Countable Ordinals Research Articles

On countable chains having decidable monadic theory

AbstractRationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.

An effective minimal encoding of uncountable sets

We propose a method for encoding sets of the countable ordinals by generic reals which preserves cardinality and enjoys the property of minimality over the encoded set.

Measurements and Gδ-Subsets of Domains

AbstractIn this paper we study domains, Scott domains, and the existence of measurements. We use a space created by D. K. Burke to show that there is a Scott domain P for which max(P) is a Gδ-subset of P and yet no measurement μ on P has ker(μ) = max(P). We also correct a mistake in the literature asserting that [0, ω1) is a space of this type. We show that if P is a Scott domain and X ⊆ max(P) is a Gδ-subset of P, then X has a Gδ-diagonal and is weakly developable. We show that if X ⊆ max(P) is a Gδ-subset of P, where P is a domain but perhaps not a Scott domain, then X is domain-representable, first-countable, and is the union of dense, completely metrizable subspaces. We also show that there is a domain P such that max(P) is the usual space of countable ordinals and is a Gδ-subset of P in the Scott topology. Finally we show that the kernel of a measurement on a Scott domain can consistently be a normal, separable, non-metrizable Moore space.

Roads to infinity: the mathematics of truth and proof

The Diagonal Argument Counting and Countability Does One Infinite Size Fit All? Cantor's Diagonal Argument Transcendental Numbers Other Uncountability Proofs Rates of Growth The Cardinality of the Continuum Historical Background Ordinals Counting Past Infinity The Countable Ordinals The Axiom of Choice The Continuum Hypothesis Induction Cantor Normal Form Goodstein's Theorem Hercules and the Hydra Historical Background Computability and Proof Formal Systems Post's Approach to Incompleteness Godel's First Incompleteness Theorem Godel's Second Incompleteness Theorem Formalization of Computability The Halting Problem The Entscheidungs problem Historical Background Logic Propositional Logic A Classical System A Cut-Free System for Propositional Logic Happy Endings Predicate Logic Completeness, Consistency, Happy Endings Historical Background Arithmetic How Might We Prove Consistency? Formal Arithmetic The Systems PA and PAomega Embedding PA in PAomega Cut Elimination in PAomega The Height of This Great Argument Roads to Infinity Historical Background Natural Unprovable Sentences A Generalized Goodstein Theorem Countable Ordinals via Natural Numbers From Generalized Goodstein to Well-Ordering Generalized and Ordinary Goodstein Provably Computable Functions Complete Disorder Is Impossible The Hardest Theorem in Graph Theory Historical Background Axioms of Infinity Set Theory without Infinity Inaccessible Cardinals The Axiom of Determinacy Largeness Axioms for Arithmetic Large Cardinals and Finite Mathematics Historical Background

Metastability in the Furstenberg–Zimmer tower

According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemeredi’s theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the ωω ω th level.

Selection over classes of ordinals expanded by monadic predicates

A monadic formula ψ ( Y ) is a selector for a monadic formula φ ( Y ) in a structure M if ψ defines in M a unique subset P of the domain and this P also satisfies φ in M . If C is a class of structures and φ is a selector for ψ in every M ∈ C , we say that φ is a selector for φ over C . For a monadic formula φ ( X , Y ) and ordinals α ≤ ω 1 and δ < ω ω , we decide whether there exists a monadic formula ψ ( X , Y ) such that for every P ⊆ α of order-type smaller than δ , ψ ( P , Y ) selects φ ( P , Y ) in ( α , < ) . If so, we construct such a ψ . We introduce a criterion for a class C of ordinals to have the property that every monadic formula φ has a selector over it. We deduce the existence of S ⊆ ω ω such that in the structure ( ω ω , < , S ) every formula has a selector. Given a monadic sentence π and a monadic formula φ ( Y ) , we decide whether φ has a selector over the class of countable ordinals satisfying π , and if so, construct one for it.

The Church Problem for Countable Ordinals

A fundamental theorem of Buchi and Landweber shows that the Church synthesis problem is computable. Buchi and Landweber reduced the Church Problem to problems about &amp;#969;-games and used the determinacy of such games as one of the main tools to show its computability. We consider a natural generalization of the Church problem to countable ordinals and investigate games of arbitrary countable length. We prove that determinacy and decidability parts of the Bu}chi and Landweber theorem hold for all countable ordinals and that its full extension holds for all ordinals &lt; \omega\^\omega.

Fruitful and helpful ordinal functions

In Simmons (Arch Math Logic 43:65–83, 2004), I described a method of producing ordinal notations ‘from below’ (for countable ordinals up to the Howard ordinal) and compared that method with the current popular ‘from above’ method which uses a collapsing function from uncountable ordinals. This ‘from below’ method employs a slight generalization of the normal function—the fruitful functions—and what seems to be a new class of functions—the helpful functions—which exist at all levels of the function space hierarchy over ordinals. Unfortunately, I was rather sparing in my description of these classes of functions. In this paper I am much more generous. I describe the properties of the helpful functions on all finite levels and, in the final section, indicate how they can be used to simplify the generation of ordinal notations. The main aim of this paper is to fill in the details missing from [7]. The secondary aim is to indicate what can be done with helpful functions. Fuller details of this development will appear elsewhere.

Selection in the monadic theory of a countable ordinal

AbstractA monadic formula ψ(Y) is a selector for a formula φ(Y) in a structure if there exists a unique subset P of which satisfies ψ and this P also satisfies φ. We show that for every ordinal α ≥ ωω there are formulas having no selector in the structure (α, &lt;). For α ≤ ω1, we decide which formulas have a selector in (α, &lt;) , and construct selectors for them. We deduce the impossibility of a full generalization of the Büchi-Landweber solvability theorem from (ω, &lt;) to (ωω, &lt;). We state a partial extension of that theorem to all countable ordinals. To each formula we assign a selection degree which measures “how difficult it is to select”. We show that in a countable ordinal all non-selectable formulas share the same degree.

The stationary set splitting game

AbstractThe stationary set splitting game is a game of perfect information of length ω1 between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ22 maximality with a predicate for the nonstationary ideal on ω1, and an example of a consistently undetermined game of length ω1 with payoff de.nable in the second‐order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

SEMI-REGULAR MASAS OF TRANSFINITE LENGTH

In 1965, Tauer produced a countably infinite family of semi-regular masas in the hyperfinite II 1 factor, no pair of which are conjugate by an automorphism. This was achieved by iterating the process of passing to the algebra generated by the normalizers and, for each n ∈ ℕ, finding masas for which this procedure terminates at the nth stage. Such masas are said to have length n. In this paper, we consider a transfinite version of this idea, giving rise to a notion of ordinal valued length. We show that all countable ordinals arise as lengths of semi-regular masas in the hyperfinite II 1 factor. Furthermore, building on work of Jones and Popa, we obtain all possible combinations of regular inclusions of irreducible subfactors in the normalizing tower.

Randomness via effective descriptive set theory

1.1 Informal introduction to Π1 relations Let 2N denote Cantor space. • A relation B ⊆ N × (2N)r is Π1 if it is obtained from an arithmetical relation by a universal quantification over sets. • If k = 1, r = 0 we have a Π1 set ⊆ N. • If k = 0, r = 1 we have a Π1 class ⊆ 2N. • The relation B is∆1 if both B and its complement are Π1. There is an equivalent representation of Π1 relations where the members are enumerated at stages that are countable ordinals. For Π1 sets (of natural numbers) these stages are in fact computable ordinals, i.e., the order types of computable well-orders.

More on partitioning triples of countable ordinals

Consider an arbitrary partition of the triples of all countable ordinals into two classes. We show that either for each finite ordinal m the first class of the partition contains all triples from a set of type ω + m, or for each finite ordinal n the second class of the partition contains all triples of an n-element set. That is, we prove that w 1 → (ω+ m, n) 3 for each pair of finite ordinals m and n.

Coding by club-sequences

Given any subset A of ω 1 there is a proper partial order which forces that the predicate x ∈ A and the predicate x ∈ ω 1 ∖ A can be expressed by ZFC -provably incompatible Σ 3 formulas over the structure 〈 H ω 2 , ∈ , N S ω 1 〉 . Also, if there is an inaccessible cardinal, then there is a proper partial order which forces the existence of a well-order of H ω 2 definable over 〈 H ω 2 , ∈ , N S ω 1 〉 by a provably antisymmetric Σ 3 formula with two free variables. The proofs of these results involve a technique for manipulating the guessing properties of club-sequences defined on stationary subsets of ω 1 at will in such a way that the Σ 3 theory of 〈 H ω 2 , ∈ , N S ω 1 〉 with countable ordinals as parameters is forced to code a prescribed subset of ω 1 . On the other hand, using theorems due to Woodin it can be shown that, in the presence of sufficiently strong large cardinals, the above results are close to optimal from the point of view of the Levy hierarchy.

Minimum model semantics for logic programs with negation-as-failure

We give a purely model-theoretic characterization of the semantics of logic programs with negation-as-failure allowed in clause bodies. In our semantics, the meaning of a program is, as in the classical case, the unique minimum model in a program-independent ordering. We use an expanded truth domain that has an uncountable linearly ordered set of truth values between False (the minimum element) and True (the maximum), with a Zero element in the middle. The truth values below Zero are ordered like the countable ordinals. The values above Zero have exactly the reverse order. Negation is interpreted as reflection about Zero followed by a step towards Zero ; the only truth value that remains unaffected by negation is Zero . We show that every program has a unique minimum model M P , and that this model can be constructed with a T P iteration which proceeds through the countable ordinals. Furthermore, we demonstrate that M P can alternatively be obtained through a construction that generalizes the well-known model intersection theorem for classical logic programming. Finally, we show that by collapsing the true and false values of the infinite-valued model M P to (the classical) True and False , we obtain a three-valued model identical to the well-founded one.

Perfect subsets of invariant CA-sets

The familiar theorem that any Σ 2 1 (a)-set X of real numbers (where a is a fixed real parameter) not containing a perfect kernel necessarily satisfies the condition X\( \subseteq \)L[a] is extended to a wider class of sets, with countable ordinals allowed as additional parameters in Σ 2 1 (a)-definitions.

A class of Banach spaces with few non-strictly singular operators

We construct a family ( X γ ) of reflexive Banach spaces with long (countable as well as uncountable) transfinite bases but with no unconditional basic sequences. The method we introduce to achieve this allows us to considerably control the structure of subspaces of the resulting spaces as well as to precisely describe the corresponding spaces on non-strictly singular operators. For example, for every pair of countable ordinals γ , β , we are able to decompose every bounded linear operator from X γ to X β as the sum of a diagonal operator and an strictly singular operator. We also show that every finite-dimensional subspace of any member X γ of our class can be moved by and ( 4 + ɛ )-isomorphism to essentially any region of any other member X δ or our class. Finally, we find subspaces X of X γ such that the operator space L ( X , X γ ) is quite rich but any bounded operator T from X into X is a strictly singular pertubation of a scalar multiple of the identity.

Between Maharam’s and von Neumann’s problems

In the context of definable algebras Maharam’s and von Neumann’s problems essentially coincide. Consequently, random forcing is the only definable ccc forcing adding a single real that does not make the ground model reals null, and the only pairs of definable ccc -ideals with the Fubini property are (meager,meager) and (null,null). In Scottish Book, von Neumann asked whether every ccc, weakly distributive complete Boolean algebra carries a strictly positive probability measure. Von Neumann’s problem naturally splits into two: (a) whether all such algebras carry a strictly positive continuous submeasure, and (b) whether every algebra that carries a strictly positive continuous submeasure carries a strictly positive measure. The latter problem is known under the names of Maharam’s Problem and Control Measure Problem (see [16], [9], [5, §393]). While von Neumann’s problem has a consistently negative answer ([16]), Maharam’s problem can be stated as a 1 statement and is therefore, by Shoenfield’s theorem, absolute between transitive models of set theory containing all countable ordinals. Theorem 0.1. Let I be a c.c.c. -ideal on Borel subsets of 2 ! that is analytic on G . The following are equivalent:

A comparison of two systems of ordinal notations

The standard method of generating countable ordinals from uncountable ordinals can be replaced by a use of fixed point extractors available in the term calculus of Howard’s system. This gives a notion of the intrinsic complexity of an ordinal analogous to the intrinsic complexity of a function described in Godel’s T.

The $\ell ^{1}$-indices of Tsirelson type spaces

If a and β are countable ordinals such that β ¬= 0, denote by T α,β the completion of c 00 with respect to the implicitly defined norm ||x|| = max{||x||S α , 1/2 sup Σ ||E i x||}, where the supremum is taken over all finite subsets E 1 ,...,E j of N such that E 1 a = ω α1 .m 1 + … +ω αn .m n in Cantor normal form and α n is not a limit ordinal, then there exists a Banach space whose l 1 -index is ω α .