Successor Ordinals Research Articles

The consistency strength of hyperstationarity

We introduce the large-cardinal notions of [Formula: see text]-greatly-Mahlo and [Formula: see text]-reflection cardinals and prove (1) in the constructible universe, [Formula: see text], the first [Formula: see text]-reflection cardinal, for [Formula: see text] a successor ordinal, is strictly between the first [Formula: see text]-greatly-Mahlo and the first [Formula: see text]-indescribable cardinals, (2) assuming the existence of a [Formula: see text]-reflection cardinal [Formula: see text] in [Formula: see text], [Formula: see text] a successor ordinal, there exists a forcing notion in [Formula: see text] that preserves cardinals and forces that [Formula: see text] is [Formula: see text]-stationary, which implies that the consistency strength of the existence of a [Formula: see text]-stationary cardinal is strictly below a [Formula: see text]-indescribable cardinal. These results generalize to all successor ordinals [Formula: see text] the original same result of Mekler–Shelah [A. Mekler and S. Shelah, The consistency strength of every stationary set reflects, Israel J. Math. 67(3) (1989) 353–365] about a [Formula: see text]-stationary cardinal, i.e. a cardinal that reflects all its stationary sets.

Countable successor ordinals as generalized ordered topological spaces

We prove the following Main Theorem: Every continuous image of a Hausdorff topological space X is a generalized ordered space if and only if X is homeomorphic to a countable successor ordinal (with the order topology). This is a generalization of E. van Douwen's result about orderable spaces.

Guessing, Mind-Changing, and the Second Ambiguous Class

In his dissertation, Wadge defined a notion of guessability on subsets of the Baire space and gave two characterizations of guessable sets. A set is guessable iff it is in the second ambiguous class (boldface Delta^0_2), iff it is eventually annihilated by a certain remainder. We simplify this remainder and give a new proof of the latter equivalence. We then introduce a notion of guessing with an ordinal limit on how often one can change one's mind. We show that for every ordinal alpha, a guessable set is annihilated by alpha applications of the simplified remainder if and only if it is guessable with fewer than alpha mind changes. We use guessability with fewer than alpha mind changes to give a semi-characterization of the Hausdorff difference hierarchy, and indicate how Wadge's notion of guessability can be generalized to higher-order guessability, providing characterizations for boldface Delta^0_alpha for all successor ordinals alpha>1.

Large Sets of Lattices without Order Embeddings

Let I and μ be an infinite index set and a cardinal, respectively, such that |I| ≤ μ and, starting from ℵ0, μ can be constructed in countably many steps by passing from a cardinal λ to 2λ at successor ordinals and forming suprema at limit ordinals. We prove that there exists a system X = {Li: i ∈ I} of complemented lattices of cardinalities less than |I| such that if i, j ∈ I and φ: Li → Lj is an order embedding, then i = j and φ is the identity map of Li. If |I| is countable, then, in addition, X consists of finite lattices of length 10. Stating the main result in other words, we prove that the category of (complemented) lattices with order embeddings has a discrete full subcategory with |I| many objects. Still in other words, the class of these lattices has large antichains (that is, antichains of size |I|) with respect to the quasiorder “embeddability.” As corollaries, we trivially obtain analogous statements for partially ordered sets and semilattices.

Preserving Categoricity and Complexity of Relations

In [3, 6, 7], it was proved that for each computable ordinal α, there is a structure that is Δ0 α categorical but not relatively Δ0 α categorical. The original examples were not familiar algebraic kinds of structures. In [11], it was shown that for α = 1, there are further examples in several familiar classes of structures, including rings and 2-step nilpotent groups. Similar example for all computable successor ordinals were constructed in [12]. In the present paper, this result is extended to computable limit ordinals. We know of an example of an algebraic field that is computably categorical but not relatively computably categorical. Here we show that for each computable limit ordinal α > ω, there is a field which is Δ0 α categorical but not relatively Δ0 α categorical. Examples on dimension and complexity of relations are given.

Symplectic capacities on surfaces

We classify capacities on the class Symo(2) of connected symplectic surfaces with at most countably many nonplanar ends. To obtain the classification we study diffeomorphism types of surfaces in Symo(2) of infinite genus with nonplanar ends; it turns out that these types are in bijective correspondence with countable successor ordinals of the form ωα · d + 1, where α is an ordinal and d ≥ 0 is an integer. It also turns out that if S1 and S2 are two open surfaces of infinite genera with at most countably many nonplanar ends, then each of the surfaces embeds into the other. Our classification implies that every capacity on the class of symplectic surfaces in Symo(2) of infinite genus differs from the Hofer–Zehnder capacity by a non-negative finite or infinite constant.

Intrinsic bounds on complexity and definability at limit levels

AbstractWe show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family). We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.

Truth by Ascent

This paper offers a lighthearted presentation of some of the chief ideas about truth that are shared by theories similar to those of Kripke, Herzberger, and Gupta. The problem is to explain the concept of truth for a language that contains its own truth predicate. The proposal of these theories is that one can unwind the tangles that threaten by invoking a transfinite series of stages of semantic reflection as one ascends the ordinals. The presentation emphasizes how each stage begins, to the extent that it can, with the fruits of semantic reflection at earlier stages. Special attention is given to clarifying how these theories about truth treat the three chief kinds of ordinal numbers: the initial ordinal, successor ordinals, and limit ordinals.

Orders, reduction graphs and spectra

In this paper, we consider Abstract Reduction Systems as the setting where we need to investigate order properties of reduction graphs. To this aim, the main tool is the notion of spectrum, which captures some essential features of reduction sequences. We show that each spectrum is a complete partial order with respect to a suitable information ordering. Then we consider linearly ordered spectra. For those of them which are not well-ordered, we obtain a graph-theoretic characterization in terms of forbidden subgraphs. For well-ordered spectra, we show that they can represent all countable successor ordinals. Then, considering constructive ordinals, we address the well-known problem of knowing which ordinals are lambda representable (that is for which ordinal α there exists a lambda term T such that α is isomorphic to the spectrum of T). We give a partial answer by showing that all successor ordinals a, with α < ε 0, are lambda-representable.

Inductive definitions over finite structures

Inductive definitions have played a central role in the foundations of mathematics for over a century. They were used in the 1970s as the backbone of one major generalization of Recursive Function Theory (Moschovakis, 1974; Aczel, 1977). In recent years the relevance of inductive definitions (in particular over finite structures) to Database Theory, to Descriptive Computational Complexity, and to Logics of programs has been recognized. A seminal paper on inductive definitions in Database Theory is Chandra and Hare1 (1982), where they define a hierarchy of queries over finite structures, within which minor steps (successor ordinals) correspond to first-order quantifier alternations, and major steps (limit ordinals) correspond to uses of lixpoints. They left open the question of whether the hierarchy remains strict above the first major step (level w). This problem was answered in the negative by Immerman (1986). Since the collection of first-order lixpoint queries over finite structures is closed under composition and first-order operations other than negation (Moschovakis, 1974), the main component of Immerman’s solution was

Some results on consecutive large cardinals II: Applications of radin forcing

Letκ be a 3 huge cardinal in a countable modelV of ZFC, and letA andB be subsets of the successor ordinals <κ so thatA ⋃B={α<κ:α is a successor ordinal}. Using techniques of Gitik, we construct a choiceless modelN A of ZF of heightκ so thatN A ╞“ZF+⌍AC ω+Forα ∈A, ℵa is a Ramsey cardinal+Forβ ∈B, ℵβ is a singular Rowbottom cardinal which carries a Rowbottom filter+Forγ a limit ordinal, ℵy is a Jonsson cardinal which carries a Jonsson filter”.