Definable Well-order Research Articles

Automorphisms and strongly invariant relations

We investigate characterizations of the Galois connection {{,textrm{Aut},}}-{{,textrm{sInv},}} between sets of finitary relations on a base set A and their automorphisms. In particular, for A=omega _1, we construct a countable set R of relations that is closed under all invariant operations on relations and under arbitrary intersections, but is not closed under {textrm{sInv Aut}}. Our structure (A, R) has an omega -categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.

-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Abstract Given an uncountable cardinal $\kappa $ , we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$ -formulas that only use the cardinal $\kappa $ and sets of hereditary cardinality less than $\kappa $ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical nondefinability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of $\kappa $ of length at least $\kappa ^+$ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of $\Sigma _1$ -definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal $\omega _1$ .

A Model in Which Well-Orderings of the Reals Appear at a Given Projective Level

The problem of the existence of analytically definable well-orderings at a given level of the projective hierarchy is considered. This problem is important as a part of the general problem of the study of the projective hierarchy in the ongoing development of descriptive set theory. We make use of a finite support product of the Jensen-type forcing notions to define a model of set theory ZFC in which, for a given n>2, there exists a good Δn1 well-ordering of the reals but there are no such well-orderings in the class Δn−11. Therefore the existence of a well-ordering of the reals at a certain level n>2 of the projective hierarchy does not imply the existence of such a well-ordering at the previous level n−1. This is a new result in such a generality (with n>2 arbitrary), and it may lead to further progress in studies of the projective hierarchy.

CLOSED AND UNBOUNDED CLASSES AND THE HÄRTIG QUANTIFIER MODEL

AbstractWe show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P, Q, {\langle L[P],\in ,P \rangle }$ and ${\langle L[Q],\in ,Q \rangle }$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. Examples of such P are Card, the class of uncountable cardinals, I the uniform indiscernibles, or for any n the class $C^{n}{=_{{\operatorname {df}}}}\{ \lambda \, | \, V_{\lambda } \prec _{{\Sigma }_{n}}V\}$ ; moreover the theory of such models is invariant under ZFC-preserving extensions. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. The inner model constructed using definability in the language augmented by the Härtig quantifier is thus also characterized.

The full basis theorem does not imply analytic wellordering

A finite support product of ω1 clones of Jensen's minimal Π21 singleton forcing is used to define a model in which any non-empty analytically definable set of reals contains an analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the reals.

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

AbstractWe study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

Simplest possible locally definable well-orders

Simplest possible locally definable well-orders

Forcing lightface definable well-orders without the GCH

For any given uncountable cardinal κ with κ<κ=κ, we present a forcing that is <κ-directed closed, has the κ+-cc and introduces a lightface definable well-order of H(κ+). We use this to define a global iteration that adds such a well-order for all such κ simultaneously and is capable of preserving the existence of many large cardinals in the universe.

Large cardinals and definable well-orders, without the GCH

We show that there is a class-sized partial order P with the property that forcing with P preserves ZFC, supercompact cardinals, inaccessible cardinals and the value of 2κ for every inaccessible cardinal κ and, if κ is an inaccessible cardinal and A is an arbitrary subset of κκ, then there is a P-generic extension of the ground model V in which A is definable in 〈H(κ+)V[G],∈〉 by a Σ1-formula with parameters.We use this result to construct a class-sized partial order with the above preservation properties that forces the existence of well-orders of H(κ+) definable in the structure 〈H(κ+),∈〉 for every inaccessible cardinal κ. Assuming the GCH, David Asperó and Sy-David Friedman showed in [1] and [2] that there is a class-sized partial order preserving ZFC and various large cardinals and forcing the existence of a well-order of the universe whose restriction to H(κ+) is definable in 〈H(κ+)V[G],∈〉 by a parameter-free formula for every uncountable regular cardinal κ. Our second result can be interpreted as a boldface version of this result in the absence of the GCH.

Co-analytic mad families and definable wellorders

We show that the existence of a $${\Pi^1_1}$$ -definable mad family is consistent with the existence of a $${\Delta^{1}_{3}}$$ -definable well-order of the reals and $${\mathfrak{b}=\mathfrak{c}=\aleph_3}$$ .

A definable failure of the singular cardinal hypothesis

We show first that it is consistent that κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(κ+). Then with further forcing we show that it is consistent that GCH fails at ℵω, ℵω strong limit, while there is a lightface definable wellorder of H(ℵω+1) (“definable failure” of the singular cardinal hypothesis at ℵω). The large cardinal hypothesis used is the existence of a κ++-strong cardinal, where κ is κ++-strong if there is an embedding j: V → M with critical point κ such that H(κ++) ⊆ M. By work of M. Gitik and W. J. Mitchell [12], [20], our large cardinal assumption is almost optimal. The techniques of proof include the “tuning-fork” method of [10] and [3], a generalisation to large cardinals of the stationary-coding of [4] and a new “definable-collapse” coding based on mutual stationarity. The fine structure of the canonical inner model L[E] for a κ++-strong cardinal is used throughout.

Indestructibility of Vopěnka’s Principle

Vopěnka’s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka’s Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) set theory, such as the Generalised Continuum Hypothesis, and the existence of a definable well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, specifically, reverse Easton iterations of increasingly directed closed partial orders.

Cardinal characteristics and projective wellorders

Using countable support iterations of S -proper posets, we show that the existence of a Δ 3 1 definable wellorder of the reals is consistent with each of the following: d < c , b < a = s and b < g .

Large cardinals and definable well-orders on the universe

AbstractWe use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle , at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.

An argument concerning the unknowable

I shall formulate and evaluate a new argument concerning the domain of the unknowable. It is an argument about (un)knowability. More specifically, it is an argument about what we can(not) know about the natural numbers. Since the domain of discourse will be the natural numbers structure, the notion of knowability can for the purposes of the argument be identified with a priori knowability or which amounts to the same thing absolute provability (as opposed to provability in an antecedently given formal system). Suppose, for a reductio, that there exists a property 0 of natural numbers such that it is provable that for some natural number n, 0(n) is true but unprovable. Then, by the least number principle, there must be a smallest such natural number n. Then there provably exists exactly one smallest number n such that 0(n) is true but unprovable. Then it is provable that the smallest n such that 0(n) is true but unprovable is true but unprovable. But then 0(n) is both provable and unprovable. But this is a contradiction. So there can be no property of natural numbers 0(x) such that it is provable that for some number «, 0(n) is true but unprovable. This argument is not specifically tied to the structure of the natural numbers. It is clear that a similar argument can be formulated for every mathematical domain for which we have a definable well-ordering. The structure of the (finite and transfinite) ordinal numbers constitutes one such domain. In the argument, no specific assumptions seem to be made about the property 0(x). If the description argument is valid, then its conclusion holds for purely arithmetical properties of natural numbers as well as for properties that also involve the notion of a priori knowability. Because reasoning about a definite description ('the smallest n such that 0(n) is true but unprovable') is a crucial component of the argumentation,

Universally Baire sets and definable well-orderings of the reals

AbstractLet n ≥ 3 be an integer. We show that it is consistent (relative to the consistency of n − 2 strong cardinals) that every Σ1n-set of reals is universally Baire yet there is a (lightface) projective well-ordering of the reals. The proof uses “David's trick” in the presence of inner models with strong cardinals.

A Δ 22 well-order of the reals and incompactness of L( Q MM)

A forcing poset of size 2 2ℵ1 which adds no new reals is described and shown to provide a Δ 2 2 definable well-order of the reals (in fact, any given relation of the reals may be so encoded in some generic extension). The encoding of this well-order is obtained by playing with products of Aronszajn trees: some products are special while other are Suslin trees. The paper also deals with the Magidor–Malitz logic: it is consistent that this logic is highly noncompact.

The perfect set theorem and definable wellorderings of the continuum

AbstractLet Γ be a collection of relations on the reals and let M be a set of reals. We call M a perfect set basis for Γ if every set in Γ with parameters from M which is not totally included in M contains a perfect subset with code in M. A simple elementary proof is given of the following result (assuming mild regularity conditions on Γ and M): If M is a perfect set basis for Γ, the field of every wellordering in Γ is contained in M. An immediate corollary is Mansfield's Theorem that the existence of a Σ21 wellordering of the reals implies that every real is constructible. Other applications and extensions of the main result are also given.

On Σ1 well-orderings of the universe

The constructible universe L of Gödel [2] has a natural well-ordering &lt;L given by the order of construction; a closer look reveals that this well-ordering is definable by a Σ1 formula. Cohen's method of forcing provides several examples of models of ZF + V ≠ L which have a definable well-ordering but none is definable by a relatively simple formula.Recently, Mansfield [7] has shown that if a set of reals (or hereditarily countable sets) has a Σ1, well-ordering then each of its elements is constructible. A question has thus arisen whether one can find a model of ZF + V ≠ L that has a Σ1 well-ordering of the universe. We answer this question in the affirmative.The main result of this paper isTheorem. There is a model of ZF + V ≠ L which has a Σ1 well-ordering.The model is a generic extension of L by adjoining a branch through a Suslin tree with certain properties. The branch is a nonconstructible subset of ℵ1. Note that by Mansfield's theorem, the model must not have nonconstructible subsets of ω.Our results can be generalized in several directions. We note that in particular, we can get a model with a Σ1 well-ordering that is not L[X] for any set X. As one might expect from a joint paper by a recursion theorist and a set theorist, the proof consists of a construction and a computation.

On Σ1 well-orderings of the universe

The constructible universe L of Gödel [2] has a natural well-ordering &lt;L given by the order of construction; a closer look reveals that this well-ordering is definable by a Σ1 formula. Cohen's method of forcing provides several examples of models of ZF + V ≠ L which have a definable well-ordering but none is definable by a relatively simple formula.Recently, Mansfield [7] has shown that if a set of reals (or hereditarily countable sets) has a Σ1, well-ordering then each of its elements is constructible. A question has thus arisen whether one can find a model of ZF + V ≠ L that has a Σ1 well-ordering of the universe. We answer this question in the affirmative.The main result of this paper isTheorem. There is a model of ZF + V ≠ L which has a Σ1 well-ordering.The model is a generic extension of L by adjoining a branch through a Suslin tree with certain properties. The branch is a nonconstructible subset of ℵ1. Note that by Mansfield's theorem, the model must not have nonconstructible subsets of ω.Our results can be generalized in several directions. We note that in particular, we can get a model with a Σ1 well-ordering that is not L[X] for any set X. As one might expect from a joint paper by a recursion theorist and a set theorist, the proof consists of a construction and a computation.