Countable Ordinals Research Articles

Strong reducibilities and set theory

We study Medvedev reducibility in the context of set theory — specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li [7], we show that the Medvedev degrees of countable ordinals are far from linearly ordered in multiple ways, our main result here being that there is a club of ordinals which is an antichain with respect to Medvedev reducibility. We then generalize these results to arbitrary “reasonably-definable” reducibilities, under appropriate set-theoretic hypotheses.We then turn from ordinals to general structures. We show that some of the results above yield characterizations of counterexamples to Vaught's conjecture; another applies to all situations, assigning an ordinal to any reasonable class of structures and “measure” on that class. We end by discussing some directions for future research.

Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal

Hardy functions are defined by transfinite recursion and provide upper bounds for the growth rate of the provably total computable functions in various formal theories, making them an essential ingredient in many proofs of independence. Their definition is contingent on a choice of fundamental sequences, which approximate limits in a ‘canonical’ way. In order to ensure that these functions behave as expected, including the aforementioned unprovability results, these fundamental sequences must enjoy certain regularity properties.In this article, we prove that Buchholz's system of fundamental sequences for the ϑ function enjoys such conditions, including the Bachmann property. We partially extend these results to variants of the ϑ function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along ϑ(εΩ+1).

Superfair Stochastic Games

A two-person zero-sum stochastic game with a nonnegative stage reward function is superfair if the value of the one-shot game at each state is at least as large as the reward function at the given state. The payoff in the game is the limit superior of the expected stage rewards taken over the directed set of all finite stop rules. If the game has countable state and action spaces and if at least one of the players has a finite action space at each state, then the game has a value. The value of the stochastic game is obtained by a transfinite algorithm on the countable ordinals.

Schreier families and -(almost) greedy bases

AbstractLet $\mathcal {F}$ be a hereditary collection of finite subsets of $\mathbb {N}$ . In this paper, we introduce and characterize $\mathcal {F}$ -(almost) greedy bases. Given such a family $\mathcal {F}$ , a basis $(e_n)_n$ for a Banach space X is called $\mathcal {F}$ -greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$ , $m \in \mathbb {N}$ , and $G_m(x)$ , we have $$ \begin{align*} \|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}. \end{align*} $$ Here, $G_m(x)$ is a greedy sum of x of order m, and $\mathbb {K}$ is the scalar field. From the definition, any $\mathcal {F}$ -greedy basis is quasi-greedy, and so the notion of being $\mathcal {F}$ -greedy lies between being greedy and being quasi-greedy. We characterize $\mathcal {F}$ -greedy bases as being $\mathcal {F}$ -unconditional, $\mathcal {F}$ -disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for $\mathcal {F}$ -almost greedy bases.Furthermore, we provide several examples of bases that are nontrivially $\mathcal {F}$ -greedy. For a countable ordinal $\alpha $ , we consider the case $\mathcal {F}=\mathcal {S}_{\alpha }$ , where $\mathcal {S}_{\alpha }$ is the Schreier family of order $\alpha $ . We show that for each $\alpha $ , there is a basis that is $\mathcal {S}_{\alpha }$ -greedy but is not $\mathcal {S}_{\alpha +1}$ -greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta $ , $$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Countable ordinals in indiscernibility spectra

Given a set $A$ of reals, the <em>indiscernibility spectrum</em> of $A$ is the set of countable ordinals which are simultaneously indiscernible in $L[a]$ for every $a^\sharp \in A$. Under large-cardinal assumptions, we construct sets of sharps with counta

$C^*$-points vs $P$-points and $P^\flat$-points

In a Tychonoff space $X$, the point $p\in X$ is called a $C^*$-point if every real-valued continuous function on $C\smallsetminus \{p\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^*$-point. A classic example is the space ${\bf W}^*=\omega_1+1$ consisting of the countable ordinals together with $\omega_1$. The point $\omega_1$ is known to be a $C^*$-point as well as a $P$-point. We supply a characterization of $C^*$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space is a $C^*$-point. This process leads to many interesting new discoveries.

Ramsey theory over partitions III: Strongly Luzin sets and partition relations

The strongest type of coloring of pairs of countable ordinals, gotten by Todorčević from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size ℵ 1 \aleph _1 . In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it. This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpiński a hundred years ago.

Entropy pair realization

AbstractWe show that the complete positive entropy (CPE) class $\alpha $ of Barbieri and García-Ramos contains a one-dimensional subshift for all countable ordinals $\alpha $ , that is, the process of alternating topological and transitive closure on the entropy pairs relation of a subshift can end on an arbitrary ordinal. This is the composition of three constructions. We first realize every ordinal as the length of an abstract ‘close-up’ process on a countable compact space. Next, we realize any abstract process on a compact zero-dimensional metrizable space as the process started from a shift-invariant relation on a subshift, the crucial construction being the implementation of every compact metrizable zero-dimensional space as an open invariant quotient of a subshift. Finally, we realize any shift-invariant relation E on a subshift X as the entropy pair relation of a supershift $Y \supset X$ , and under strong technical assumptions, we can make the CPE process on Y end on the same ordinal as the close-up process of E.

Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals

We define a collection of topological Ramsey spaces consisting of equivalence relations on ω with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of ω. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) in order to obtain a dual Ramsey theorem for such ordinals.

The size of the class of countable sequences of ordinals

Assume Z F + A D + D C R \mathsf {ZF} + \mathsf {AD} + \mathsf {DC}_\mathbb {R} . There is no injection of &gt; ω 1 ω 1 {}^{&gt;\omega _{1}}{\omega _{1}} (the set of countable length sequences of countable ordinals) into ω O N {}^\omega \mathrm {ON} (the class of ω \omega length sequences of ordinals). There is no injection of [ ω 1 ] ω 1 [{\omega _{1}}]^{{\omega _{1}}} (the powerset of ω 1 {\omega _{1}} ) into &gt; ω 1 O N {}^{&gt;{\omega _{1}}}\mathrm {ON} (the class of countable length sequences of ordinals).

Reducing ω-model reflection to iterated syntactic reflection

In mathematical logic there are two seemingly distinct kinds of principles called “reflection principles.” Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic reflection principles assert that every provable sentence from some complexity class is true. In this paper, we study connections between these two kinds of reflection principles in the setting of second-order arithmetic. We prove that, for a large swathe of theories, [Formula: see text]-model reflection is equivalent to the claim that arbitrary iterations of uniform [Formula: see text] reflection along countable well-orderings are [Formula: see text]-sound. This result yields uniform ordinal analyzes of theories with strength between [Formula: see text] and [Formula: see text]. The main technical novelty of our analysis is the introduction of the notion of the proof-theoretic dilator of a theory [Formula: see text], which is the operator on countable ordinals that maps the order-type of [Formula: see text] to the proof-theoretic ordinal of [Formula: see text]. We obtain precise results about the growth of proof-theoretic dilators as a function of provable [Formula: see text]-model reflection. This approach enables us to simultaneously obtain not only [Formula: see text], [Formula: see text] and [Formula: see text] ordinals but also reverse-mathematical theorems for well-ordering principles.

Countable Ordinals and Big Ramsey Degrees

In this paper we consider big Ramsey degrees of finite chains in countable ordinals. We prove that a countable ordinal has finite big Ramsey degrees if and only if it is smaller than ωω. Big Ramsey degrees of finite chains in all other countable ordinals are infinite.

Infinite-dimensional manifolds related to C-spaces

Haver's property C turns out to be related to Borst's transfinite extension of the covering dimension. We prove that, for a uncountably many countable ordinals β there exists a strongly universal kω-space for the class of spaces of transfinite covering dimension &lt;β. In some sense, our result is a kω-counterpart of Radul's theorem on existence of absorbing sets of given transfinite covering dimension.

Borel reducibility and symmetric models

We develop a correspondence between Borel equivalence relations induced by closed subgroups of S ∞ S_\infty and weak choice principles, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998). For example, we show that the equivalence relation ≅ ω + 1 , 0 ∗ \cong ^\ast _{\omega +1,0} is strictly below ≅ ω + 1 , &gt; ω ∗ \cong ^\ast _{\omega +1,&gt;\omega } in Borel reducibility. By results of Hjorth-Kechris-Louveau, ≅ ω + 1 , &gt; ω ∗ \cong ^\ast _{\omega +1,&gt;\omega } provides invariants for Σ ω + 1 0 \Sigma ^0_{\omega +1} equivalence relations induced by actions of S ∞ S_\infty , while ≅ ω + 1 , 0 ∗ \cong ^\ast _{\omega +1,0} provides invariants for Σ ω + 1 0 \Sigma ^0_{\omega +1} equivalence relations induced by actions of abelian closed subgroups of S ∞ S_\infty . We further apply these techniques to study the Friedman-Stanley jumps. For example, we find an equivalence relation F F , Borel bireducible with = + + =^{++} , so that F ↾ C F\restriction C is not Borel reducible to = + =^{+} for any non-meager set C C . This answers a question of Zapletal, arising from the results of Kanovei-Sabok-Zapletal (2013). For these proofs we analyze the symmetric models M n M_n , n &gt; ω n&gt;\omega , developed by Monro (1973), and extend the construction past ω \omega , through all countable ordinals. This answers a question of Karagila (2019).

Partition properties for simply definable colourings

We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and forcing axioms imply that there is a homogeneous closed unbounded subset of ω1 for every colouring of the finite sets of countable ordinals that is definable by a Σ1-formula that only uses the cardinal ω1 and real numbers as parameters. Moreover, it is shown that certain large cardinal properties cause analogous partition properties to hold at the given large cardinal and these implications yield natural examples of inaccessible cardinals that possess strong partition properties for Σ1-definable colourings and are not weakly compact. In contrast, we show that Σ1- definability behaves fundamentally different at ω2 by showing that various large cardinal assumptions and Martin’s Maximum are compatible with the existence of a colouring of pairs of elements of ω2 that is definable by a Σ1-formula with parameter ω2 and has no uncountable homogeneous set. Our results will also allow us to derive tight bounds for the consistency strengths of various partition properties for definable colourings. Finally, we use the developed theory to study the question whether certain homeomorphisms that witness failures of weak compactness at small cardinals can be simply definable.

Topological partition relations for countable ordinals

Topological partition relations for countable ordinals

Even more on partitioning triples of countable ordinals

We prove that $\omega _1 \to (\omega + \omega + 1, n)^3$ for all $n < \omega$.

The order on the light cone and its induced topology

In this paper, we first correct a recent misconception about a topology that was suggested by Zeeman as a possible alternative to his fine topology. This misconception appeared while trying to establish the causality in the ambient boundary-ambient space cosmological model. We then show that this topology is actually the intersection topology (in the sense of Reed [The intersection topology w.r.t. the real line and the countable ordinals, Trans. Am. Math. Soc. 297(2) (1986) 509–520]) between the Euclidean topology on [Formula: see text] and the order topology whose order, namely horismos, is defined on the light cone and we show that the order topology from horismos belongs to the class of Zeeman topologies. These results accelerate the need for a deeper and more systematic study of the global topological properties of spacetime manifolds.

Some Applications of Order-Embeddings of Countable Ordinals into the Real Line

It is a well-known fact that an ordinal \( \alpha \) can be embedded into the real line \( \mathbb{R} \) in an order-preserving manner if and only if \( \alpha \) is countable. However, it would seem that outside of set theory, this fact has not yet found any concrete applications. The goal of this paper is to present some applications. More precisely, we show how two classical results, one in point-set topology and the other in real analysis, can be proven by defining specific order-embeddings of countable ordinals into \( \mathbb{R} \).