Descriptive properties of I2-embeddings
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913570927673344 |
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| author | Dimonte, Vincenzo Iannella, Martina Lücke, Philipp |
| author_facet | Dimonte, Vincenzo Iannella, Martina Lücke, Philipp |
| contents | We contribute to the study of generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set $P(λ)$ of a singular cardinal $λ$ of countable cofinality or products $\prod_{i<ω}λ_i$ for a strictly increasing sequence $\langleλ_i ~ \vert ~ i<ω\rangle$ of cardinals. We consider the question under which large cardinal hypotheses classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms and classes of sets definable by $Σ_1$-formulas with parameters from various collections of sets. We prove that $ω$-many measurable cardinals, while sufficient to prove the Perfect Set Property of all $Σ_1$-definable sets with parameters in $V_λ\cup\{V_λ\}$, are not enough to prove it if there is a cofinal sequence in $λ$ in the parameters. For this conclusion, the existence of an I2-embedding is enough, but there are parameters in $V_{λ+1}$ for which I2 is still not enough. The situation is similar for the Baire Property: under I2 all sets that are $Σ_1$-definable using elements of $V_λ$ and a cofinal sequence as parameters have the Baire property, but I2 is not enough for some parameter in $V_{λ+1}$. Finally, the existence of an I0-embedding implies that all sets that are $Σ^1_n$-definable with parameters in $V_{λ+1}$ have the Baire property. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_00376 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Descriptive properties of I2-embeddings Dimonte, Vincenzo Iannella, Martina Lücke, Philipp Logic 03E55, 03E35, 03E45, 03E47 We contribute to the study of generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set $P(λ)$ of a singular cardinal $λ$ of countable cofinality or products $\prod_{i<ω}λ_i$ for a strictly increasing sequence $\langleλ_i ~ \vert ~ i<ω\rangle$ of cardinals. We consider the question under which large cardinal hypotheses classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms and classes of sets definable by $Σ_1$-formulas with parameters from various collections of sets. We prove that $ω$-many measurable cardinals, while sufficient to prove the Perfect Set Property of all $Σ_1$-definable sets with parameters in $V_λ\cup\{V_λ\}$, are not enough to prove it if there is a cofinal sequence in $λ$ in the parameters. For this conclusion, the existence of an I2-embedding is enough, but there are parameters in $V_{λ+1}$ for which I2 is still not enough. The situation is similar for the Baire Property: under I2 all sets that are $Σ_1$-definable using elements of $V_λ$ and a cofinal sequence as parameters have the Baire property, but I2 is not enough for some parameter in $V_{λ+1}$. Finally, the existence of an I0-embedding implies that all sets that are $Σ^1_n$-definable with parameters in $V_{λ+1}$ have the Baire property. |
| title | Descriptive properties of I2-embeddings |
| topic | Logic 03E55, 03E35, 03E45, 03E47 |
| url | https://arxiv.org/abs/2311.00376 |