arXiv subjects
Publications and source records attributed to Linda Westrick.
Many statements studied in reverse mathematics can be seen as mathematical problems, formulated in terms of instances and solutions. We develop a framework of typicality encompassing measure and genericity, and we classify the reverse mathematics zoo in terms of which problems admit typical solutions. It turns out that even very weak problems do not admit typical solutions.
For any subset $Z \subseteq \mathbb{Q}$, consider the set $S_Z$ of subfields $L\subseteq \overline{\mathbb{Q}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in $L$ such that $C \cap \mathbb{Q}=Z$. Placing a natural topology on the set $\text{Sub}(\overline{\mathbb{Q}})$ of subfields of $\overline{\mathbb{Q}}$, we show that if $Z$ is not thin in $\mathbb{Q}$, then $S_Z$ is meager in $\text{Sub}(\overline{\mathbb{Q}})$. Here, thin and meager both mean "small", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields $L$ have the property that the ring of algebraic integers $\mathcal{O}_L$ is universally definable in $L$. The main tools are Hilbert's Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists$-definable subset of an algebraic extension of $\mathbb Q$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.
Let At denote the set of infinite sequences of effective dimension t. We determine both how close and how far an infinite sequence of dimension s can be from one of dimension t, measured using the Besicovitch pseudometric. We also identify classes of sequences for which these infima and suprema are realized as minima and maxima. When t < s, we find d(X,At) is minimized when X is a Bernoulli p-random, where H(p)=s, and maximized when X belongs to a class of infinite sequences that we call s-codewords. When s < t, the situation is reversed.
We characterize the completely determined Borel subsets of HYP as exactly the omega_1^{ck} subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon & the third author.
We consider the reverse math strength of the statement $\mathsf{C\text-DM}$:"Every completely determined Borel set is measurable." Over $\mathsf{WWKL}_0$, we obtain the following results analogous to the previously studied category case. $\mathsf{C\text-DM}$ lies strictly between $\mathsf{ATR}_0$ and $\mathsf{L}_{ω_1,ω}\text-\mathsf{CA}$. Whenever $M\subseteq 2^ω$ is the second-order part of an $ω$-model of $\mathsf{C\text-DM}$, then for every $Z \in M$, there is a $R \in M$ such that $R$ is $Δ^1_1$-random relative to $Z$. On the other hand, without $\mathsf{WWKL}_0$, all sets have measure zero (as measured according to $\mathsf{C\text-DM}$), and it follows vacuously that $\neg \mathsf{WWKL}_0$ implies $\mathsf{C\text-DM}$ over $\mathsf{RCA}_0$.
We show that a computable function $f:\mathbb R\rightarrow\mathbb R$ has Luzin's property (N) if and only if it reflects $Π^1_1$-randomnes, if and only if it reflects $Δ^1_1(\mathcal O)$-randomness, and if and only if it reflects $\mathcal O$-Kurtz randomness, but reflecting Martin-Löf randomness or weak-2-randomness does not suffice. Here a function $f$ is said to reflect a randomness notion $R$ if whenever $f(x)$ is $R$-random, then $x$ is $R$-random as well. If additionally $f$ is known to have bounded variation, then we show $f$ has Luzin's (N) if and only if it reflects weak-2-randomness, and if and only if it reflects $\emptyset'$-Kurtz randomness. This links classical real analysis with algorithmic randomness.
We show that if $1 \leq_W F$ and $F \star F \leq_W F$, then $F^\diamond \leq_W F$, where $\star$ and $\diamond$ are the following operations in the Weihrauch lattice: $\star$ is the compositional product, which allows the use of two principles in sequence, while the diamond operator $\diamond$ allows an arbitrary but finite number of uses of the given principle in sequence. This answers a question of Pauly.
We construct Z^2-SFTs at every computable level of the hierarchy of topological completely positive entropy (TCPE), answering Barbieri and García-Ramos, who asked if there was one at level 3. Furthermore, we show the property of TCPE in Z^2-SFTs is coanalytic complete. Thus there is no simpler description of TCPE in Z^2-SFTs than in the general case.