Sequence Of Ordinals Research Articles

Functors and ordinal notations. I: A functorial construction of the veblen hierarchy

This paper is the first of several works: the matter is the systematic rewriting of usual results of ordinal notations, using Π½-logic. The purpose of these articles is already realized in Π½-logic, in a relatively abstract form; also, they are examples for Π½-logic. The first and second articles in this series are complete versions of unpublished results that one of authors had announced in 1976.Veblen introduced his hierarchy in 1908. The hierarchy has since been widely used in proof theory; in fact, what is actually needed is the more natural hierarchy: , with (not necessarily a Cantor normal form) and K the Veblen hierarchy.We shall show:i) φ is in fact a two-variables functor (denoted by V in the text) commuting with direct limits and with pull-backs.ii) The property of commutation with pull-backs permits one to give a representation of each ordinal x < φα(0) by means of a finite sequence of ordinals < α. This representation is independent of α it is ⊆ x, and can be used in the building up of notations systems. This representation is the “blowing-up” of the Cantor normal form, and it corresponds exactly to the denotation of ordinals < φα(0) by means of the functions (φβ)β<α.

Adding closed cofinal sequences to large cardinals

If κ is measurable, Prikry's forcing adds a sequence of ordinals of order type ω cofinal in κ. This destroys the regularity of κ but κ does remain uncountable. Magidor has a forcing notion generalizing Prikry's which adds a closed cofinal sequence of ordinals through a large cardinal. The cardinal remains uncountable but uts regularity is still destroyed. We obtain a forcing notion which adds a closed cofinal sequence of ordinals (and more complex objects) through a large cardinal κ, of order type κ, and keeps κ regular. In fact κ remains measurable after the forcing. Our forcing shares certain properties with Prikry's forcing. Closed cofinal sebsequences of generic sequences are generic (under appropriate interpretations). Archetypical generic sequences can be generated by taking the critical points of iterated elementary embeddings.

$C(\alpha )$ preserving operators on $C(K)$ spaces

Let A: C(K) -» X be a bounded linear operator where if is a compact Hausdorff space and X is a separable Banach space. Sufficient conditions are given for A to be an isomorphism (into) when restricted to a subspace Y of C(K), such that Y is isometrically isomorphic to a space C(a) of continuous functions on the space of ordinal numbers less than or equal to the countable ordinal a. 1. For a compact Hausdorff space K, C(K) will denote the Banach space of continuous real-valued functions on K with the supremum norm. The Greek letters a, s and y will be reserved for countable ordinal numbers. We consider the set [0, a] = {y:0 X > 0. For each countable ordinal a we will define a family Pa(X, B) of open subsets G and associated measures ju,G G B. Let P0(X, B) = {(G, jttG): G is an open set in K, ixG G B and | /?G | (G) > X}. If a s + 1, then Pa(X, B) = {(G, nc): (G, jiiG) G P0(X, B) and there is a disjoint sequence {G„} of open subsets of G and associated measures /xG such that (G„, ftG ) G Ps(X, B), ju,Gn -> juG and UnG C G). If a is a limit ordinal,then ^(X, B) = {(G, /xG) G P0(X, B): there is a disjoint sequence of open subsets {Gn} of G with associated measures /xG such that fic — nc, UnG„ C G and (G„, jUG ) G .Pa(X, B) where an/a). The notation a„ /a means that a„ is an increasing sequence of ordinals with a = sup„ an. A simple example illustrating the sets Pa(X, B) is contained in §4. Received by the editors March 20, 1979 and, in revised form, July 30, 1981. 1980 Mathematics Subject Classification. Primary 46E15. 'Research on this paper was partially supported by the National Science Foundation grant number MCS 77-01690. ©1982 American Mathematical Society O0O2-9939/82/OO0O-O726/SO5.OO

Commutativity in Series of Ordinals: A Study of Invariants

It is well known that two ordinals are additively commutative if and only if they are finite multiples of some given ordinal, and it is very easy to extend this result to any finite sequence of ordinals. However, no necessary and sufficient conditions for the commutativity of a series of ordinals seem to be known when the length of that series is infinite, although sufficient conditions for certain cases have been given by Sierpiński and Ginsburg. In this paper we present such necessary and sufficient conditions. The general problem is split into five distinct cases: those in which the length of the series is a regular initial ordinal, a singular initial ordinal, an infinite, noninitial prime component, an infinite successor ordinal, and an infinite limit ordinal that is not a prime component. These are dealt with respectively in the second through to the sixth sections of the paper, and it turns out that in every case our criteria can be expressed in terms of an ordinal parameter, which is in fact an invariant of the series in question. This concept of invariance is introduced in the first section, which also contains several lemmas and a slight strengthening of the original Sierpiński-Ginsburg result. The final section of this paper differs from the preceding four sections in two aspects. Firstly, the proofs of its two main results are merely sketched, since they contain no arguments that have not previously appeared in some form or other. Secondly, we have not given any explicit determination of the ordinal parameter introduced in this section, since we felt that such a determination would prolong the paper intolerably and encroach upon work done by J. A. H. Anderson: we have therefore simply referred to Anderson’s interesting paper.

Commutativity in series of ordinals: a study of invariants

It is well known that two ordinals are additively commutative if and only if they are finite multiples of some given ordinal, and it is very easy to extend this result to any finite sequence of ordinals. However, no necessary and sufficient conditions for the commutativity of a series of ordinals seem to be known when the length of that series is infinite, although sufficient conditions for certain cases have been given by Sierpiński and Ginsburg. In this paper we present such necessary and sufficient conditions. The general problem is split into five distinct cases: those in which the length of the series is a regular initial ordinal, a singular initial ordinal, an infinite, noninitial prime component, an infinite successor ordinal, and an infinite limit ordinal that is not a prime component. These are dealt with respectively in the second through to the sixth sections of the paper, and it turns out that in every case our criteria can be expressed in terms of an ordinal parameter, which is in fact an invariant of the series in question. This concept of invariance is introduced in the first section, which also contains several lemmas and a slight strengthening of the original Sierpiński-Ginsburg result. The final section of this paper differs from the preceding four sections in two aspects. Firstly, the proofs of its two main results are merely sketched, since they contain no arguments that have not previously appeared in some form or other. Secondly, we have not given any explicit determination of the ordinal parameter introduced in this section, since we felt that such a determination would prolong the paper intolerably and encroach upon work done by J. A. H. Anderson: we have therefore simply referred to Anderson’s interesting paper.

On sequences generic in the sense of Prikry

I establish here a criterion for a sequence of ordinals to be generic over a transitive model of ZFC with respect to a notation of forcing first considered by Prikry in his Doctoral dissertation [2]. In Section 0 I review some notation, in Section 1 I list some facts about measurable cardinals, and in Section 2, after giving Prikry's result, I state and prove mine.

A souslin operation for Π 2 1

Throughout this paper we assume the existence of a measureable cardinal. Membership in a Π21 set of reals is shown to be equivalent to the existence of an infinite path in a tree ℐ* of pairs of finite sequences of natural numbers and ordinals. This is used to prove that every Π21 relation can be uniformized by a Π21 function. One gets that under certain assumptions the Shoenfield Absoluteness Theorem holds for Σ31 statements, that Π31 sets include perfect subsets and that Σ31 sets are Lebesgue measureable and have the Baire Property.

A new proof of the paracompactness of metric spaces

The purpose of this paper is to give a new proof of A. H. Stone's Theorem that every metric space is paracompact.' We feel that the combinatorics involved are of some interest in their own right and that our method may be useful in other contexts. The statement that a space X is paracompact means that every open cover of X has a locally finite open refinement that covers X. (An open refinement of a collection of open sets is a new collection of open sets such that each set in the new collection is contained in some set of the original collection. A cover is said to be locally finite if each point has a neighborhood intersecting only a finite number of elements in the cover.) PROOF. (A) We shall first get a refinement such that each point is covered only a finite number of times. To do this we will order the elements of the given cover. Call the cover { C,} . Let R(x, m) be the open circle of radius 1/2m about x. A chosen circle (with respect to C,) is a circle R(x, n.+1) such that (i) R(x, n) CC,C,, (ii) n. is the smallest integer for which (i) holds and (iii) R(x, n,)CCC for some f3 m then y<?. Hence the C, that contain x form a descending sequence of ordinals and therefore only a finite number are distinct.

On the restricted ordinal theorem

The proposition that a decreasing sequence of ordinals necessarily terminates has been given a new, and perhaps unexpected, importance by the rôle which it plays in Gentzen's proof of the freedom from contradiction of the “reine Zahlentheorie.” Gödel's construction of non-demonstrable propositions and the establishment of the impossibility of a proof of freedom from contradiction, within the framework of a certain type of formal system, showed that a proof of freedom from contradiction could be found only by transcending the axioms and proof processes of that formal system. Gentzen's proof succeeds by utilisingtransfinite inductionto prove that certain sequences of reduction processes, enumerated by ordinals less than ε (the first ordinal to satisfy ε = ὡ) are finite. Were it possible to provethe restricted ordinal theorem, that a descending sequence of ordinals, less thanε, is finite, in Gentzen's “reine Zahlentheorie,” then it would be possible to determine a contradiction in that number system. In his paper, Gentzen proves the theorem of transfinite induction, which he requires, by an intuitive argument. There is also a method of reducing transfinite induction, for ordinals less than ε, to a number-theoretic principle given by Hilbert and Bernays, and a similar method by Ackermann. None of these proofs of transfinite induction isfinitist.

On notation for ordinal numbers

Consider a system of formal notations for ordinal numbers in the first and second number classes, with the following properties. Given a notation for an ordinal, it can be decided effectively whether the ordinal is zero, or the successor of an ordinal, or the limit of an increasing sequence of ordinals. In the second case, a notation for the preceding ordinal can be determined effectively. In the third case, notations for the ordinals of an increasing sequence of type ω with the given ordinal as limit can be determined effectively.Are there systems of this sort which extend farthest into the second number class? When the conditions for the systems have been made precise, the question will be answered in the affirmative. There is an ordinal ω1in the second number class such that there are systems of notations of the sort described which extend to all ordinals less than ω1, but none in which ω1itself is assigned a notation.1. An effective or constructive operation on the objects of an enumerable class is one for which a fixed set of instructions can be chosen such that, for each of the infinitely many objects (orn-tuples of objects), the operation can be completed by a finite process in accordance with the instructions. This notion is made exact by specifying the nature of the process and set of instructions. It appears possible to do so without loss of generality.