Limit Ordinals Research Articles

Periodicity in the cumulative hierarchy

We investigate the structure of rank-to-rank elementary embeddings at successor rank, working in \mathrm{ZF} set theory without the Axiom of Choice. Recall that the set-theoretic universe is naturally stratified by the cumulative hierarchy, whose levels V_\alpha are defined via iterated application of the power set operation, starting from V_0=\emptyset , setting V_{\alpha+1}=\mathcal{P}(V_\alpha) , and taking unions at limit stages. Assuming that j:V_{\alpha+1}\to V_{\alpha+1} is a (non-trivial) elementary embedding, we show that V_\alpha is fundamentally different from V_{\alpha+1} : we show that j is definable from parameters over V_{\alpha+1} iff \alpha+1 is an odd ordinal. The definability is uniform in odd \alpha+1 and j . We also give a characterization of elementary j:V_{\alpha+2}\to V_{\alpha+2} in terms of ultrapower maps via certain ultrafilters. For limit ordinals \lambda , we prove that if j:V_\lambda\to V_\lambda is \Sigma_1 -elementary, then j is not definable over V_\lambda from parameters, and if \beta<\lambda and j:V_\beta\to V_\lambda is fully elementary and \in -cofinal, then j is likewise not definable. If there is a Reinhardt cardinal, then for all sufficiently large ordinals \alpha , there is indeed an elementary j:V_\alpha\to V_\alpha , and therefore the cumulative hierarchy is eventually periodic (with period 2).

Constructing Initial Algebras Using Inflationary Iteration

An old theorem of Ad\'amek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using "inflationary" iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor's constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich class of endofunctors to which the new theorem applies, provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk, van den Berg and Palmgren, and which is known to hold in the internal constructive logic of many kinds of topos.

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.

Inductive limits of ideals

G. Debs and J. Saint Raymond in 2009 defined the Borel separation rank of an analytic ideal I (rk(I)) as minimal ordinal α<ω1 such that there is S∈Σ1+α0 with I⊆S and I⋆∩S=∅, where I⋆ is the filter dual to the ideal I. Moreover, they introduced ideals Finα, for all α<ω1, and conjectured that rk(I)≥α if and only if I contains an isomorphic copy of Finα (Finα⊑I). To define Finα in the case of limit ordinals 0<α<ω1, G. Debs and J. Saint Raymond introduced inductive limits of ideals.We show that the above conjecture is false in the case of α=ω by constructing an ideal Finω′ of rank ω such that Finω⋢Finω′. However, we show that Finω′⊑I is equivalent to ∀n∈ωFinn⊑I. We discuss (indicated by the above result) possible modification of the original conjecture for limit ordinals.

Clocked population protocols

We define an extension to the standard population protocol that provides each agent with a clock signal that indicates when the agent has waited long enough for the protocol to have converged. We represent “long enough” as an infinite time interval, and treat the protocol as operating over transfinite time. Over finite time intervals, the protocol behaves as in the standard model. At nonzero limit ordinals, corresponding to clock ticks, the protocol switches to a limit of previous configurations supplemented by a clock signal appearing as an extra component in some of the agents' states. Fairness generalizes to transfinite executions straightforwardly. We show that a clocked population protocol running in less than ωk time for any fixed k≥2 is equivalent in power to a nondeterministic Turing machine with space complexity logarithmic in the population size, and can be simulated using only finitely many clock ticks.

When Pκ(λ) (vaguely) resembles κ

Let μ<κ<λ be three infinite cardinals, the first two being regular. Assuming the existence of a cofinal subset of (Pκ(λ),⊆) of size λ, we define an ideal on Pκ(λ) which we argue to be the analog for Pκ(λ) of NSκ|Eμκ (the restriction of the nonstationary ideal on κ to the set of limit ordinals less than κ of cofinality μ). We show that this ideal on Pκ(λ) is the ideal dual to the well-known μ-club filter, thus answering a question raised by Krueger and Foreman.

DEGREES OF CATEGORICITY ON A CONE VIAη-SYSTEMS

AbstractWe investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees. We show that, on a cone, every structure has a strong degree of categoricity, and that degree of categoricity is${\rm{\Delta }}_\alpha ^0 $-complete for someα. To prove this, we extend Montalbán’sη-system framework to deal with limit ordinals in a more general way. We also show that, for any fixed computable structure, there is an ordinalαand a cone in the Turing degrees such that the exact complexity of computing an isomorphism between the given structure and another copy${\cal B}$in the cone is a c.e. degree in${\rm{\Delta }}_\alpha ^0\left( {\cal B} \right)$. In each of our theorems the cone in question is clearly described in the beginning of the proof, so it is easy to see how the theorems can be viewed as general theorems with certain effectiveness conditions.

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.

Generalizations of basic and large submodules of $$QTAG$$ -modules

A \(QTAG\)-module \(M\) over an associative ring \(R\) with unity is \(k\)-projective if \(H_k(M)=0\) and for a limit ordinal \(\sigma ,\) it is \(\sigma \)-projective if there exists a submodule \(N\) bounded by \(\sigma \) such that \(M/N\) is a direct sum of uniserial modules. \(M\) is totally projective if it is \(\sigma \)-projective for all limit ordinals \(\sigma .\) If \(\alpha \) denotes the class of all \(QTAG\)-modules \(M\) such that \(M/H_\beta (M)\) is totally projective for every ordinal \(\beta <\alpha ,\) then these modules are called \(\alpha \)-modules. Here we study these \(\alpha \)-modules and generalize the concept of basic submodules as \(\alpha \)-basic submodules. It is found that every \(\alpha \)-module \(M\) contains an \(\alpha \)-basic submodule and any two \(\alpha \)-basic submodules of \(M\) are isomorphic. A submodule \(L\) of an \(\alpha \)-module is \(\alpha \)-large if \(M=L+B,\) for any \(\alpha \)-basic submodule \(B\) of \(M.\) Many other interesting properties of \(\alpha \)-basic, \(\alpha \)-large and \(\alpha \)-modules are studied.

Indecomposable modules over pure semisimple hereditary rings

If R is a hereditary left artinian ring, then R is left pure semisimple if and only if the family R-ind of all finitely generated indecomposable left R-modules has a (unique) Ext-injective partition R-ind=⋃α⩽δUα. This partition is used to give a complete description of the distribution of all indecomposable modules over a left pure semisimple hereditary indecomposable ring R of infinite representation type. More precisely, R-ind is the disjoint union of the countable set of all preinjective modules and the finite set of all preprojective modules, and countable sets of Auslander–Reiten components of the form ⋃k<ωUα+k, for all limit ordinals α, constructed from the Ext-injective partition of R-ind. In particular, we show that an indecomposable left R-module M is not the source of a left almost split morphism in R-mod if and only if M belongs to Uα, where α is an infinite limit ordinal; and the direct sum of modules in Uα is not endofinite for each infinite limit ordinal α. Moreover, the endomorphism ring of each indecomposable left R-module is a division ring.

Transfinite ranges and the local spectrum

Let T be a Banach space operator. For ordinal numbers α we define the α-ranges Rα(T) which generalize the ranges of powers R(Tn). The intersection ⋂αRα(T) is the coeur algébrique of T. Moreover, the coeur algébrique (and more generally the α-ranges for limit ordinals α) have similar properties as the coeur analytique of T. So it is possible to introduce algebraic local spectra which have properties analogous to those of classical (analytic) local spectra studied in local spectral theory.

On the Unit Groups of Commutative Modular Group Algebras of KT-Groups

Let G be an abelian group and let R be a commutative indecomposable p-perfect ring with identity of prime characteristic p. Denote by RG the group ring of G over R and by V(RG) the group of normalized units (i.e., the units of augmentation 1) of the group ring RG. Warfield [16] introduces the concept KT-module A over a discrete valuation ring R and invariants h(α, A) of this module for limit ordinals α. We compute the invariants h(α, V(RG)) of the group V(RG). Our result correct essential mistakes of the result in this direction of Danchev [4].

Remarks on the space [formula omitted] in ZF

We show: (1) ℵ 1 with the order topology is effectively normal. i.e., there is a function associating to every pair ( A , B ) of disjoint closed subsets of ℵ 1 a pair ( U , V ) of disjoint open sets with A ⊆ U and B ⊆ V . (2) For every countable ordinal α the ordered space α is metrizable. Hence, every closed subset of α is a zero set and consequently the Čech–Stone extension of α coincides with its Wallman extension. (3) In the Feferman–Levy model where ℵ 1 is singular, the ordinal space ℵ 1 is base-Lindelöf but not Lindelöf. (4) The Čech–Stone extension β ℵ 1 of ℵ 1 is compact iff its Wallman extension W ( ℵ 1 ) is compact. (5) The set L of all limit ordinals of ℵ 1 is not a zero set.

The automorphism tower of a centerless group without Choice

For a centerless group G, we can define its automorphism tower. We define G α : G 0 = G, G α+1 = Aut(G α ) and for limit ordinals $${G^{\delta}=\bigcup_{\alpha<\delta}G^{\alpha}}$$ . Let τ G be the ordinal when the sequence stabilizes. Thomas’ celebrated theorem says $${\tau_{G}<(2^{|G|})^{+}}$$ and more. If we consider Thomas’ proof too set theoretical (using Fodor’s lemma), we have here a more direct proof with little set theory. However, set theoretically we get a parallel theorem without the Axiom of Choice. Moreover, we give a descriptive set theoretic approach for calculating an upper bound for τ G for all countable groups G (better than the one an analysis of Thomas’ proof gives). We attach to every element in G α , the αth member of the automorphism tower of G, a unique quantifier free type over G (which is a set of words from $${G*\langle x\rangle}$$ ). This situation is generalized by defining “(G, A) is a special pair”.

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.

Branch space representations of lines

Todorčevic noted that any linearly ordered set ( X , < ) is isomorphic to the branch space of some tree, provided the tree is allowed to be as complicated as X itself. In this paper we investigate representing linearly ordered sets as branch spaces of trees where the trees satisfy certain natural restrictions designed to make them less complicated than the set we are seeking to represent. We show that any uncountable order-complete linearly ordered set X can be represented as the branch space of a tree T that is more simple than X and that if X is representable as a branch space, then any G δ -subset of X is also representable as a branch space. However, we show that even though the usual set R of real numbers can be represented as the branch space of a tree with countable height and countable levels, most subsets of R cannot be represented in this way. We characterize those limit ordinals λ that can be represented by trees whose nodes do not contain copies of λ or λ ∗ . We also study topological properties of branch spaces. Our results show that if T is a tree of height ω 1 that does not contain any ω 1 -branches, then the branch space of T must be hereditarily paracompact. We investigate branch spaces of Aronszajn trees, showing that any such branch space is hereditarily paracompact, first-countable, Lindelöf, non-separable, non-metrizable, and provided T does not contain any Souslin subtree, then its branch space is not perfect (i.e., has a closed subset that is not a G δ -set). We also study the existence of σ-disjoint π bases and σ-disjoint bases in branch spaces of trees.

Uniformization and anti-uniformization properties of ladder systems

Let S denote a stationary subset of limit ordinals of ω1. A ladder system on S is a sequence {Lα : α ∈ S} such that each Lα is an unbounded subset of α of order type ω. A ladder system is uniformizable if for each sequence 〈fα : α ∈ S〉 of functions fα : Lα → ω there is an F : ω1 → ω such that F 1 Lα =∗ fα for each α ∈ S. I.e., for each α ∈ S, {β ∈ Lα : F (β) 6= fα(β)} is finite. We now formulate natural weakenings of uniformizable denoted, for each n ∈ ω, by Pn: A ladder system is said to satisfy Pn if for each f : S → ω there is an F : ω1 → [ω] such that for each α ∈ S (a) F 1 Lα is eventually constant with value sα, and (b) f(α) ∈ sα. Note that P0 is equivalent to the version of uniformizable obtained by considering only sequences of constant functions fα. We will say that a ladder system satisfies P<ω if for each f : S → ω there is an F : ω1 → [ω] satisfying (a) and (b) above. If we drop the requirement that the restrictions F 1 Lα are eventually constant we obtain uniformization properties that we denote Mn and M<ω. E.g., a ladder system is said to satisfy M<ω if for each f : S → ω there is an F : ω1 → [ω] such that for each α ∈ S, f(α) ∈ F (β) for all but finitely many β ∈ Lα. Most of these uniformization properties can be characterized in terms of properties of a certain topological space naturally associated to any ladder system. If L is a ladder system, let XL denote the topology space ω1 × {0} ∪ S × {1} where every point (α, 0) is isolated and for each α ∈ S, a basic neighborhood of (α, 1) consists of {(α, 1)} along with a cofinite subset of Lα×{0}. Such a space is always first countable and locally compact. The stationarity of S implies that it is not collectionwise Hausdorff. Spaces XL have been considered by many to construct examples of normal not collectionwise Hausdorff spaces (see [11] and [2]). It is folklore that a ladder system L satisfies P0 if and only if XL is normal. The property M<ω is characterized by XL being countably metacompact. For this reason, we will say that a ladder system L is countably metacompact in the case that it satisfies M<ω.

Γ0 May Be Minimal Subrecursively Inaccessible

Let T be the standard Veblen 1908 ordinal notation system for Γ0 as defined, for example, in Schutte's 1977 textbook [13] on Proof Theory. We define a slight modification of the standard assignment of fundamental sequences for the limit ordinals in T and prove that Γ0 is subrecursively inaccessible for this assignment, i.e. the induced slow and fast growing hierarchy match up at Γ0 for the first time.The results of this paper also indicate that φe00 may be considered as a new slow growing ordinal of PA in the sense that the induced slow growing hierarchy up to φe00 classifies the PA-provablyrecursive functions.We show how the results of this paper can be used to bui d a subrecursive hierarchyover the predicative ordinals in the spirit of Wainer's 1970 abstract [16 ].

An Elementary Approach to the Fine Structure of L

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α&lt;λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some , &lt; ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L.