arXiv subjects
Publications and source records attributed to Paul B. Larson.
Using a variation of Woodin's $\mathbb{P}_{\mathrm{max}}$ forcing, we force over a model of the Axiom of Determinacy to produce a model of ZFC containing a very strongly increasing sequence of length $\omega_{2}$ consisting of functions from $\omega$ to $\omega$. We also show that there can be no such sequence of length $\omega_{4}$.
The consistency of the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is universally Baire'' is proved relative to $\mathsf{ZFC} + {}$``there is a cardinal that is a limit of Woodin cardinals and of strong cardinals.'' The proof is based on the derived model construction, which was used by Woodin to show that the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is Suslin'' is consistent relative to $\mathsf{ZFC} + {}$``there is a cardinal $\lambda$ that is a limit of Woodin cardinals and of $\mathord{<}\lambda$-strong cardinals.'' The $\Sigma^2_1$ reflection property of our model is proved using genericity iterations as used by Neeman and Steel.
We introduce a hierarchy of models of the Axiom of Determinacy called \emph{Nairian models}. Forcing over the simplest Nairian model, we obtain a model of ${\sf{ZFC}}+{\sf{MM^{++}}}(c)+\neg\square_{\omega_3}+\neg\square(\omega_3)$. Then, fixing $n\in [3, \omega)$, we design a Nairian model and force over it to produce a model of ${\sf{ZFC}}+{\sf{MM^{++}}}(c)+\forall i\in [2, n]\, \neg\square(\omega_i)$. We also build a Nairian model that satisfies ${\sf{ZF}}+"\omega_1$ is a supercompact cardinal." We obtain as corollaries of these constructions (1) the consistent failure of the Iterability Conjecture for the Mitchell-Schindler $\sf{K}^{c}$ construction, (2) the consistent failure of the Iterability Conjecture for the $\sf{K}^{c}$ construction using $2^{2^{\dots 2^{\omega}}}$-complete (for any finite stack of exponents) background extenders, answering a strong version of a question asked by Steel, and (3) a negative answer to Trang's question whether ${\sf{ZF}}+"\omega_1$ is a supercompact cardinal" is equiconsistent with ${\sf{ZFC}}+"$there is a proper class of Woodin cardinals that are limits of Woodin cardinals." These corollaries identify obstructions to extending the methods of (descriptive) inner model theory past a Woodin cardinal which is a limit of Woodin cardinals.
In this paper, we study some new examples of ideals on $ω$ with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order -- known as the "weak Rudin-Keisler order" -- and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin on the Tukey order, we also show that there is an analytic P-ideal above all other analytic P-ideals in the weak Rudin-Keisler. Acknowledgment: this preprint has not undergone peer review or any post-submission improvements or corrections. The Version of Record of this article is published in the Archive for Mathematical Logic, and is available online at https://doi.org/10.1007/s00153-023-00897-z.
We produce a forcing extension of the constructible universe $\bL$ in which every universally measurable set of reals is $\uTDelta^{1}_{2}$, partially answering question CG from David Fremlin's problem list. The analogous result for category holds in the same model.
We show that the statements $\square(ω_{3})$ and $\square(ω_{4})$ both fail in the $\mathbb{P}_{\mathrm{max}}$ extension of a variation of the Chang model introduced by Sargsyan.
Given a unilateral shift $B_w$ (determined by a bounded sequence $w$), a sequence $x \in \ell^2$ is "hypercyclic" for $w$ iff the forward iterates of $x$ under $B_w$ are dense in $\ell^2$. We show that it is possible to make the set of $x \in \ell^2$ which are simultaneously hypercyclic for all $w$ in a fixed $W \subseteq \ell^\infty$ arbitrarily complicated by choosing $W$ appropriately.
We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$. This shows that a nonprincipal ultrafilter on the integers is not sufficient to construct a discontinuous automorphism of the complex field, confirming a conjecture of Simon Thomas. This is an improved version of our paper in the Proceedings of the American Mathematical Society, which used a weak version of the Axiom of Choice for the same result.
Suppose $\mathcal I$ and $\mathcal J$ are proper ideals on some set $X$. We say that $\mathcal I$ and $\mathcal J$ are incompatible if $\mathcal I \cup \mathcal J$ does not generate a proper ideal. Equivalently, $\mathcal I$ and $\mathcal J$ are incompatible if there is some $A \subseteq X$ such that $A \in \mathcal I$ and $X \setminus A \in \mathcal J$. If some $B \subseteq X$ is either in $\mathcal I \setminus \mathcal J$ or in $\mathcal J \setminus \mathcal I$, then we say that $B$ chooses between $\mathcal I$ and $\mathcal J$. We consider the following Ramsey-theoretic problem: Given several pairs $(\mathcal I_1,\mathcal J_1), (\mathcal I_2,\mathcal J_2), \dots, (\mathcal I_k,\mathcal J_k)$ of incompatible ideals on a set $X$, find some $A \subseteq X$ that chooses between as many of these pairs of ideals as possible. The main theorem is that for every $n \in \mathbb N$, there is some $I(n) \in \mathbb N$ such that given at least $I(n)$ pairs of incompatible ideals on any set $X$, there is some $A \subseteq X$ choosing between at least $n$ of them. This theorem is proved in two main steps. The first step is to identify a (purely finitary) problem in extremal combinatorics, and to show that our problem concerning ideals is equivalent to this combinatorial problem. The second step is to analyze the combinatorial problem in order to show that the number $I(n)$ described above exists, and to put bounds on it. We show $\textstyle \frac{1}{2}n \log_2 n - O(n) \,<\, I(n) \,<\, n \ln n + O(n).$ The upper bound is proved by considering a different but closely related combinatorial problem involving hypergraphs, which may be of independent interest. We also investigate some applications of this theorem to a problem concerning conditionally convergent series.
How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We define the \emph{rearrangement number}, a new cardinal characteristic of the continuum, as the answer to this question. We compare the rearrangement number with several natural variants, for example one obtained by requiring the rearranged series to still converge but to a new, finite limit. We also compare the rearrangement number with several well-studied cardinal characteristics of the continuum. We present some new forcing constructions designed to add permutations that rearrange series from the ground model in particular ways, thereby obtaining consistency results going beyond those that follow from comparisons with familiar cardinal characteristics. Finally, we deal briefly with some variants concerning rearrangements by a special sort of permutation and with rearranging some divergent series to become (conditionally) convergent.
We show that for any nonprincipal ultrafilter $U$ on the positive integers, then probability measure induced by the $U$-limit of asymptotic density is not a universally measurable function.
The Levy-Steinitz theorem characterizes the values that a conditionally convergent sequence in of real numbers can attain under permutations. We extend this analysis to sequences of countable sequences of real numbers, under pointwise convergence, reproving a theorem of Stanimir Troyanski.
A countable structure is said to be extendible if it has the same Scott sentence as some uncountable structure. Rigid structures are not extendible. We give an example of an extendible model with a rigid elementary extension.
The Scott process of a relational structure $M$ is the sequence of sets of formulas given by the Scott analysis of $M$. We present axioms for the class of Scott processes of structures in a relational vocabulary $τ$, and use them to give a proof of an unpublished theorem of Leo Harrington from the 1970's, showing that a counterexample to Vaught's Conjecture has models of cofinally many Scott ranks below $ω_{2}$. Our approach also gives a theorem of Harnik and Makkai, showing that if there exists a counterexample to Vaught's Conjecture, then there is a counterexample whose uncountable models have the same $\mathcal{L}_{ω_{1}, ω}(τ)$-theory, and which has a model of Scott rank $ω_{1}$. Moreover, we show that if $ϕ$ is a sentence of $\mathcal{L}_{ω_{1}, ω}(τ)$ giving rise to a counterexample to Vaught's Conjecture, then for every limit ordinal $α$ greater than the quantifier depth of $ϕ$ and below $ω_{2}$, $ϕ$ has a model of Scott rank $α$.
A function of two variables F(x,y)is universal iff for every other function G(x,y) there exists functions h(x) and k(y) with G(x,y) = F(h(x),k(y)) Sierpinski showed that assuming the continuum hypothesis there exists a Borel function F(x,y) which is universal. Assuming Martin's Axiom there is a universal function of Baire class 2. A universal function cannot be of Baire class 1. We show that it is consistent that for each countable ordinal alpha>2 there is a universal function of class alpha but none of smaller class. We show that it is consistent with ZFC that there is no universal function (Borel or not) on the reals, and we show that it is consistent that there is a universal function but no Borel universal function. We also prove some results concerning higher arity universal functions. For example, the existence of an F such that for every G there are unary h,k,j such that G(x,y,z) = F(h(x),k(y),j(z)) is equivalent to the existence of a 2-ary universal F. However the existence of an F such that for every G there are h,k,j such that G(x,y,z) = F(h(x,y),k(x,z),j(y,z)) follows from a 2-ary universal F but is strictly weaker. Results obtained Mar-June 2009, Nov 2010. Last revised April 2012 LaTex2e: 28 pages Latest version at: www.math.wisc.edu/~miller
The \emph{Filter Dichotomy} says that every uniform nonmeager filter on the integers is mapped by a finite-to-one function to an ultrafilter. The consistency of this principle was proved by Blass and Laflamme. A function between topological spaces is \emph{universally measurable} if the preimage of %every open subset of the codomain is measured by every Borel measure on the domain. A \emph{medial limit} is a universally measurable function from $\mathcal{P}(ω)$ to the unit interval [0,1] which is finitely additive for disjoint sets, and maps singletons to 0and $ω$ to 1. Christensen and Mokobodzki independently showed that the Continuum Hypothesis implies the existence of medial limits. We show that the Filter Dichotomy implies that there are no medial limits.