denote the closure ordinal of Γ$\Gamma$‐inductive definitions. We give upper bounds on the values of |Σ2n+11,mon|$|\Sigma ^{1,mon}_{2n+1}|$ and |Π2n+21,mon|$|\Pi ^{1,mon}_{2n+2}|$ under the assumption that all projective sets of reals are determined, significantly improving the known results. In particular, the bounds show that |Πn1,mon| |
| doi_str_mv | 10.1112/blms.13194 |
| format | Article |
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P. ; Welch, P. D.</creator><creatorcontrib>Aguilera, J. P. ; Welch, P. D.</creatorcontrib><description>For a class of operators Γ$\Gamma$, let |Γ|$|\Gamma |$ denote the closure ordinal of Γ$\Gamma$‐inductive definitions. We give upper bounds on the values of |Σ2n+11,mon|$|\Sigma ^{1,mon}_{2n+1}|$ and |Π2n+21,mon|$|\Pi ^{1,mon}_{2n+2}|$ under the assumption that all projective sets of reals are determined, significantly improving the known results. In particular, the bounds show that |Πn1,mon|<|Πn1|$|\Pi ^{1,mon}_{n}| < |\Pi ^1_{n}|$ and |Σn1,mon|<|Σn1|$|\Sigma ^{1,mon}_{n}| < |\Sigma ^1_{n}|$ hold for 2⩽n$2\leqslant n$ under the assumption of projective determinacy. Some of these inequalities were obtained by Aanderaa in the 70s via recursion‐theoretic methods but never appeared in print. 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| title | Monotone versus non‐monotone projective operators |
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