Bounded ultraimaginary independence and its total Morley sequences
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866909184387186688 |
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| author | Hanson, James |
| author_facet | Hanson, James |
| contents | We investigate the following model-theoretic independence relation: $\def\indbu{{\rlap{\hspace11.9mu\vert}\lower7.5mu\smile}^{\!\mathrm{bu}}} b \indbu_A\hspace3mu c$ if and only if $\mathrm{bdd}^u(Ab)\cap \mathrm{bdd}^u(Ac) = \mathrm{bdd}^u(A)$, where $\mathrm{bdd}^u(X)$ is the class of all ultraimaginaries bounded over $X$. In particular, we sharpen a result of Wagner to show that $b \indbu_A\hspace3mu c$ if and only if $\langle \mathrm{Autf}(\mathbb{M}/Ab)\cup\mathrm{Autf}(\mathbb{M}/Ac) \rangle = \mathrm{Autf}(\mathbb{M}/A)$, and we establish full existence over hyperimaginary parameters (i.e., for any set of hyperimaginaries $A$ and ultraimaginaries $b$ and $c$, there is a $b' \equiv_A b$ such that $b' \indbu_A\hspace3mu c$). Extension then follows as an immediate corollary.
We also study total $\hspace-5mu\indbu$-Morley sequences (i.e., $A$-indiscernible sequences $I$ satisfying $J \indbu_A\hspace3mu K$ for any $J$ and $K$ with $J + K \equiv^{\mathrm{EM}}_A I$), and we prove that an $A$-indiscernible sequence $I$ is a total $\hspace-5mu\indbu$-Morley sequence over $A$ if and only if whenever $I$ and $I'$ have the same Lascar strong type over $A$, $I$ and $I'$ are related by the transitive, symmetric closure of the relation '$J+K$ is $A$-indiscernible.' This is also equivalent to $I$ being 'based on' $A$ in a sense defined by Shelah in his early study of simple unstable theories.
Finally, we show that for any $A$ and $b$ in any theory $T$, if there is an Erdös cardinal $κ(α)$ with $|Ab|+|T| < κ(α)$, then there is a total $\hspace-5mu\indbu$-Morley sequence $(b_i)_{i<ω}$ over $A$ with $b_0 = b$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_03631 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bounded ultraimaginary independence and its total Morley sequences Hanson, James Logic 03C45 We investigate the following model-theoretic independence relation: $\def\indbu{{\rlap{\hspace11.9mu\vert}\lower7.5mu\smile}^{\!\mathrm{bu}}} b \indbu_A\hspace3mu c$ if and only if $\mathrm{bdd}^u(Ab)\cap \mathrm{bdd}^u(Ac) = \mathrm{bdd}^u(A)$, where $\mathrm{bdd}^u(X)$ is the class of all ultraimaginaries bounded over $X$. In particular, we sharpen a result of Wagner to show that $b \indbu_A\hspace3mu c$ if and only if $\langle \mathrm{Autf}(\mathbb{M}/Ab)\cup\mathrm{Autf}(\mathbb{M}/Ac) \rangle = \mathrm{Autf}(\mathbb{M}/A)$, and we establish full existence over hyperimaginary parameters (i.e., for any set of hyperimaginaries $A$ and ultraimaginaries $b$ and $c$, there is a $b' \equiv_A b$ such that $b' \indbu_A\hspace3mu c$). Extension then follows as an immediate corollary. We also study total $\hspace-5mu\indbu$-Morley sequences (i.e., $A$-indiscernible sequences $I$ satisfying $J \indbu_A\hspace3mu K$ for any $J$ and $K$ with $J + K \equiv^{\mathrm{EM}}_A I$), and we prove that an $A$-indiscernible sequence $I$ is a total $\hspace-5mu\indbu$-Morley sequence over $A$ if and only if whenever $I$ and $I'$ have the same Lascar strong type over $A$, $I$ and $I'$ are related by the transitive, symmetric closure of the relation '$J+K$ is $A$-indiscernible.' This is also equivalent to $I$ being 'based on' $A$ in a sense defined by Shelah in his early study of simple unstable theories. Finally, we show that for any $A$ and $b$ in any theory $T$, if there is an Erdös cardinal $κ(α)$ with $|Ab|+|T| < κ(α)$, then there is a total $\hspace-5mu\indbu$-Morley sequence $(b_i)_{i<ω}$ over $A$ with $b_0 = b$. |
| title | Bounded ultraimaginary independence and its total Morley sequences |
| topic | Logic 03C45 |
| url | https://arxiv.org/abs/2201.03631 |