The Axiom of Real Determinacy and the Axiom of Real Blackwell Determinacy
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914401245724672 |
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| author | Ikegami, Daisuke Woodin, W. Hugh |
| author_facet | Ikegami, Daisuke Woodin, W. Hugh |
| contents | We show that the Axiom of Real Determinacy $\mathsf{AD}_{\mathbb{R}}$ and the Axiom of Real Blackwell Determinacy $\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ are equivalent in $\mathsf{ZF}$+$\mathsf{DC}$. This answers the question of Löwe [15, Question 53]. While we do not know whether they are equivalent in $\mathsf{ZF}$+$\mathsf{AC}_ω (\mathbb{R})$, we show that the theories $\mathsf{ZF}$+$\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{ZF}$+$\mathsf{AC}_ω (\mathbb{R})$+$\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ are equiconsistent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11034 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Axiom of Real Determinacy and the Axiom of Real Blackwell Determinacy Ikegami, Daisuke Woodin, W. Hugh Logic 03E60, 03E15, 28A05 We show that the Axiom of Real Determinacy $\mathsf{AD}_{\mathbb{R}}$ and the Axiom of Real Blackwell Determinacy $\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ are equivalent in $\mathsf{ZF}$+$\mathsf{DC}$. This answers the question of Löwe [15, Question 53]. While we do not know whether they are equivalent in $\mathsf{ZF}$+$\mathsf{AC}_ω (\mathbb{R})$, we show that the theories $\mathsf{ZF}$+$\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{ZF}$+$\mathsf{AC}_ω (\mathbb{R})$+$\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ are equiconsistent. |
| title | The Axiom of Real Determinacy and the Axiom of Real Blackwell Determinacy |
| topic | Logic 03E60, 03E15, 28A05 |
| url | https://arxiv.org/abs/2510.11034 |