arXiv subjects

Samson Leung

Publications and source records attributed to Samson Leung.

4 recordsLinked to original sources

arXiv2022

Categoricity transfer for short AECs with amalgamation over sets

Let ${\bf K}$ be an $\mathrm{LS}({\bf K})$-short abstract elementary class and assume more than the existence of a monster model (amalgamation over sets and arbitrarily large models). Suppose ${\bf K}$ is categorical in some $\mu>\mathrm{LS}({\bf K})$, then it is categorical in all $\mu'\geq\mu$. Our result removes the successor requirement of $\mu$ made by Grossberg-VanDieren, at the cost of using shortness instead of tameness; and of using amalgamation over sets instead of over models. It also removes the primes requirement by Vasey which assumes tameness and amalgamation over models. As a corollary, we obtain an alternative proof of the upward categoricity transfer for first-order theories by Morley and Shelah. In our construction, we simplify Vasey's results to build a weakly successful frame. This allows us to use Shelah-Vasey's argument to obtain primes for sufficiently saturated models. If we replace the categoricity assumption by $\mathrm{LS}({\bf K})$-superstability, ${\bf K}$ is already excellent for sufficiently saturated models. This sheds light on the investigation of the main gap theorem for uncountable first-order theories within ZFC.

math.LO↗

arXiv2021

Stability results assuming tameness, monster model and continuity of nonsplitting

Assuming the existence of a monster model, tameness and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>LS({\bf K})$ be a regular stability cardinal and let $\chi$ be the local character of $\mu$-nonsplitting. The following holds: 1. When $\mu$-nonforking is restricted to $(\mu,\geq\chi)$-limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension and continuity. It also has local character $\chi$. This generalizes Vasey's result which assumed $\mu$-superstability to obtain same properties but with local character $\aleph_0$. 2. There is $\lambda\in[\mu,h(\mu))$ such that if ${\bf K}$ is stable in every cardinal between $\mu$ and $\lambda$, then ${\bf K}$ has $\mu$-symmetry while $\mu$-nonforking in (1) has symmetry. In this case (a) ${\bf K}$ has the uniqueness of $(\mu,\geq\chi)$-limit models: if $M_1,M_2$ are both $(\mu,\geq\chi)$-limit over some $M_0\in K_\mu$, then $M_1\cong_{M_0}M_2$; (b) any increasing chain of $\mu^+$-saturated models of length $\geq\chi$ has a $\mu^+$-saturated union. These generalize VanDieren-Vasey's result and remove the symmetry assumption in Boney-VanDieren and Vasey's result. Under $(<\mu)$-tameness, the conclusions of (1), (2)(a)(b) are equivalent to ${\bf K}$ having the $\chi$-local character of $\mu$-nonsplitting. Grossberg and Vasey gave eventual superstability criteria for tame AECs with a monster model. We remove the high cardinal threshold and reduce the cardinal jump between equivalent superstability criteria. We also add two new superstability criteria to the list: a weaker version of solvability and the boundedness of the $U$-rank.

math.LO↗

arXiv2021

Axiomatizing AECs and applications

For any abstract elementary class (AEC) ${\bf K}$ with $\lambda=LS({\bf K})$, the following holds: 1. $K$ has an axiomatization in $L_{(2^\lambda)^+,\lambda^+}$, allowing game quantification. If ${\bf K}$ has arbitrarily large models, the $\lambda$-amalgamation property and is categorical both in $\lambda$ and $\lambda^+$, then it has an axiomatization in $L_{\lambda^{+},\lambda^{+}}$ with game quantification. These extend Kueker's result which assumes finite character and $\lambda=\aleph_0$. 2. If $K$ is universal and categorical in $\lambda$, then it is axiomatizable in $L_{\lambda^+,\lambda^+}$. 3. Shelah's celebrated presentation theorem asserts that for any AEC ${\bf K}$ there is a first-order theory in an expansion of $L({\bf K})$, and a set $\Gamma$ of $2^\lambda$ many $T$-types such that $K=PC(T,\Gamma,L({\bf K}))$. We provide a better bound on $|\Gamma|$ in terms of $I_2(\lambda,{\bf K})$. 4. We present additional applications which extend, simplify and generalize results of Shelah and Shelah-Vasey. Some of our main results generalize to $\mu$-AECs.

math.LO↗

arXiv2021

Hanf number of the first stability cardinal in AECs

We show that $\beth_{(2^{\operatorname{LS}({\bf K})})^+}$ is the lower bound to the Hanf numbers for the length of the order property and for stability in stable abstract elementary classes (AECs). Our examples satisfy the joint embedding property, no maximal model, $(<\aleph_0)$-tameness but not necessarily the amalgamation property. We also define variations on the order and syntactic order properties by allowing the index set to be linearly ordered rather than well-ordered. Combining with Shelah's stability theorem, we deduce that our examples can have the order property up to any $\mu<\beth_{(2^{\operatorname{LS}({\bf K})})^+}$. Boney conjectured that the need for joint embedding property for two type-counting lemmas is necessary. We solved the conjecture by showing it is independent of ZFC. Using Galois Morleyization, we give syntactic proofs to known stability results assuming a monster model.

math.LO↗