Axiom Of Choice Research Articles

Set and revealed preference axioms for multi-valued choice

We consider choice correspondences that assign a subset to every choice set of alternatives, where the total set of alternatives is an arbitrary finite or infinite set. We focus on the relations between several extensions of the condition of independence of irrelevant alternatives on one hand, and conditions on the revealed preference relation on sets, notably the weak axiom of revealed preference, on the other hand. We also establish the connection between the condition of independence of irrelevant alternatives and so-called strong sets; the latter characterize a social choice correspondence satisfying independence of irrelevant alternatives.

The Axiom of Choice and the Road Paved by Sierpiński

From 1908 to 1916, articles supporting the axiom of choice were scant. The situation changed in 1916, when Waclaw Sierpinski published a series of articles reviving the debate. The posterity of the...

A NOTE ON CHOICE PRINCIPLES IN SECOND-ORDER LOGIC

AbstractZermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to Hilbert and Ackermann, which neither implies nor is implied by the well-ordering principle.

A note on the Knaster–Tarski Fixpoint Theorem

This note shows that several statements about fixpoints in order theory are equivalent to Knaster–Tarski Fixpoint Theorem for complete lattices. All proofs have been done in Zermelo–Fraenkel set theory without the Axiom of Choice.

Dual choice axiom and probabilistic choice

A decision maker chooses in a probabilistic manner if she does not necessarily prefer the same choice alternative when repeatedly presented with the same choice set. Probabilistic choice may occur for a variety of reasons such as unobserved attributes of choice alternatives, imprecision of preferences, or random errors/noise in decisions. The Luce choice model (also known as strict utility or multinomial logit) is derived from the choice axiom (also known as the independence from irrelevant alternatives). This axiom postulates that the relative likelihood of choosing one choice alternative A over another choice alternative B is not affected by the presence or absence of other choice alternatives in the choice set. This paper presents a dual choice axiom: the relative probability of NOT choosing A over the probability of NOT choosing B is independent from irrelevant alternatives. A new model of probabilistic choice is derived from this dual axiom. This model coincides with Luce’s choice model only in the case of a binary choice. The new model has similar properties as the Luce choice model: the higher is the utility of a choice alternative, the higher is the probability that a decision maker chooses this alternative and the lower is the probability that he or she chooses any other alternative. The new model differs from the Luce choice model in two aspects: utility of choice alternatives is bounded (from above and below) and choice probabilities are more sensitive to differences in utility of choice alternatives.

The Axiom of Choice and the Partition Principle from Dialectica Categories

Abstract The method of morphisms is a well-known application of Dialectica categories to set theory (more precisely, to the theory of cardinal invariants of the continuum). In a previous work, Valeria de Paiva and the author have asked how much of the Axiom of Choice is needed in order to carry out the referred applications of such method. In this paper, we show that, when considered in their full generality, those applications of Dialectica categories give rise to equivalents (within $\textbf{ZF}$) of either the Axiom of Choice ($\textbf{AC}$) or Partition Principle ($\textbf{PP}$)—which is a consequence of $\textbf{AC}$ whose precise status of its relationship with$\textbf{AC}$ itself is an open problem for more than a hundred years.

Set-theoretic reflection is equivalent to induction over well-founded classes

We show that induction over $\Delta(\mathbb R)$-definable well-founded classes is equivalent to the reflection principle which asserts that any true formula of first order set theory with real parameters holds in some transitive set. The equivalence is proved in primitive recursive set theory (which is weaker than Kripke-Platek set theory) extended by the axiom of dependent choice.

Indeterminism in Quantum Mechanics: Beyond and/or Within Causation

The problem of indeterminism in quantum mechanics usually being considered as a generalization determinism of classical mechanics and physics for the case of discrete (quantum) changes is interpreted as an only mathematical problem referring to the relation of a set of independent choices to a well-ordered series therefore regulated by the equivalence of the axiom of choice and the well-ordering “theorem”. The former corresponds to quantum indeterminism, and the latter, to classical determinism. No other premises (besides the above only mathematical equivalence) are necessary to explain how the probabilistic causation of quantum mechanics refers to the unambiguous determinism of classical physics. The same equivalence underlies the mathematical formalism of quantum mechanics. It merged the well-ordered components of the vectors of Heisenberg’s matrix mechanics and the non-ordered members of the wave functions of Schrodinger’s undulatory mechanics. The mathematical condition of that merging is just the equivalence of the axiom of choice and the well-ordering theorem implying in turn Max Born’s probabilistic interpretation of quantum mechanics. Particularly, energy conservation is justified differently than classical physics. It is due to the equivalence at issue rather than to the principle of least action. One may involve two forms of energy conservation corresponding whether to the smooth changes of classical physics or to the discrete changes of quantum mechanics. Further both kinds of changes can be equated to each other under the unified energy conservation as well as the conditions for the violation of energy conservation to be investigated therefore directing to a certain generalization of energy conservation.

Renamings and a Condition-free Formalization of Kronecker’s Construction

Summary In [7], [9], [10] we presented a formalization of Kronecker’s construction of a field extension E for a field F in which a given polynomial p ∈ F [X]\F has a root [5], [6], [3]. A drawback of our formalization was that it works only for polynomial-disjoint fields, that is for fields F with F ∩ F [X] = ∅. The main purpose of Kronecker’s construction is that by induction one gets a field extension of F in which p splits into linear factors. For our formalization this means that the constructed field extension E again has to be polynomial-disjoint. In this article, by means of Mizar system [2], [1], we first analyze whether our formalization can be extended that way. Using the field of polynomials over F with degree smaller than the degree of p to construct the field extension E does not work: In this case E is polynomial-disjoint if and only if p is linear. Using F [X]/<p> one can show that for F = ℚ and F = ℤ n the constructed field extension E is again polynomial-disjoint, so that in particular algebraic number fields can be handled. For the general case we then introduce renamings of sets X as injective functions f with dom(f) = X and rng(f) ∩ (X ∪ Z) = ∅ for an arbitrary set Z. This, finally, allows to construct a field extension E of an arbitrary field F in which a given polynomial p ∈ F [X]\F splits into linear factors. Note, however, that to prove the existence of renamings we had to rely on the axiom of choice.

On Dominated Extension of Linear Operators

An ordered topological vector space has the countable dominated extension property if any linear operator ranging in this space, defined on a subspace of a separable metrizable topological vector space, and dominated there by a continuous sublinear operator admits extension to the entire space with preservation of linearity and domination. Our main result is that the strong $$\sigma$$ -interpolation property is a necessary and sufficient condition for a sequentially complete topological vector space ordered by a closed normal reproducing cone to have the countable dominated extension property. Moreover, this fact can be proved in Zermelo–Fraenkel set theory with the axiom of countable choice.

How to have more things by forgetting how to count them†.

Cohen’s first model is a model of Zermelo–Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal κ. In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set A which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of ‘Adding a Cohen subset’ by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, 2A is extremally disconnected, or [A]<ω is Dedekind-finite.

СУПЕРЕЧЛИВІСТЬ ТЕОРІЇ МНОЖИН ЦЕРМЕЛО-ФРЕНКЕЛЯ З АК-СІОМОЮ ВИБОРУ

By using a counterexample to the known equivalent of the axiom of choice (Krull’s theorem about maximal ideal existence) contradictions within Zermelo–Fraenkel set theory with axiom of choice was shown. Ring ideals set which satisfies the Zorn’s lemma conditions, but with no maximal ele-ment was built.

Why the Axiom of Choice Sometimes Fails

Abstract The early controversies surrounding the axiom of choice are well known, as are the many results that followed concerning its (in)dependence from, and equivalence to, other mathematical propositions. This paper focuses not on the logical status of the axiom but rather on showing why it fails in certain categories.

Mass at Rest after Quantum Information

The way, in which quantum information can unify quantum mechanics (and therefore the Standard model) and general relativity, is investigated. Quantum information is defined as the generalization of the concept of information as to the choice among infinite sets of alternatives. Relevantly, the axiom of choice is necessary in general. The unit of quantum information, a qubit is interpreted as a relevant elementary choice among an infinite set of alternatives generalizing that of a bit. The invariance to the axiom of choice shared by quantum mechanics is introduced: It constitutes quantum information as the relation of any state unorderable in principle (e.g. any coherent quantum state before measurement) and the same state already well-ordered (e.g. the well-ordered statistical ensemble of the measurement of the quantum system at issue). This allows of equating the classical and quantum time correspondingly as the well-ordering of any physical quantity or quantities and their coherent superposition. That equating is interpretable as the isomorphism of Minkowski space and Hilbert space. Quantum information is the structure interpretable in both ways and thus underlying their unification. Its deformation is representable correspondingly as gravitation in the deformed pseudo-Riemannian space of general relativity and the entanglement of two or more quantum systems. The Standard model studies a single quantum system and thus privileges a single reference frame turning out to be inertial for the generalized symmetry U(1)XSU(2)XSU(3) “gauging” the Standard model. As the Standard model refers to a single quantum system, it is necessarily linear and thus the corresponding privileged reference frame is necessary inertial. The Higgs mechanism U(1) → U(1)XSU(2) confirmed enough already experimentally describes exactly the choice of the initial position of a privileged reference frame as the corresponding breaking of the symmetry. The Standard model defines ‘mass at rest’ linearly and absolutely, but general relativity non-linearly and relatively. The “Big Bang” hypothesis is additional interpreting that position as that of the “Big Bang”. It serves also in order to reconcile the linear Standard model in the singularity of the “Big Bang” with the observed nonlinearity of the further expansion of the universe described very well by general relativity. Quantum information links the Standard model and general relativity in another way by mediation of entanglement. The linearity and absoluteness of the former and the nonlinearity and relativeness of the latter can be considered as the relation of a whole and the same whole divided into parts entangled in general.

EXACT COMPLETION AND CONSTRUCTIVE THEORIES OF SETS

AbstractIn the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects (i.e., objects satisfying the axiom of choice). Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS.

On Loeb and sequential spaces in ZF

A topological space is called Loeb if the collection of all its non-empty closed sets has a choice function. Loeb and sequential spaces are investigated in the absence of the axiom of choice. Among other results, the following theorems are proved in ZF: (i) if X is a Cantor completely metrizable second-countable space, then Xω is Loeb; (ii) if a sequential, locally sequentially compact space X has the property that every infinite countable collection of non-empty closed subsets of X has a choice function, then the Cartesian product X×Y of X with any sequential space Y is sequential; (iii) if R is sequential, then every second-countable compact Hausdorff space is sequential.Several statements on Loeb and sequential spaces are shown to be independent of ZF. Open problems are posed.

Between Atomism and Superatomism

There are at least three vaguely atomistic principles that have come up in the literature, two explicitly and one implicitly. First, standard atomism is the claim that everything is composed of atoms, and is very often how atomism is characterized in the literature. Second, superatomism is the claim that parthood is well-founded, which implies that every proper parthood chain terminates, and has been discussed as a stronger alternative to standard atomism. Third, there is a principle that lies between these two theses in terms of its relative strength: strong atomism, the claim that every maximal proper parthood chain terminates. Although strong atomism is equivalent to superatomism in classical extensional mereology, it is strictly weaker than it in strictly weaker systems in which parthood is a partial order. And it is strictly stronger than standard atomism in classical extensional mereology and, given the axiom of choice, in such strictly weaker systems as well. Though strong atomism has not, to my knowledge, been explicitly identified, Shiver appears to have it in mind, though it is unclear whether he recognizes that it is not equivalent to standard atomism in each of the mereologies he considers. I prove these logical relationships which hold amongst these three atomistic principles, and argue that, whether one adopts classical extensional mereology or a system strictly weaker than it in which parthood is a partial order, standard atomism is a more defensible addition to one’s mereology than either of the other two principles, and it should be regarded as the best formulation of the atomistic thesis.

A Rigid Hyperfinite Type II1 Factor

Abstract We show that it is relatively consistent with Zermelo-Fraenkel set theory with the axiom of choice that there exists a hyperfinite type $\textrm{II}_1$-factor of density character $\aleph _1$ that is not isomorphic to its opposite, does not have any outer automorphisms, and has trivial fundamental group.

Correction to: Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice

Correction to: Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice

Towards a Unified Theory of Learning and Information.

In this paper, we introduce the notion of “learning capacity” for algorithms that learn from data, which is analogous to the Shannon channel capacity for communication systems. We show how “learning capacity” bridges the gap between statistical learning theory and information theory, and we will use it to derive generalization bounds for finite hypothesis spaces, differential privacy, and countable domains, among others. Moreover, we prove that under the Axiom of Choice, the existence of an empirical risk minimization (ERM) rule that has a vanishing learning capacity is equivalent to the assertion that the hypothesis space has a finite Vapnik–Chervonenkis (VC) dimension, thus establishing an equivalence relation between two of the most fundamental concepts in statistical learning theory and information theory. In addition, we show how the learning capacity of an algorithm provides important qualitative results, such as on the relation between generalization and algorithmic stability, information leakage, and data processing. Finally, we conclude by listing some open problems and suggesting future directions of research.