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Douglas Ulrich

Publications and source records attributed to Douglas Ulrich.

12 recordsLinked to original sources
arXiv2018

Cardinal Characteristics of Models of Set Theory

We continue our investigation =of Shelah's interpretability orders $\trianglelefteq^*_κ$ as well as the new orders $\trianglelefteq^\times_κ$. In particular, we give streamlined proofs of the existence of minimal unstable, unsimple and nonlow theories in these orders, and we give a similar analysis of the hypergraph examples $T_{n, k}$ of Hrushovski. We also prove that if $\mathcal{B}$ is a complete Boolean algebra with the $λ$-c.c., then no nonprincipal ultrafilter on $\mathcal{U}$ $λ^+$-saturates any unsimple theory.

math.LO↗
arXiv2018

Pseudosaturation and the Interpretability Orders

We streamline treatments of the interpretability orders $\trianglelefteq^*_κ$ of Shelah, the key new notion being that of pseudosaturation. Extending work of Malliaris and Shelah, we classify the interpretability orders on the stable theories. As a further application, we prove that for all countable theories $T_0, T_1$, if $T_1$ is unsupersimple, then $T_0 \trianglelefteq^*_1 T_1$ if and only if $T_0 \trianglelefteq^*_{\aleph_1} T_1$. We thus deduce that simplicity is a dividing line in $\trianglelefteq^*_{\aleph_1}$, and that consistently, $SOP_2$ characterizes maximality in $\trianglelefteq^*_{\aleph_1}$; previously these results were only known for $\trianglelefteq^*_1$.

math.LO↗
arXiv2018

A Streamlined Proof of $\mathfrak{p}=\mathfrak{t}$

We streamline Malliaris and Shelah's proof that $\mathfrak{p} = \mathfrak{t}$. In particular, we replace cofinality spectrum problems with models of $ZFC^-$, and we eliminate the use of peculiar cuts.

math.LO↗
arXiv2018

Keisler's Order and Full Boolean-Valued Models

We prove a compactness theorem for full Boolean-valued models. As an application, we show that if $T$ is a complete countable theory and $\mathcal{B}$ is a complete Boolean algebra, then $λ^+$-saturated $\mathcal{B}$-valued models of $T$ exist. Moreover, if $\mathcal{U}$ is an ultrafilter on $T$ and $\mathbf{M}$ is a $λ^+$-saturated $\mathcal{B}$-valued model of $T$, then whether or not $\mathbf{M}/\mathcal{U}$ is $λ^+$-saturated just depends on $\mathcal{U}$ and $T$; we say that $\mathcal{U}$ $λ^+$-saturates $T$ in this case. We show that Keisler's order can be formulated as follows: $T_0 \trianglelefteq T_1$ if and only if for every cardinal $λ$, for every complete Boolean algebra $\mathcal{B}$ with the $λ^+$-c.c., and for every ultrafilter $\mathcal{U}$ on $\mathcal{B}$, if $\mathcal{U}$ $λ^+$-saturates $T_1$, then $\mathcal{U}$ $λ^+$-saturates $T_0$.

math.LO↗
arXiv2018

Distinct Volume Subsets via Indiscernibles

Erdös proved that for every infinite $X \subseteq \mathbb{R}^d$ there is $Y \subseteq X$ with $|Y|=|X|$, such that all pairs of points from $Y$ have distinct distances, and he gave partial results for general $a$-ary volume. In this paper, we search for the strongest possible canonization results for $a$-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose $T$ is a stable theory, $Δ$ is a finite set of formulas of $T$, $M \models T$, and $X$ is an infinite subset of $M$. Then there is $Y \subseteq X$ with $|Y| = |X|$ and an equivalence relation $E$ on $Y$ with infinitely many classes, each class infinite, such that $Y$ is $(Δ, E)$-indiscernible. We also consider the definable version of these problems, for example we assume $X \subseteq \mathbb{R}^d$ is perfect (in the topological sense) and we find some perfect $Y \subseteq X$ with all distances distinct. Finally we show that Erdös's theorem requires some use of the axiom of choice.

math.LO↗
arXiv2018

Torsion-Free Abelian Groups are Consistently $a Δ^1_2$-complete

Let $\mbox{TFAG}$ be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of $ZFC^- + κ(ω)$ exists, then $\mbox{TFAG}$ is $a Δ^1_2$-complete; in particular, this is consistent with $ZFC$. We define the $α$-ary Schröder- Bernstein property, and show that $\mbox{TFAG}$ fails the $α$-ary Schröder-Bernstein property for every $α< κ(ω)$. We leave open whether or not $\mbox{TFAG}$ can have the $κ(ω)$-ary Schröder-Bernstein property; if it did, then it would not be $a Δ^1_2$-complete, and hence not Borel complete.

math.LO↗
arXiv2017

Low is a Dividing Line in Keisler's Order

We show in $ZFC$ that the class of low theories forms a dividing line in Keisler's order. That is, if $T$ is low and $T' \trianglelefteq T$ then $T'$ is low. We also show there is a minimal nonlow theory $T_{cas}$.

math.LO↗
arXiv2016

Borel Complexity and Potential Canonical Scott Sentences

We define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Phi|| for sentences Phi of $L_{ω_1,ω}$, which counts the number of sentences of $L_{\infty,ω}$ that, in some forcing extension, become a canonical Scott sentence of a model of Phi. We show this cardinal bounds the complexity of (Mod(Phi), iso), the class of models of Phi with universe omega, by proving that (Mod(Phi),iso) is not Borel reducible to (Mod(Psi),iso) whenever ||Psi|| < ||Phi||. Using these tools, we analyze the complexity of the class of countable models of four complete, first-order theories T for which (Mod(T),iso) is properly analytic, yet admit very different behavior. We prove that both `Binary splitting, refining equivalence relations' and Koerwien's example of an eni-depth 2, omega-stable theory have (Mod(T),iso) non-Borel, yet neither is Borel complete. We give a slight modification of Koerwien's example that also is omega-stable, eni-depth 2, but is Borel complete. Additionally, we prove that I_{\infty,ω}(Phi)<\beth_{ω_1} whenever (Mod(Phi),iso) is Borel.

math.LO↗
arXiv2016

The Number of Atomic Models of Uncountable Theories

We show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with $ZFC + \aleph_1 < 2^{\aleph_0}$ that there is a complete theory in a language of size $\aleph_1$ possessing a unique atomic model which is not constructible. Finally we show it is consistent with $ZFC + \aleph_1 < 2^{\aleph_0}$ that for every complete theory $T$ in a language of size $\aleph_1$, if $T$ has uncountable atomic models but no constructible models, then $T$ has $2^{\aleph_1}$ atomic models of size $\aleph_1$.

math.LO↗
arXiv2015

NIM with Cash

Let A be a finite subset of $\nat$. Then NIM(A;n) is the following 2-player game: initially there are $n$ stones on the board and the players alternate removing $a\in A$ stones. The first player who cannot move loses. This game has been well studied. We investigate an extension of the game where Player I starts out with d dollars, Player II starts out with e dollars, and when a player removes a\in A he loses a dollars. The first player who cannot move loses; however, note this can happen for two different reasons: (1) the number of stones is less than min(A), (2) the player has less than $\min(A)$ dollars. This game leads to more complex win conditions then standard NIM. We prove some general theorems from which we can obtain win conditions for a large variety of finite sets A. We then apply them to the sets A={1,L}, and A={1,L,L+1}.

math.CO↗
arXiv2015

Distinct volume subsets

Suppose that $a$ and $d$ are positive integers with $a \geq 2$. Let $h_{a,d}(n)$ be the largest integer $t$ such that any set of $n$ points in $\mathbb{R}^d$ contains a subset of $t$ points for which all the non-zero volumes of the ${t \choose a}$ subsets of order $a$ are distinct. Beginning with Erdős in 1957, the function $h_{2,d}(n)$ has been closely studied and is known to be at least a power of $n$. We improve the best known bound for $h_{2,d}(n)$ and show that $h_{a,d}(n)$ is at least a power of $n$ for all $a$ and $d$.

math.CO↗