Local mantles of $L[x]$
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2021
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908360877539328 |
|---|---|
| author | Schlutzenberg, Farmer |
| author_facet | Schlutzenberg, Farmer |
| contents | Assume ZFC. Let $κ$ be a cardinal. Recall that a ${<κ}$-ground is a transitive proper class $W$ modelling ZFC such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality ${<κ}$, and the $κ$-mantle is the intersection of all ${<κ}$-grounds.
Assume there is a Woodin cardinal and a proper class of measurables, and let $x$ be a real of sufficiently high Turing degree. Let $κ$ be a limit cardinal of $L[x]$ of uncountable cofinality in $L[x]$. Using methods from Woodin's analysis of $\mathrm{HOD}^{L[x,G]}$, we analyze the $κ$-mantle of $L[x]$, and show that it models ZFC + GCH + "There is a Woodin cardinal". Moreover, we show that it is a fully iterable strategy mouse (analogous to $\mathrm{HOD}^{L[x,G]}$). We also analyze another form of "local mantle", partly assuming also a weak form of Turing determinacy. We also compute bounds on how much iteration strategy can be added to $M_1$ before $M_1^\#$ is added. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_12925 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Local mantles of $L[x]$ Schlutzenberg, Farmer Logic 03E45, 03E55, 03E40, 03E25 Assume ZFC. Let $κ$ be a cardinal. Recall that a ${<κ}$-ground is a transitive proper class $W$ modelling ZFC such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality ${<κ}$, and the $κ$-mantle is the intersection of all ${<κ}$-grounds. Assume there is a Woodin cardinal and a proper class of measurables, and let $x$ be a real of sufficiently high Turing degree. Let $κ$ be a limit cardinal of $L[x]$ of uncountable cofinality in $L[x]$. Using methods from Woodin's analysis of $\mathrm{HOD}^{L[x,G]}$, we analyze the $κ$-mantle of $L[x]$, and show that it models ZFC + GCH + "There is a Woodin cardinal". Moreover, we show that it is a fully iterable strategy mouse (analogous to $\mathrm{HOD}^{L[x,G]}$). We also analyze another form of "local mantle", partly assuming also a weak form of Turing determinacy. We also compute bounds on how much iteration strategy can be added to $M_1$ before $M_1^\#$ is added. |
| title | Local mantles of $L[x]$ |
| topic | Logic 03E45, 03E55, 03E40, 03E25 |
| url | https://arxiv.org/abs/2103.12925 |