Related papers: The Rigid Relation Principle, a New Weak Choice Pr…
A folk theorem says higher order arithmetic has the proof theoretic strength of set theory with limited power set. This paper makes the theorem precise in terms of several axiom system based on ZF.
A cardinal is weakly Reinhardt if it is the critical point of an elementary embedding from the universe of sets into a model that contains the double powerset of every ordinal. This note establishes the equiconsistency of a proper class of…
Earlier papers \cite{VB2022,VB2023a,VB2023b} introduced the notions of a core and an index of a relation (an index being a special case of a core). A limited form of the axiom of choice was postulated -- specifically that all partial…
Recent work on the logical structure of non-locality has constructed scenarios where observations of multi-partite systems cannot be adequately described by compositions of non-signaling subsystems. In this paper we apply these frameworks…
Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well…
Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and G\"odel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed…
We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$.…
In this paper, we describe the formalization of the axiom of choice and several of its famous equivalent theorems in Morse-Kelley set theory. These theorems include Tukey's lemma, the Hausdorff maximal principle, the maximal principle,…
The weak axiom of revealed preference (WARP) ensures that the revealed preference (i) is a preference relation (i.e., it is complete and transitive) and (ii) rationalizes the choices. However, when WARP fails, either one of these two…
Matthias Schr\"oder has asked the question whether there is a weakest discontinuous problem in the continuous version of the Weihrauch lattice. Such a problem can be considered as the weakest unsolvable problem. We introduce the…
We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside…
We prove that any strongly mixing action of a countable abelian group on a probability space has higher order mixing properties. This is achieved via introducing and utilizing $\mathcal R$-limits, a notion of convergence which is based on…
The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle/ZF. The proof builds upon a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the…
The replacement (or collection or choice) axiom scheme asserts bounded quantifier exchange. We prove the independence of this scheme from various weak theories of arithmetic, sometimes under a complexity assumption.
The purpose of this paper is to outline a simple set of axioms for basic set theory from which most fundamental facts can be derived. The key to the whole project is a new axiom of set theory which I dubbed "The Law of Extremes". It allows…
We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections).…
We study mechanism which operate on ordinal preference information (i.e., rank ordered lists of alternatives) on the full domain of weak preferences that admits indifferences. We present a novel decomposition of strategyproofness into three…
This paper is dedicated to a cautious learning methodology for predicting preferences between alternatives characterized by binary attributes (formally, each alternative is seen as a subset of attributes). By "cautious", we mean that the…
We investigate Hindman- and Owings-type Ramsey-theoretic statements in Zermelo-Fraenkel set theory without the Axiom of Choice, with some occasional extra assumptions (such as the Axiom of Dependent Choice and/or the Axiom of Determinacy).…
We show that first-order logic can be translated into a very simple and weak logic, and thus set theory can be formalized in this weak logic. This weak logical system is equivalent to the equational theory of Boolean algebras with three…