Axiom Of Choice Research Articles

Kısmi Rasyonel Bireyler İçin Basit Bir Seçim Modeli

We analyze a restricted pro-con random choice model of Dogan and Yildiz (2018) in which there are only two observable orderings relevant for choice such as price and quality. An agent is endowed with two orderings that both serve as pro-orderings and con-orderings. For each choice set, if an alternative is the top-ranked (bottom-ranked) by one of these orderings, then this is a pro (con) for choosing that alternative. Each pro or con item has a weight reflecting its salience. Each alternative is chosen with a probability proportional to the total weight of its pros and cons. In our analysis, we provide a set of choice axioms, which guarantee that the observed choices can be generated via a pro-con random choice model with respect to two given observable orderings. A notable aspect of this characterization is that the associated axioms discipline the choice behavior when choice sets present attraction effect scenarios. This indicates the tight connection between the model and the attraction effect phenomena.

Fuzzy Zorn\u2019s lemma with applications

We introduced the fuzzy axioms of choice, fuzzy Zorn’s lemma and fuzzy well-ordering principle, which are the fuzzy versions of the axioms of choice, Zorn’s lemma and well-ordering principle, and discussed the relations among them. As an application of fuzzy Zorn’s lemma, we got the following results: (1) Every proper fuzzy ideal of a ring was contained in a maximal fuzzy ideal. (2) Every nonzero ring contained a fuzzy maximal ideal. (3) Introduced the notion of fuzzy nilpotent elements in a ring R, and proved that the intersection of all fuzzy prime ideals in a commutative ring R is the union of all fuzzy nilpotent elements in R. (4) Proposed the fuzzy version of Tychonoff Theorem and by use of fuzzy Zorn’s lemma, we proved the fuzzy Tychonoff Theorem.

Influence of memory processes on choice-consistency.

Choice-consistency is considered a hallmark of rational value-based choice. However, because the cognitive apparatus supporting decision-making is imperfect, real decision-makers often show some degree of choice inconsistency. Cognitive models are necessary to complement idealized choice axioms with attention, perception and memory processes. Specifically, compelling theoretical work suggests that the (imperfect) retention of choice-relevant memories might be important for choice-consistency, but this hypothesis has not been tested directly. We used a novel multi-attribute visual choice paradigm to experimentally test the influence of memory retrieval of exemplars on choice-consistency. Our manipulation check confirmed that our retention interval manipulation successfully reduced memory representation strength. Given this, we found strong evidence against our hypothesis that choice-consistency decreases with increasing retention time. However, quality controls indicated that the choice-consistency of our participants was non-discernable from random behaviour. In addition, an exploratory analysis showed essentially no test–retest reliability of choice-consistency between two observations. Taken together, this suggests the presence of a floor effect in our data and, thus, low data quality for conclusively evaluating our hypotheses. Further exploration tentatively suggested a high difficulty of discriminating between the choice objects driving this floor effect.

Finite $\epsilon$-unit distance graphs

In 2005, Exoo posed the following question. Fix $\epsilon$ with $0\leq\epsilon<1$. Let $G_\epsilon$ be the graph whose vertex set is the Euclidean plane, where two vertices are adjacent iff the Euclidean distance between them lies in the closed interval $[1-\epsilon,1+\epsilon]$. What is the chromatic number $\chi(G_\epsilon)$ of this graph? The case $\epsilon=0$ is precisely the classical ``chromatic number of the plane'' problem. In a 2018 preprint, de Grey shows that $\chi(G_0)\geq 5$; the proof relies heavily on machine computation. In 2016, Grytczuk et al. proved a weaker result with a human-comprehensible but nonconstructive proof: whenever $0<\epsilon<1$, we have that $\chi(G_\epsilon)\geq 5$. (This lower bound of $5$ was later improved by Currie and Eggleton to $6$.) The De Bruijn - Erd\H{o}s theorem (which relies on the axiom of choice) then guarantees the existence, for each $\epsilon$, of a finite subgraph $H_\epsilon$ of $G_\epsilon$ such that $\chi(H_\epsilon)\geq 5$. In this paper, we explicitly construct such finite graphs $H_\epsilon$. We find that the number of vertices needed to create such a graph is no more than $2\pi(15+14\epsilon^{-1})^2$. Our proof can be done by hand without the aid of a computer.

A program for the full axiom of choice

The theory of classical realizability is a framework for the Curry-Howard correspondence which enables to associate a program with each proof in Zermelo-Fraenkel set theory. But, almost all the applications of mathematics in physics, probability, statistics, etc. use Analysis i.e. the axiom of dependent choice (DC) or even the (full) axiom of choice (AC). It is therefore important to find explicit programs for these axioms. Various solutions have been found for DC, for instance the lambda-term called "bar recursion" or the instruction "quote" of LISP. We present here the first program for AC.

Effective randomness for continuous measures

We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n n , all but countably many reals are n n -random for such a measure, where n n indicates the arithmetical complexity of the Martin-Löf tests allowed. The proof rests upon an application of Borel determinacy. Therefore, the proof presupposes the existence of infinitely many iterates of the power set of the natural numbers. In the second part of the paper we present a metamathematical analysis showing that this assumption is indeed necessary. More precisely, there exists a computable function G G such that, for any n n , the statement “All but countably many reals are G ( n ) G(n) -random with respect to a continuous probability measure” cannot be proved in Z F C n − \mathsf {ZFC}^-_n . Here Z F C n − \mathsf {ZFC}^-_n stands for Zermelo-Fraenkel set theory with the Axiom of Choice, where the Power Set Axiom is replaced by the existence of n n -many iterates of the power set of the natural numbers. The proof of the latter fact rests on a very general obstruction to randomness, namely the presence of an internal definability structure.

An Algorithm for Linearizing the Collatz Convergence

The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers for which the inflation propensity remains so unpredictable it could be used to generate reliable proof-of-work algorithms for the cryptocurrency industry; it has so far resisted every attempt at linearizing its behavior. Here, we establish an ad hoc equivalent of modular arithmetics for Collatz sequences based on five arithmetic rules that we prove apply to the entire Collatz dynamical system and for which the iterations exactly define the full basin of attractions leading to any odd number. We further simulate these rules to gain insight into their quiver geometry and computational properties and observe that they linearize the proof of convergence of the full rows of the binary tree over odd numbers in their natural order, a result which, along with the full description of the basin of any odd number, has never been achieved before. We then provide two theoretical programs to explain why the five rules linearize Collatz convergence, one specifically dependent upon the Axiom of Choice and one on Peano arithmetic.

Microfoundations of Discounting

An important question in economics is how people choose between different payments in the future. The classical normative model predicts that a decision maker discounts a later payment relative to an earlier one by an exponential function of the time between them. Descriptive models use nonexponential functions to fit observed behavioral phenomena, such as preference reversal. Here we propose a model of discounting, consistent with standard axioms of choice, in which decision makers maximize the growth rate of their wealth. Four specifications of the model produce four forms of discounting—no discounting, exponential discounting, hyperbolic discounting, and a hybrid of exponential and hyperbolic discounting—two of which predict preference reversal. Our model requires no assumption of behavioral bias or payment risk.

A Choice-Free Cardinal Equality

For a cardinal a, let fin(a) be the cardinality of the set of all finite subsets of a set which is of cardinality a. It is proved without the aid of the axiom of choice that, for all infinite cardinals a and all natural numbers n, 2fin(a)n=2[fin(a)]n. On the other hand, it is proved consistent with ZF that there exists an infinite cardinal a such that 2fin(a)<2fin(a)2<2fin(a)3<⋯<2fin(fin(a)).

Several results on compact metrizable spaces in mathbf {ZF}

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in mathbf {ZF}, some are shown to be independent of mathbf {ZF}. For independence results, distinct models of mathbf {ZF} and permutation models of mathbf {ZFA} with transfer theorems of Pincus are applied. New symmetric models of mathbf {ZF} are constructed in each of which the power set of mathbb {R} is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube [0, 1]^{mathbb {R}}.

A note on the asymptotic expressiveness of ZF and ZFC

Abstract We investigate the asymptotic densities of theorems provable in Zermelo–Fraenkel set theory zf and its extension zfc including the axiom of choice. Assuming a canonical De Bruijn representation of formulae, we construct asymptotically large sets of sentences unprovable within zf, yet provable in zfc. Furthermore, we link the asymptotic density of zfc theorems with the provable consistency of zfc itself. Consequently, if zfc is consistent, it is not possible to refute the existence of the asymptotic density of zfc theorems within zfc. Both these results address a recent question by Zaionc regarding the asymptotic equivalence of zf and zfc.

A canon of probabilistic rationality

We prove that a random choice rule satisfies Luce's Choice Axiom if and only if its support is a choice correspondence that satisfies the Weak Axiom of Revealed Preference, thus it consists of alternatives that are optimal according to some preference, and then random choice occurs according to a tie-breaking mechanism among such alternatives that satisfies Renyi's Conditioning Axiom.Our result shows that the Choice Axiom is, in a precise formal sense, a probabilistic version of the Weak Axiom. It thus supports Luce's view of his own axiom as a “canon of probabilistic rationality.”

The Brain and the New Foundations of Mathematics

Many concepts in mathematics are not fully defined, and their properties are implicit, which leads to paradoxes. New foundations of mathematics were formulated based on the concept of innate programs of behavior and thinking. The basic axiom of mathematics is proposed, according to which any mathematical object has a physical carrier. This carrier can store and process only a finite amount of information. As a result of the D-procedure (encoding of any mathematical objects and operations on them in the form of qubits), a mathematical object is digitized. As a consequence, the basis of mathematics is the interaction of brain qubits, which can only implement arithmetic operations on numbers. A proof in mathematics is an algorithm for finding the correct statement from a list of already-existing statements. Some mathematical paradoxes (e.g., Banach–Tarski and Russell) and Smale’s 18th problem are solved by means of the D-procedure. The axiom of choice is a consequence of the equivalence of physical states, the choice among which can be made randomly. The proposed mathematics is constructive in the sense that any mathematical object exists if it is physically realized. The consistency of mathematics is due to directed evolution, which results in effective structures. Computing with qubits is based on the nontrivial quantum effects of biologically important molecules in neurons and the brain.

Second-countable compact Hausdorff spaces as remainders in ZF and two new notions of infiniteness

In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in ZF. Among other independence results, the characterization of locally compact Hausdorff spaces having all non-empty metrizable compact spaces as remainders, obtained by Hatzenbuhler and Mattson in ZFC, is proved to be independent of ZF. Urysohn's Metrization Theorem is generalized. New concepts of a strongly filterbase infinite set and a dyadically filterbase infinite set are introduced, both stemming from the investigations on compactifications. Set-theoretic and topological definitions of the new concepts are given, and their relationship with certain known notions of infinite sets is investigated in ZF. A new permutation model is introduced in which there exists a strongly filterbase infinite set which is weakly Dedekind-finite.

The finite subsets and the permutations with finitely many non‐fixed points of a set

AbstractWe write and for the cardinalities of the set of finite subsets and the set of permutations with finitely many non‐fixed points, respectively, of a set which is of cardinality . In this paper, we investigate relationships between and for an infinite cardinal in the absence of the Axiom of Choice. We give conditions that make and comparable as well as give related consistency results.

Influence of divided attention on the attraction effect in multialternative choice

AbstractThe attraction effect in multialternative decision making reflects the context-dependent violation of rational choice axioms. This study examined the effect of concurrent divided attention in three-alternative visual choice tasks. The concurrent divided attention task is considered to consume the mental resources available for the choice task. There were three conditions: (a) the task-relevant condition, in which the auditory task should consume resources across multiple levels; (b) the task-irrelevant condition, in which the auditory stimuli should consume perceptual resources; (c) and the control condition, in which the resources should not be consumed by auditory stimuli. Thirty-three participants solved 24 hypothetical purchase problems with three alternatives that differed in terms of two attributes. The results indicated that the choice proportion of the target was significantly higher in the task-relevant condition than in the task-irrelevant and control conditions, thereby suggesting that a reduction in cognitive (and/or response) resources is critical for the attraction effect.

DEDUCTIVE CARDINALITY RESULTS AND NUISANCE-LIKE PRINCIPLES

AbstractTheinjectiveversion of Cantor’s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of choice—a result hitherto unestablished. It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali–Fortian method of proof.

TD implies [formula omitted

Assuming ZF, we prove that Turing determinacy (TD) implies the countable choice axiom for sets of reals (CCR).

Eilenberg–Jachymski collections and its first consequences for the fixed point theory

In this paper, we introduce the concept of Eilenberg–Jachymski collection on a nonempty set. Then, we establish three results equivalent to Bourbaki–Kneser’s fixed point theorem, and, therefore, to the axiom of choice. As consequences, we present new fixed point theorems in compact topological spaces, which extend and unify those of Nemytskii–Edelstein, Liepins and Suzuki.

Portfolio Selection by the Axiom of Choice: Post Mean-Variance Analysis

Portfolio Selection by the Axiom of Choice: Post Mean-Variance Analysis