Strong Axiom Research Articles

From real-life to very strong axioms. Classification problems in Descriptive Set Theory and regularity properties in Generalized Descriptive Set Theory

AbstractThis thesis is divided into three parts, the first and second ones focused on combinatorics and classification problems on discrete and geometrical objects in the context of descriptive set theory, and the third one on generalized descriptive set theory at singular cardinals of countable cofinality.Descriptive Set Theory (briefly: DST) is the study of definable subsets of Polish spaces, i.e., separable completely metrizable spaces. One of the major branches of DST is Borel reducibility, successfully used in the last 30 years to solve and compare many classification problems. One of our goals is the classification of knots, very familiar and tangible objects in everyday life, which also play an important role in modern mathematics. The study of knots and their properties is known as knot theory. Our plan is to gain insight into knots using discrete objects, such as linear and circular orders. This approach was already exploited in [6]. The first part of this work is therefore devoted to countable linear orders and the study of the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results. We also expand our research to the case of circular orders.Another objective of this first part is to extend the notion of convex embeddability on countable linear orders. We provide a family of quasi-orders of which embeddability is a particular case as well. We study these quasi-orders from a combinatorial point of view and analyse their complexity with respect to Borel reducibility. Furthermore, we extend the analysis of these quasi-orders to the set of uncountable linear orders.The second part of the project deals with classification problems on knots and $3$ -manifolds. The goal here is to apply the results obtained in the first part to the study of proper arcs and knots, establishing lower bounds for the complexity of some natural relations between these geometrical objects. We also obtain some combinatorial results which are particularly interesting when we restrict to the set of wild proper arcs and wild knots, classes which haven’t received much attention so far. These parts are included in the two preprints [4, 5] in collaboration with my supervisor Alberto Marcone, Luca Motto Ros (University of Torino), and Vadim Weinstein (University of Oulu).The second part of this work also includes the classification of non-compact $3$ -manifolds up to homeomorphism (the case of compact $3$ -manifolds has already been solved: indeed, there are only countably many $3$ -manifolds up to homeomorphism; see [7]), and that of Cantor sets of $\mathbb {R}^3$ up to conjugation (answering to Question 5.5 of [3]). Here we resort to algebraic tools. Stone duality gives a neat way to go back-and-forth between totally disconnected Polish spaces and countable Boolean algebras (see [1]). The main ingredient is the Stone space of all ultrafilters on a Boolean algebra. In this work we introduce a weaker concept which we call “blurry filter”. Using blurry filters instead of ultrafilters enables one to extend the class of spaces under consideration beyond totally disconnected. As an application of this method, we show that both homeomorphism on non-compact $3$ -manifolds and conjugation of Cantor sets in $\mathbb {R}^3$ are completely classifiable by countable structures. These results are part of an upcoming paper in collaboration with Vadim Weinstein.The last part of this thesis concerns the natural generalization of descriptive set theory that occurs when countable is replaced by uncountable, called Generalized Descriptive Set Theory (briefly: GDST). In particular, we focus on the case of GDST for a singular cardinal $\kappa $ of countable cofinality. The goal here is to study the generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set $\mathcal {P}(\kappa )$ . We consider the question under which large cardinal hypotheses classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms, like I2, and classes of sets definable by $\Sigma _1$ -formulas with parameters from various collections of sets. The obtained results are included in [2], a joint work with my co-supervisor Vincenzo Dimonte and Philipp Lücke (University of Hamburg).Abstract prepared by Martina IannellaE-mail: martina.iannella@tuwien.ac.at.URL: https://air.uniud.it/handle/11390/1262824.

On pseudo-Riemannian quartics in Finsler geometry

Finsler geometry usually describes an extension of Riemannian geometry into a direction-dependent geometric structure. Historically, the well-known Riemann quartic length element example served as the inspiration for this construction. Surprisingly, the same quartic expression emerges as a fundamental dispersion relationcovariant Fresnel equationin solid-state electrodynamics. As a result, it is possible to conceive of the Riemann quartic length expression as a mathematical representation of a well-known physical phenomenon. This paper provides a number of Riemann quartic examples that show Finsler geometry to be overly constrictive for many applications, even when the signature space is positive definite in the Euclidean sense. The strong axioms of Finsler geometry are broken down on many more singular hypersurfaces for the spaces having an indefinite (Minkowski) signature. We suggest a more flexible definition of a Finsler structure that only has to hold for open subsets of a manifold’s tangent bundle. We demonstrate the distinctive singular hypersurfaces connected to the Riemann quartic and discuss the potential physics explanations for them. As an illustration of the pseudo-Riemannian quartic, we took into consideration the dispersion relation that appears in electromagnetic wave propagation in uniaxial crystal. Our analysis suggests that the signature of the Finsler measure may be altered for large anisotropy factors.

Pseudo-Riemann’s quartics in Finsler’s geometry—two-dimensional case

Finsler’s geometry usually describes an extension of Riemmann’s geometry into a direction-dependent geometric structure. Historically, the well-known Riemann’s quartic length element example served as the inspiration for this construction. Surprisingly, the covariant Fresnel equation—a fundamental dispersion relation in solid-state electrodynamics—emerges as the exact same quartic expression. As a result, Riemann’s quartic length expression can be regarded of as a mathematical representation of a well-known physical phenomenon. In this study, we offer numerous Riemann’s quartic examples that show Finsler’s geometry, even in the situation of a positive definite Euclidean signature space, is too restrictive for many applications. The strong axioms of Finsler’s geometry are violated in a substantially greater number of distinctive subsets for the spaces having an indefinite (Minkowski) signature. We suggest a weaker characterization of Finsler’s structure based on explicitly calculated two-dimensional examples. In tangential vector space, this concept permits singular subsets. Only open subsets of a manifold’s tangent bundle are required to satisfy the strong axioms of Finsler’s geometry. We demonstrate the distinctive unique subsets of the Riemann’s quartic in two dimensions and briefly discuss their possible physical origins.

Approval-based apportionment

In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters can support multiple parties by casting approval ballots. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates instead of parties. Using techniques from both apportionment and multiwinner elections, we identify rules that generalize the D’Hondt apportionment method and that satisfy strong axioms which are generalizations of properties commonly studied in the apportionment literature. In fact, the rules we discuss provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Second, we demonstrate that extended justified representation is compatible with committee monotonicity (also known as house monotonicity).

DETERMINACY OF SCHMIDT’S GAME AND OTHER INTERSECTION GAMES

AbstractSchmidt’s game and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, $\mathsf {AD}_{\mathbb R}$ , which is a much stronger axiom than that asserting all integer games are determined, $\mathsf {AD}$ . One of our main results is a general theorem which under the hypothesis $\mathsf {AD}$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt’s $(\alpha ,\beta ,\rho )$ game on $\mathbb R$ is determined from $\mathsf {AD}$ alone, but on $\mathbb R^n$ for $n \geq 3$ we show that $\mathsf {AD}$ does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt’s $(\alpha , \beta , \rho )$ game on $\mathbb {R}$ has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt’s game. These results highlight the obstacles in obtaining the determinacy of Schmidt’s game from $\mathsf {AD}$ .

AN UPPER BOUND ON THE DEGREE OF CHOICE REGULARITIES ASSOCIATED WITH SARP-LIKE REQUIREMENTS

Yildiz (2016a) proposes a novel tool, namely choice regularities, to classify choice theories according to the syntactic structure of the underlying choice axioms. We focus on the regularity of the SARP-like acyclicity requirements. Axiomatic characterizations of many choice theories contain conditions formulated in the vein of the strong axiom of revealed preference (SARP), such as Masatlioglu et al. (2012)and Yildiz (2016b), in which a binary relation –based on the observed choice behavior– is formulated and its acyclicity is required. We focus on the regularity of the SARP-like acyclicity requirements. For those binary relations that can be represented in a specific form, we find an upper bound on the degree of the free regularities that impose the binary relation to be acyclic.

Rationalizable choice functions

We study choice functions which arise in the context of matching with contracts. A choice function is rationalizable if it coincides with a choice function induced by a preference relation over sets of contracts. We show that a choice function is rationalizable if and only if it satisfies the Strong Axiom of Revealed Preference (SARP), and give an algorithm for verifying the rationalizability of a given choice function. Consequently, we prove that every path independent choice function is rationalizable. This implies that under the substitutability assumption, the rationalizability condition and the irrelevance of rejected contracts (IRC) condition are equivalent.

Similarities in axiomatizations: equal surplus division value and first-price auctions

We compare axiomatizations between a value for cooperative games with transferable utilities (TU games), and a rule for auctions. The equal surplus division value on the set of zero-monotonic TU games is characterized by the following: individual rationality, Pareto efficiency, and equal effect of players’ nullification on others. Meanwhile, first-price auctions, on the general preference domain, are characterized by individual rationality, envy-freeness, and weak equal effect of buyers’ nullification on others. Here, envy-freeness implies Pareto efficiency in the model of auctions. Given the agents’ general preferences in the auction model, the characteristic of a weak equal effect of buyers’ nullification on others weakens the requirement of equal effect of players’ nullification on others. Although the two models are different, the corresponding axioms in both models require conditions corresponding to each other. In particular, individual rationality requires voluntary participation of agents, Pareto efficiency (or its stronger axiom of envy-freeness in the model of auctions) requires outcomes with no waste, and (weak) equal effect of players’ (buyers’) nullification on others requires equal treatment of agents when an agent is nullified. Therefore, in terms of axiomatizations, we can similarly interpret the equal surplus division value and first-price auctions.

A General Equilibrium Approach to Taxonomy of Institutional Proxies for Innovation Outcomes

This study develops a new formal theoretical rubric for ranking of proxies for institutional factors that, conceptually, have taxonomy of either of facilitators of, or constraints on economic development. Illustrative implementations of the formal theoretical rubric reveal that spending on each of elementary or secondary education has taxonomy, not of social capital, but of `Productive Capability'. Conditional on optimal spending on primary and secondary education, stock markets are beneficial for welfare of individuals, and are shown to have taxonomy, not of investment capability, but of `Innovation Capability'. An hitherto unexplored factor shown to rank high as facilitator of economic development is `Policy Capability' that incentivizes export orientation within a domestic economy. Alongside spending on education, and stock market size, this policy capability is shown to be more important for realization of economic development than the stock of physical capital within an economy. All rank preferences are shown to satisfy the Strong Axiom of Revealed Preference (SARP). Empirical tests reveal importance of availability and application of formal theoretical rubrics to characterization and interpretation of effects of institutional factors that, conceptually, have taxonomy of facilitators of, or constraints on economic development.

Income Elasticities without Parameters

This paper proposes a simple non-parametric framework to calculate income elasticities from a data set of observed prices and consumed quantities without having to estimate any parameters. The framework can be applied when the price-quantity data satisfies a revealed preference axiom called the strong version of the strong axiom of revealed preference (SSARP). The framework is used to calculate income elasticities for food and non-alcoholic drinks from a rich panel of household expenditures. For this category, it is found that households are rather homogeneous in their demand responses.

Specializing Aronszajn Trees with Strong Axiom A and Halving

We construct creature forcings with strong Axiom A that specialize a given Aronszajn tree. We work with tree creature forcing. The creatures that live on the Aronszajn tree are normed and have the halving property. We show that our models fulfill ℵ1=d<unif(M)=ℵ2=2ω.

Choice on the simplex domain

One unit of a good has to be divided among a group of agents who each are entitled to a minimal share, and these shares sum up to less than one. The associated set of choice problems consists of the unit simplex and all its full-dimensional subsimplices with the same orientation. We characterize all choice rules that are independent of irrelevant alternatives, continuous, and monotonic; the last condition means that if an agent receives its minimal share and that share increases, then no other agent benefits. In line with Kıbrıs (2012) we show that these rules are rationalizable and representable by a real-valued function. On the issue of rationalizability, we also consider weakenings of our conditions. In particular, we show that in general, excluding cycles of any fixed length does not imply the strong axiom of revealed preference, that is, the exclusion of cycles of any length. For continuous three-agent choice rules, however, excluding cycles of length three implies the strong axiom of revealed preference.

Descriptive complexity and revealed preference theory

This paper formalizes revealed preference theory using the notion of Ramsey eliminability in logic, and shows how the language required to state a revealed preference axiom for some choice theory is closely connected with the computational tractability of testing that theory. The connection is made through results in descriptive complexity theory, a relatively new field in finite model theory. It is shown that checking whether observed choices of players in normal form games are Nash rationalizable is an NP-complete problem. This also means that there does not exist an analogue (in a precise sense) of the Strong Axiom of Revealed Preference for Nash equilibrium.

Quantum groups under very strong axioms

We study the intermediate quantum groups $H_N\subset G\subset U_N^+$. The basic examples are $H_N,K_N,O_N,U_N,H_N^+,K_N^+,O_N^+,U_N^+$, which form a cube. Any other example $G$ sits inside the cube, and by using standard operations, namely intersection $\cap$ and generation $ $, can be projected on the faces and edges. We prove that under the strongest possible axioms, namely (1) easiness, (2) uniformity, and (3) geometric coherence of the various projection operations, the 8 basic solutions are the only ones.

Arrow’s decisive coalitions

In his classic monograph, Social Choice and Individual Values, Arrow introduced the notion of a decisive coalition of voters as part of his mathematical framework for social choice theory. The subsequent literature on Arrow’s Impossibility Theorem has shown the importance for social choice theory of reasoning about coalitions of voters with different grades of decisiveness. The goal of this paper is a fine-grained analysis of reasoning about decisive coalitions, formalizing how the concept of a decisive coalition gives rise to a social choice theoretic language and logic all of its own. We show that given Arrow’s axioms of the Independence of Irrelevant Alternatives and Universal Domain, rationality postulates for social preference correspond to strong axioms about decisive coalitions. We demonstrate this correspondence with results of a kind familiar in economics—representation theorems—as well as results of a kind coming from mathematical logic—completeness theorems. We present a complete logic for reasoning about decisive coalitions, along with formal proofs of Arrow’s and Wilson’s theorems. In addition, we prove the correctness of an algorithm for calculating, given any social rationality postulate of a certain form in the language of binary preference, the corresponding axiom in the language of decisive coalitions. These results suggest for social choice theory new perspectives and tools from logic.

The Abridgement Paradox

When axiomatizing a body of truths, one first concentrates on obtaining a set of axioms that entail all and only those truths. The theorist expects that this complete system will have some needlessly strong axioms that can later be weakened or even deleted. What happens if the theorist includes in his system recognition of this superfluity? Contradiction! Even admitting the possibility of superfluity dashes all hope of consistency. Any suspicion of superfluous information must be voiced from outside the system.

THE RAMIFIED ANALYTICAL HIERARCHY USING EXTENDED LOGICS

AbstractThe use of Extended Logics to replace ordinary second order definability in Kleene’s Ramified Analytical Hierarchy is investigated. This mirrors a similar investigation of Kennedy, Magidor and Väänänen [11] where Gödel’s universe L of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain minimal correct models of analysis. A wide spectrum of models can be so generated from abstract definability notions: one may take an abstract Spector Class and extract an extended logic for it. The resultant structure is then a minimal model of the given kind of definability.

Unitary easy quantum groups: Geometric aspects

We discuss the classification problem for the unitary easy quantum groups, under strong axioms, of noncommutative geometric nature. Our main results concern the intermediate easy quantum groups $O_N\subset G\subset U_N^+$. To any such quantum group we associate its Schur-Weyl twist $\bar{G}$, two noncommutative spheres $S,\bar{S}$, a noncommutative torus $T$, and a quantum reflection group $K$. Studying $(S,\bar{S},T,K,G,\bar{G})$ leads then to some natural axioms, which can be used in order to investigate $G$ itself. We prove that the main examples are covered by our formalism, and we conjecture that in what concerns the case $U_N\subset G\subset U_N^+$, our axioms should restrict the list of known examples.

Transitivity of preferences: When does it matter?

We define the empirical conditions on prices and incomes under which transitivityof preferences has specific testable implications. In particular, we set out necessaryand sufficient requirements for budget sets under which consumption choices canviolate SARP (Strong Axiom of Revealed Preferences) but not WARP (Weak Axiomof Revealed Preferences). As SARP extends WARP by additionally imposingtransitive preferences, this effectively defines the conditions under which transitivityis separately testable. Our findings have considerable practical relevance, as transitivityconditions are known to substantially aggravate the computational burden ofempirical revealed preference analysis. Our characterization takes the form of triangularconditions that must hold for all three-element subsets of normalized prices,and which are easy to verify in practice. We demonstrate their practical use throughtwo short empirical applications.

The axiomatic characterizations on L-fuzzy covering-based approximation operators

Axiomatic characterization is the foundation of L-fuzzy rough set theory: the axiom sets of approximation operators guarantee the existence of L-fuzzy relations or L-fuzzy coverings that reproduce the approximation operators. Axiomatic characterizations of approximation operators based on L-fuzzy coverings have not been fully explored, although those based on L-fuzzy relations have been studied thoroughly. Focusing on three pairs of widely used L-fuzzy covering-based approximation operators, we establish an axiom set for each of them, and their independence is examined. It should be noted that the axiom set of each L-fuzzy covering-based approximation operator is different from its crisp counterpart, with an either new or stronger axiom included in the L-fuzzy version.