Sequence Of Ordinals Research Articles

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 > ω 1 ω 1 {}^{>\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 > ω 1 O N {}^{>{\omega _{1}}}\mathrm {ON} (the class of countable length sequences of ordinals).

Homomorphism Order of Connected Monounary Algebras

A homomorphism quasi-order between algebraic structures $\mathcal {A}$ and ${\mathcal{B}}$ of the same type is defined as follows: ${\mathscr{A}}\leq {\mathscr{B}}$ if there is a homomorphism of ${\mathscr{A}}$ to ${\mathscr{B}}$ . The paper deals with the class ${\mathcal{L}}$ of all connected monounary algebras. Let $\sim $ be the equivalence corresponding to the relation ≤ and let $\mathbb {L}={\mathcal{L}}\sim $ . We prove that $\mathbb {L}$ is a lattice. Next it is shown that the lattice $\mathbb {L}$ is distributive. For the purpose of the proof, to each connected monounary algebra without a cycle, an increasing sequence of ordinals is assigned. Finally, we deal with the special case of increasing sequences, namely, such that the sequence contains only positive integers.

Some transfinite natural sums

AbstractWe study a transfinite iteration of the ordinal Hessenberg natural sum obtained by taking suprema at limit stages. We show that such an iterated natural sum differs from the more usual transfinite ordinal sum only for a finite number of iteration steps. The iterated natural sum of a sequence of ordinals can be obtained as a mixed sum (in an order‐theoretical sense) of the ordinals in the sequence; in fact, it is the largest mixed sum which satisfies a finiteness condition. We introduce other infinite natural sums which are invariant under permutations and show that all the sums under consideration coincide in the countable case.

An Infinite Natural Product

Abstract We study a countably infinite iteration of the natural product between ordinals.We present an “effective” way to compute this countable natural product; in the non trivial cases the result depends only on the natural sum of the degrees of the factors, where the degree of a nonzero ordinal is the largest exponent in its Cantor normal form representation. Thus we are able to lift former results about infinitary sums to infinitary products. Finally, we provide an order-theoretical characterization of the infinite natural product; this characterization merges in a nontrivial way a theorem by Carruth describing the natural product of two ordinals and a known description of the ordinal product of a possibly infinite sequence of ordinals.

Well-orders in the transfinite Japaridze algebra

This paper studies the transfinite propositional provability logics $\glp_\Lambda$ and their corresponding algebras. These logics have for each ordinal $\xi< \Lambda$ a modality $\la \alpha \ra$. We will focus on the closed fragment of $\glp_\Lambda$ (i.e., where no propositional variables occur) and \emph{worms} therein. Worms are iterated consistency expressions of the form $\la \xi_n\ra \ldots \la \xi_1 \ra \top$. Beklemishev has defined well-orderings $<_\xi$ on worms whose modalities are all at least $\xi$ and presented a calculus to compute the respective order-types. In the current paper we present a generalization of the original $<_\xi$ orderings and provide a calculus for the corresponding generalized order-types $o_\xi$. Our calculus is based on so-called {\em hyperations} which are transfinite iterations of normal functions. Finally, we give two different characterizations of those sequences of ordinals which are of the form $\la {\formerOmega}_\xi (A) \ra_{\xi \in \ord}$ for some worm $A$. One of these characterizations is in terms of a second kind of transfinite iteration called {\em cohyperation.}

Distributive proper forcing axiom and cardinal invariants

In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ? sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.

Iteratively Changing the Heights of Automorphism Towers

We extend the results of Hamkins and Thomas concerning the malleability of automorphism tower heights of groups by forcing. We show that any reasonable sequence of ordinals can be realized as the automorphism tower heights of a certain group in consecutive forcing extensions or ground models, as desired. For example, it is possible to increase the height of the automorphism tower by passing to a forcing extension, then increase it further by passing to a ground model, and then decrease it by passing to a further forcing extension, and so on, transfinitely. We make sense of the limit models occurring in such a sequence of models. At limit stages, the automorphism tower height will always be 1.

A Topological Study of the Closed Fragment of GLP

In this article, we study the canonical model for the closed fragment of GLP and establish its precise relationship with a universal model constructed by Ignatiev. In particular, we effectively characterize the canonical model in terms of a coordinate system based on sequences of ordinals up to e0.We then define a simple topological model of this logic by defining a natural polytopology on the ordinal e0 itself.

Phase transitions for Gödel incompleteness

Gödel’s first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers have been looking for natural examples of such assertions and breakthroughs were obtained in the seventies by Jeff Paris [Some independence results for Peano arithmetic. J. Symbolic Logic 43 (1978) 725–731] (in part jointly with Leo Harrington [J. Paris, L. Harrington, A mathematical incompleteness in Peano arithmetic, in: J. Barwise (Ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977] and Laurie Kirby [L. Kirby, Jeff Paris, Accessible independence results for Peano Arithmetic, Bull. of the LMS 14 (1982) 285–293]) and Harvey Friedman [S.G. Simpson, Non-provability of certain combinatorial properties of finite trees, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 87–117; R. Smith, The consistency strength of some finite forms of the Higman and Kruskal theorems, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 119–136] who produced the first mathematically interesting independence results in Ramsey theory (Paris) and well-order and well-quasi-order theory (Friedman). In this article we investigate Friedman-style principles of combinatorial well-foundedness for the ordinals below ε 0 . These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers. For these independence principles we classify (as a part of a general research program) their phase transitions, i.e. we classify exactly the bounding conditions which lead from provability to unprovability in the induced combinatorial well-foundedness principles. As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel’s coding of finite sequences from his classical 1931 paper on his incompleteness results. This choice makes the investigation highly non-trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions. For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem of Parameswaran which apparently is far remote from Gödel’s theorem.

A cardinal defined by a polarized partition relation

We study the first cardinalκ satisfying a partition relation defined on the set of finite sequences of smaller ordinals. We show that the fact that this cardinal is ℵω is equiconsistent with the existence of a measurable cardinal. Under GCH, this cardinal must be inaccessible if it has uncountable cofinality. It is shown that the GCH assumption is necessary here.

The universe constructed from a sequence of ordinals

We prove that if $V = L [s]$ where $s$ is an $\omega$ -sequence of ordinals then the GCH holds.

Ackermann functions and transfinite ordinals

A set of binary operators are defined and shown to be equivalent to Ackermann functions. The same set of operators are used to develop a notation for writing the sequence of transfinite ordinals.

Coherent sequences versus Radin sequences

We attempt to make a connection between the sequences of measures used to define Radin forcing and the coherent sequences of extenders which are the basis of modern inner model theory. We show that in certain circumstances we can read off sequences of measures as defined by Radin from coherent sequences of extenders, and that we can define Radin forcing directly from a coherent extender sequence and a sequence of ordinals; this generalises Mitchell's construction of Radin forcing from a coherent sequence of measures.

The theorem of the means for cardinal and ordinal numbers

AbstractThe theorem that the arithmetic mean is greater than or equal to the geometric mean is investigated for cardinal and ordinal numbers. It is shown that whereas the theorem of the means can be proved for n pairwise comparable cardinal numbers without the axiom of choice, the inequality a2 + b2 ≥ 2ab is equivalent to the axiom of choice. For ordinal numbers, the inequality α2 + β2 ≥ 2αβ is established and the conditions for equality are derived; stronger inequalities are obtained for finite and infinite sequences of ordinals under suitable monotonicity hypotheses. MSC: 03E10, 04A10, 03E25, 04A25.

Some results on higher Suslin trees

Higher Suslin trees have become a tool in some forcing constructions in set theory (see, for example, [D1] and [D2]). Most of the constructions using ω1 Suslin trees can be extended to κ+ Suslin trees for any regular cardinal κ. Some of these are given in §1.In many such constructions, sequences of Suslin trees are used. In §II we show, in various ways, that the generalization to sequences, even ω-sequences, of κ+ Suslin trees cannot be done.In these constructions the Suslin trees are used as forcing poset (the forcing adds a branch in the tree). There is another way to kill a Suslin tree, namely by adding a big antichain. Some results on this forcing are given in §III.Our notation is standard. If T is a tree and x ∈ T, then ∣x∣ is the height of x in T. We define Tα (or T(α)) = {x ∈ T: ∣x∣ = α} and T∣α = {x ∈ T: ∣x∣ &lt; α}.If p and q are forcing conditions, p ≤ q means that p has more information than q.If (Tα: α ∈ I) is a sequence of trees, Π Tα will always mean the set of (xα: α ∈ I) such that xα ∈ Tα and ∣xα∣ = ∣xβ∣ for α, β ∈ I.For functions b, T,… we denote by b ∣α, T∣ α,… their restriction to α.If x is a sequence of ordinals and α is an ordinal, x∧α is the sequence obtained by concatenating α at the end of x.

Generalizations of the one-dimensional version of the Kruskal-Friedman theorems

The paper [Schütte + Simpson] deals with the following one-dimensional case of Friedman's extension (see in [Simpson 1]) of Kruskal's theorem ([Kruskal]). Given a natural number n, let Sn+1 be the set of all finite sequences of natural numbers &lt;n + 1. If s1 = (a0,…,ak) ∈Sn+1 and s2 = (b0,…,bm) ∈Sn + 1, then a strictly monotone function f: {0,…, k} → {0,…, m} is called an embedding of s1 into s2 if the following two assertions are satisfied:1) ai, = bf(i), for all i &lt; k;2) if f(i) &lt; j &lt; f(i + 1) then bj &gt; bf(i+1), for all i &lt; k, j &lt; m.Then for every infinite sequence s1, s2,…,sk,… of elements of Sn + 1 there exist indices i &lt; j and an embedding of si into Sj. That is, Sn+1 forms a well-quasi-ordering (wqo) with respect to embeddability. For each n, this statement W(Sn+1) is provable in the standard second order conservative extension of Peano arithmetic. On the other hand, the proof-theoretic strength of the statements W(Sn+1) grows so fast that this formal theory cannot prove the limit statement ∀nW(Sn+1). The appropriate first order -versions of these combinatory statements preserve their proof-theoretic strength, so that actually one can speak in terms of provability in Peano arithmetic. These are the main conclusions from [Schütte + Simpson].We wish to extend this into the transfinite. That is, we take an arbitrary countable ordinal τ &gt; 0 instead of n + 1 and try to obtain an analogous “strong” combinatory statement about finite sequences of ordinals &lt; τ.

A hierarchy of filters on regular uncountable cardinals

AbstractWe introduce a well-founded relation &lt; between filters on the space of descending sequences of ordinals. For each regular uncountable cardinal κ, the length of the relation is an ordinal o(κ) ≤ (2κ)+.

Combinatorial problems on trees: Partitions, δ-systems and large free subtrees

We prove partition theorems on trees and generalize to a setting of trees the theorems of Erdös and Rado on δ-systems and the theorems of Fodor and Hajnal on free sets. Let μ be an infinite cardinal and T μ be the tree of finite sequences of ordinals <μ, with the partial ordering of being an initial segment. α⩽ β denotes that α is an initial segment of β. A subtree of T μ is a nonempty subset of T μ closed under initial segments. T⩽ T μ means that T is a subtree of T μ and 〈T, ⩽〉 ≊ T μ . The following are extracts from Section 2, 3 and 4. Theorem 1 (Shelah). A partition theorem. Suppose cf(λ) ≠ cf(μ), F : T μ → λ, and for every branch b of T μ Sup({F( α) ∥ α ϵ b}) < λ, then there is T ⩽ T μ such that Sup({F( α) ∥ α ϵ T}) < λ. Theorem 2 (Rubin). A theorem on large free subtrees. Let λ + ⩽ μ, F : T μ → P( T μ ), for every branch b of T μ:∥ϒ {F( α) ∥ α ϵ b} ∥ < λ, and for every α ϵ T μ and β ϵ F( α), β | ̌ α; then there is T ⩽ T μ such that for every α, β ϵ T : β ϵ F( α) . Let P λ ( C) denote the ideal in P( C) of all subsets of C whose power is less than λ. Let cov(μ, λ) mean that μ is regular, λ < μ, and for every κ < μ there is D ∪ P λ ( κ) such that ∥ D∥ < μ, and D generates the ideal P λ ( κ) of P(κ). Note that if for every κ < μ κ < λ < cf( μ) = μ, then cov(μ, λ) holds. Let α ∧ β denote the maximal common initial segment of α and β. Theorem 3 (Shelah). A theorem on δ-systems. Suppose Cov(μ, λ) holds, F : T μ → P( C) and for every branch b of T μ: ∥⊂ {F( α) ∥ α ϵ β} ∥ < λ, then there is T ⩽ T μ and a function K : T → P λ(C) such that for every incomparable α, β ϵ T : F( α) ⊃ F( β) ⊆ K( α ∧ β) . In 4.12, 4.13, we almost get that K( α) ⊂K( β) = K( α ⊂ β) .

On the Kleene degrees of Π11 sets

AbstractLet A and B be subsets of the reals. Say that AK≥ B, if there is a real a such that the relation “x ∈ B” is uniformly ⊿1 (a, A) in . This reducibility induces an equivalence relation ≡K on the sets of reals; the ≡K-equivalence class of a set is called its Kleene degree. Let be the structure that consists of the Kleene degrees and the induced partial order ≥. A substructure of that is of interest is , the Kleene degrees of the sets of reals. If sharps exist, then there is not much to , as Steel [9] has shown that the existence of sharps implies that has only two elements: the degree of the empty set and the degree of the complete set. Legrand [4] used the hypothesis that there is a real whose sharp does not exist to show that there are incomparable elements in ; in the context of V = L, Hrbáček has shown that is dense and has no minimal pairs. The Hrbáček results led Simpson [6] to make the following conjecture: if V = L, then forms a universal homogeneous upper semilattice with 0 and 1. Simpson's conjecture is shown to be false by showing that if V = L, then Gödel's maximal thin set is the infimum of two strictly larger elements of .The second main result deals with the notion of jump in . Let A′ be the complete Kleene enumerable set relative to A. Say that A is low-n if A(n) has the same degree as ⊘(n), and A is high-n if A(n) has the same degree as ⊘(n+1). Simpson and Weitkamp [7] have shown that there is a high (high-1) incomplete set in L. They have also shown that various other sets are neither high nor low in L. Legrand [5] extended their results by showing that, if there is a real x such that x# does not exist, then there is an element of that, for all n, is neither low-n nor high-n. In §2, ZFC is used to show that, for all n, if A is and low-n then A is Borel. The proof uses a strengthened version of Jensen's theorem on sequences of admissible ordinals that appears in [7, Simpson-Weitkamp].

On the Kleene degrees of Π11 sets

AbstractLet A and B be subsets of the reals. Say that AK≥ B, if there is a real a such that the relation “x ∈ B” is uniformly ⊿1 (a, A) in . This reducibility induces an equivalence relation ≡K on the sets of reals; the ≡K-equivalence class of a set is called its Kleene degree. Let be the structure that consists of the Kleene degrees and the induced partial order ≥. A substructure of that is of interest is , the Kleene degrees of the sets of reals. If sharps exist, then there is not much to , as Steel [9] has shown that the existence of sharps implies that has only two elements: the degree of the empty set and the degree of the complete set. Legrand [4] used the hypothesis that there is a real whose sharp does not exist to show that there are incomparable elements in ; in the context of V = L, Hrbáček has shown that is dense and has no minimal pairs. The Hrbáček results led Simpson [6] to make the following conjecture: if V = L, then forms a universal homogeneous upper semilattice with 0 and 1. Simpson's conjecture is shown to be false by showing that if V = L, then Gödel's maximal thin set is the infimum of two strictly larger elements of .The second main result deals with the notion of jump in . Let A′ be the complete Kleene enumerable set relative to A. Say that A is low-n if A(n) has the same degree as ⊘(n), and A is high-n if A(n) has the same degree as ⊘(n+1). Simpson and Weitkamp [7] have shown that there is a high (high-1) incomplete set in L. They have also shown that various other sets are neither high nor low in L. Legrand [5] extended their results by showing that, if there is a real x such that x# does not exist, then there is an element of that, for all n, is neither low-n nor high-n. In §2, ZFC is used to show that, for all n, if A is and low-n then A is Borel. The proof uses a strengthened version of Jensen's theorem on sequences of admissible ordinals that appears in [7, Simpson-Weitkamp].