Inner models from extended logics and the Delta-operation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918367011536896 |
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| author | Väänänen, Jouko Ya'ar, Ur |
| author_facet | Väänänen, Jouko Ya'ar, Ur |
| contents | If $\mathcal{L}$ is an abstract logic (a.k.a. model theoretic logic), we can define the inner model $C(\mathcal{L})$ by replacing first order logic with $\mathcal{L}$ in Gödel's definition of the inner model $L$ of constructible sets. Set theoretic properties of such inner models $C(\mathcal{L})$ have been investigated recently and a spectrum of new inner models is emerging between $L$ and $\mathrm{HOD}$. The topic of this paper is the effect on $C(\mathcal{L})$ of a slight modification of $\mathcal{L}$ i.e. how sensitive is $C(\mathcal{L})$ on the exact definition of $\mathcal{L}$? The $Δ$-extension $Δ(\mathcal{L})$ of a logic is generally considered a "mild" extension of $\mathcal{L}$. We give examples of logics $\mathcal{L}$ for which the inner model $C(\mathcal{L})$ is consistently strictly smaller than the inner model $C(Δ(\mathcal{L}))$, and in one case we show this follows from the existence of $0^{\sharp}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07892 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inner models from extended logics and the Delta-operation Väänänen, Jouko Ya'ar, Ur Logic 03E45 (Primary) 03C95 (Secondary) If $\mathcal{L}$ is an abstract logic (a.k.a. model theoretic logic), we can define the inner model $C(\mathcal{L})$ by replacing first order logic with $\mathcal{L}$ in Gödel's definition of the inner model $L$ of constructible sets. Set theoretic properties of such inner models $C(\mathcal{L})$ have been investigated recently and a spectrum of new inner models is emerging between $L$ and $\mathrm{HOD}$. The topic of this paper is the effect on $C(\mathcal{L})$ of a slight modification of $\mathcal{L}$ i.e. how sensitive is $C(\mathcal{L})$ on the exact definition of $\mathcal{L}$? The $Δ$-extension $Δ(\mathcal{L})$ of a logic is generally considered a "mild" extension of $\mathcal{L}$. We give examples of logics $\mathcal{L}$ for which the inner model $C(\mathcal{L})$ is consistently strictly smaller than the inner model $C(Δ(\mathcal{L}))$, and in one case we show this follows from the existence of $0^{\sharp}$. |
| title | Inner models from extended logics and the Delta-operation |
| topic | Logic 03E45 (Primary) 03C95 (Secondary) |
| url | https://arxiv.org/abs/2508.07892 |