Mouse scales
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909608020279296 |
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| author | Schlutzenberg, Farmer |
| author_facet | Schlutzenberg, Farmer |
| contents | We give a construction of scales (in the descriptive set theoretic sense) directly from mouse existence hypotheses, without using any determinacy arguments. The construction is related to the Martin-Solovay construction for scales on $Π^1_2$ sets. The prewellorders of the scales compare reals $x$ and $y$ by comparing features of certain kinds of fully backgrounded $L[E,x]$- and $L[E,y]$-constructions executed in mice $P$ with $x,y \in P$. In this way we produce an inner model theoretic proof of the scale property for many pointclasses, for which the scale property was classically established using determinacy arguments (for example, $Π^1_3$). Moreover, it also yields many further pointclasses with the scale property, for example intermediate between $Π^1_{2n+1}$ and $Σ^1_{2n+2}$, and also instances of complexity well beyond projective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19764 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mouse scales Schlutzenberg, Farmer Logic 03E45, 03E55, 03E15 We give a construction of scales (in the descriptive set theoretic sense) directly from mouse existence hypotheses, without using any determinacy arguments. The construction is related to the Martin-Solovay construction for scales on $Π^1_2$ sets. The prewellorders of the scales compare reals $x$ and $y$ by comparing features of certain kinds of fully backgrounded $L[E,x]$- and $L[E,y]$-constructions executed in mice $P$ with $x,y \in P$. In this way we produce an inner model theoretic proof of the scale property for many pointclasses, for which the scale property was classically established using determinacy arguments (for example, $Π^1_3$). Moreover, it also yields many further pointclasses with the scale property, for example intermediate between $Π^1_{2n+1}$ and $Σ^1_{2n+2}$, and also instances of complexity well beyond projective. |
| title | Mouse scales |
| topic | Logic 03E45, 03E55, 03E15 |
| url | https://arxiv.org/abs/2310.19764 |