Countable partition ordinals

The structure of ordinals of the form ω^ωβ for countable β is studied. The main result is:

Theorem 1. Ifβ<ω₁is the sum of one or two indecomposable ordinals, then

ω^ωβ→(ω^ωβ,3)².     

Cleavability over ordinals

In this paper we show that if X is an infinite compactum cleavable over an ordinal, then X must be homeomorphic to an ordinal. X must also therefore be a LOTS. This answers two fundamental questions in the area of cleavability. We also leave it as an open question whether cleavability of an infinite compactum X over an ordinal λ implies X is embeddable into λ.     

Register computations on ordinals

We generalize ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. Ordinal register machines are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets of ordinals in a model of the Zermelo-Fraenkel axioms ZFC. This allows the following characterization of computable sets: a set of ordinals is ordinal register computable if and only if it is an element of Gödel’s constructible universe L.     

Computable shuffle sums of ordinals

The main result is that for sets , the following are equivalent:

On perfect images of ordinals

In the first section of this paper, the ordinal space [1, ω] is characterized. Then its quotients are completely described in easy terms. In the second section, the ideas of a well-ordered sum of an arbitrary collection of T₂spaces and a Chandelier space are introduced. The Chandelier spaces are obtained by taking compact ordinals and forming their interweaves and well ordered sums and one-point compactification of such spaces. Finally it is proved that the Hausdorff quotients of compact ordinals are precisely the Chandelier spaces. A problem of M. Hu ek on quotients of [1, ω] is thus solved.     

Partition relations for denumerable ordinals

The principal result of this paper provides a nearly complete answer to the following question. For which integers k, m, n is it true that whenever we have a graph G whose set of vertices is an ordered set of order type ωⁿ, then either this graph contains a complete subgraph with k vertices or the complement of $\text{G,}\text{G}\text{?}$ contains a complete subgraph whose set of vertices has order type ω^m.     

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.     

Superautonomous numerations and projected ordinals

Translated from Algebra i Logika, Vol. 30, No. 1, pp. 28–47, January–February, 1991.     

A partition theorem for ordinals

In this note we show that α →^trans(α, m)² for all m < ω and every α of the form ω^β, where α →^trans(α, m)² is a weakening of the usual partition property obtained by considering only partitions whose first member is a transitive relation.     

Partition relations for uncountable ordinals

Partition relations of the form α→(α,m)², where α is an ordinal andm is a positive integer, are considered. Let κ be a cardinal. The following are proved: If κ is singular and 2^K=K ⁺ then (K⁺)²?((K⁺)²,3)². If κ is a strong limit cardinal, then², iff ((cfκ)²→((cfκ)²,m)². If κ is regular and K²→(K²,3)², then the κ-Souslin hypothesis holds. If K^ω<α+ and cfα=cfκ>ω, then α?(α,3)².     

Reverse mathematics of prime factorization of ordinals

One of the earliest applications of Cantor's Normal Form Theorem is Jacobstahl's proof of the existence of prime factorizations of ordinals. Applying the techniques of reverse mathematics, we show that the full strength of the Normal Form Theorem is used in this proof. Received: 13 August 1997 / Revised version: 3 November 1997     

Order-two continuous Hausdorff images of compact ordinals

Theorem. LetK be a Hausdorff space. The following conditions are equivalent: (a)K is homeomorphic to a compact scattered ordered space; (b)K is an order-two image of a compact ordinal.     

Injections into function spaces over ordinals

We investigate when and how function spaces over subspaces of ordinals admit continuous injections into each other. To formulate our results let τ be an uncountable regular cardinal. We prove, in particular, that: (1) If A and B are disjoint stationary subsets of τ then Cp(A) does not admit a continuous injection into Cp(B); (2) For A⊂ω₁, admits a continuous injection into iff A is countable or ω₁ embeds into A (which, in its turn, is equivalent to the statement “ embeds into ”).     

Dual properties of subspaces in products of ordinals

Let μ and ν be two ordinals. If X is a subspace of μ×ν, then X is dually scattered of rank?2. If X is a subspace of countable extent of μ×ν, then X is dually discrete.