It is proved that the forcing apparatus can be built and set to work in ZFCA (=ZFC minus foundation plus the antifoundation axiom AFA). The key tools for this construction are greatest fixed points of continuous operators (a method sometimes referred to as “corecursion”). As an application it is shown that the generic extensions of standard models of ZFCA are models of ZFCA again.

The still unsettled decision problem for the restricted purely universal formulae 0-formulae) of the first order set-theoretic language based over =, ∈ is discussed in relation with the adoption or rejection of the axiom of foundation. Assuming the axiom of foundation, the related finite set-satisfiability problem for the very significant subclass of the 0-formulae consisting of the formulae involving only nested variables of level 1 is proved to be semidecidable on the ground of a reflection property over the hereditarily finite (...) sets, and various extensions of this result are obtained. When variables are restricted to range only over sets, in universes with infinitely many urelements the set-satisfiability problem is shown to be solvable provided the axiom of foundation is assumed; if it is not, then the decidability of a related derivability problem still holds. That, in turn, suggests the alternative adoption of an antifoundation axiom under which the set-satisfiability problem is also solvable . Turning to set theory without urelements, assuming a form of Boffa's antifoundation axiom, the complement of the set-satisfiability problem for the full class of Δ0-formulae is shown to be semidecidable; a result that is known not to hold, for the set-satisfiability problem itself, even for a very restricted subclass of the Δ0-formulae. (shrink)

We show that the Axiom of Foundation, as well as the Antifoundation Axiom AFA, plays a crucial role in determining the decidability of the following problem. Given a first-order theory T over the language , and a sentence F of the form with quantifier-free in the same language, are there models of T in which F is true? Furthermore we show that the Extensionality Axiom is quite irrelevant in that respect.

In Zermelo-Fraenkel set theory without the Axiom of Foundation we study the schema version of the principle of dependent choices in connection with Aczel's antifoundation axiom , Boffa's anti-foundation axiom, and axiom of collection.

In Aczel [1], the existence of largest fixed points of set continuous operators is proved assuming the schema version of dependent choices in Zermelo-Fraenkel set theory without the axiom of Foundation. In the present paper, we study whether the existence of largest fixed points of set continuous operators is provable without the schema version of dependent choices, using Boffa's weak antifoundation axioms.

We adapt C. Freiling's axioms of symmetry 190–200) to models of set theory with classes by identifying small classes with sets getting thus a sequence of principles An, for n2, of increasing strength. Several equivalents of A2 are given. A2 is incompatible both with the foundation axiom and the antifoundation axioms AFA considered in Aczel . A hierarchy of symmetry degrees of preorderings is introduced and compared with An. Models are presented in which this hierarchy is strict. (...) The main result of the paper is that a class X satisfies ¬An iff it has symmetry degree n−2. (shrink)

In this paper, we show that non-well-founded sets can be defined constructively by formalizing Hallnäs' limit definition of these within Martin-Löf's theory of types. A system is a type W together with an assignment of ᾱ ∈ U and α̃ ∈ ᾱ → W to each α ∈ W. We show that for any system W we can define an equivalence relation = w such that α = w β ∈ U and = w is the maximal bisimulation. Aczel's proof (...) that CZF can be interpreted in the type V of iterative sets shows that if the system W satisfies an additional condition (*), then we can interpret CZF minus the set induction scheme in W. W is then extended to a complete system W * by taking limits of approximation chains. We show that in W * the antifoundation axiom AFA holds as well as the axioms of CFZ -. (shrink)

Aczel's theory of hypersets provides an interesting alternative to the standard view of sets as inductively constructed, well-founded objects, thus providing a convienent formalism in which to consider non-well-founded versions of classically well-founded constructions, such as the "circular logic" of [3]. This theory and ZFC are mutually interpretable; in particular, any model of ZFC has a canonical "extension" to a non-well-founded universe. The construction of this model does not immediately generalize to weaker set theories such as the theory of admissible (...) sets. In this paper, we formulate a version of Aczel's antifoundation axiom suitable for the theory of admissible sets. We investigate the properties of models of the axiom system KPU - , that is, KPU with foundation replaced by an appropriate strengthening of the extensionality axiom. Finally, we forge connections between "non-wellfounded sets over the admissible set A" and the fragment L A of the modal language L ∞. (shrink)

We establish the decidability, with respect to open formulas in the first order language with equality =, the membership relation , the constant for the empty set, and a binary operation w which, applied to any two sets x and y, yields the results of adding y as an element to x, of the theory NW having the obvious axioms for and w. Furthermore we establish the completeness with respect to purely universal sentences of the theory , obtained from (...) NW by adding the Extensionality Axiom E and the Regularity Axiom R, and of the theory obtained by adding to NW (a slight variant of) the Antifoundation Axiom AFA. (shrink)

The role of foundation with respect to transitive closure in the Zermelo system Z has been investigated by Boffa; our aim is to explore the role of antifoundation. We start by showing the consistency of "Z antifoundation transitive closure" relative to Z (by a technique well known for ZF). Further, we introduce a "weak replacement principle" (deductible from antifoundation and transitive closure) and study the relations among these three statements in Z via interpretations. Finally, we give some (...) adaptations for ZF without infinity. (shrink)

Asperó and Schindler have completely solved the Axiom [Formula: see text] vs. [Formula: see text] problem. They have proved that if [Formula: see text] holds then Axiom [Formula: see text] holds, with no additional assumptions. The key question now concerns the relationship between [Formula: see text] and Axiom [Formula: see text]. This is because the foundational issues raised by the problem of Axiom [Formula: see text] vs. [Formula: see text] arguably persist in the problem of Axiom [Formula: see text] vs. (...) [Formula: see text]. The first of our two main theorems is that Axiom [Formula: see text] is equivalent to Axiom [Formula: see text], and as a corollary we show that Axiom [Formula: see text] fails in all the known models of [Formula: see text]. This suggests that [Formula: see text] actually refutes Axiom [Formula: see text]. Our second main theorem is that the [Formula: see text] Conjecture holds assuming [Formula: see text]. This is the strongest partial result known on this conjecture which is one of the central open problems of [Formula: see text]-theory and [Formula: see text]-logic. These results identify a fundamental asymmetry between the Continuum Hypothesis and any axiom which is both [Formula: see text]-expressible and which implies [Formula: see text], on the basis of generic absoluteness for the simplest of the nontrivial sentences of Third-Order Number Theory. These are the [Formula: see text]-sentences with no parameters. Such sentences are those which simply assert the existence of a set [Formula: see text] for which some property involving only quantification over [Formula: see text] holds. (shrink)

We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal $$\delta $$ δ -separated sets in metric and pseudometric spaces from the point of (...) view the Axiom of Choice and its weaker forms. (shrink)

We analyze axioms and postulates as speech acts. After a brief historical appraisal of the concept of axiom in Euclid, Frege, and Hilbert, we evaluate contemporary axiomatics from a linguistic perspective. Our reading is inspired by Hilbert and is meant to account for the assertive, directive, and declarative components of modern axiomatics. We will do this by describing the constitutive and regulative roles that axioms possess with respect to the linguistic practice of mathematics.

This paper relates to a formal statement of the mechanisms that are thought minimally necessary to underpin consciousness. This is expressed in the form of axioms. We deem this to be useful if there is ever to be clarity in answering questions about whether this or the other organism is or is not conscious. As usual, axioms are ways of making formal statements of intuitive beliefs and looking, again formally, at the consequences of such beliefs. The use of (...) this style of exposition does not entail a claim to provide a mathematically rigorous formal deductive system. Conventional mathematical notation is used to achieve clarity, although this is elaborated with natural language in an attempt to reduce terseness. In our view, making the approach axiomatic is synonymous with building clear usable tests for consciousness and is therefore a central feature of the paper. The extended scope of this approach is to lay down some essential properties that should be considered when designing machines that could be said to be conscious. In the broader discussion about the nature of consciousness and its neurological mechanisms, it may seem to some that axiomatisation is premature and continues to beg many questions. However, the approach is meant to be open- ended so that others can build further axiomatic clarifications that address the very large number of questions which, in the search for a formal basis for consciousness, still remain to be answered. Of course, in discussions about consciousness many will also argue that the subject is not one that may ever be formally addressed by means of axioms. The view taken in this paper is 'let's try to do it and see how far it gets'. (shrink)

We will give a simple philosophical "proof" of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpinski and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will (...) in fact show why there must be an infinity of cardinalities between the integers and the reals. We will also show why Martin's Axiom must be false, and we will prove the extension of Fubini's Theorem for Lebesgue measure where joint measurability is not assumed. Following the philosophy--if you reject CH you are only two steps away from rejecting the axiom of choice (AC)--we will point out along the way some extensions of our intuition which contradict AC. (shrink)

Current dynamic epistemic logics often become cumbersome and opaque when common knowledge is added. In this paper we propose new versions that extend the underlying static epistemic language in such a way that dynamic completeness proofs can be obtained by perspicuous reduction axioms.

Current dynamic epistemic logics often become cumbersome and opaque when common knowledge is added. In this paper we propose new versions that extend the underlying static epistemic language in such a way that dynamic completeness proofs can be obtained by perspicuous reduction axioms.

We consider forcing axioms for suitable families of μ‐complete ‐c.c. forcing notions. We show that some form of the condition “ have a in ” is necessary. We also show some versions are really stronger than others.

While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski’s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating “rigorously” with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as “drawn formulas”, and formulas as (...) “written diagrams”, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski’s diagrammatic methods in number theory prompted Hilbert’s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert’s assessment of Minkowski’s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics. (shrink)

Dependencies are identified in two recently proposed first-order axiom systems for plane hyperbolic geometry. Since the dependencies do not specifically concern hyperbolic geometry, our results yield two simpler axiom systems for absolute geometry.

Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive axiomatization allows solutions (...) to geometric problems to be effected as computer programs. I present a formalization of the axiomatization in type theory. This formalization works directly as a computer implementation of geometry. (shrink)

We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable [Formula: see text]-algebras with the metric approximation property and an increasing approximate identity of projections.

We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable [Formula: see text]-algebras with the metric approximation property and an increasing approximate identity of projections.

The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...) the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences — many indispensable, some startling — and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. (shrink)

The axiom of reducibility plays an important role in the logic of Principia Mathematica, but has generally been condemned as an ad hoc non-logical axiom which was added simply because the ramified type theory without it would not yield all the required theorems. In this paper I examine the status of the axiom of reducibility. Whether the axiom can plausibly be included as a logical axiom will depend in no small part on the understanding of propositional functions. If we understand (...) propositional functions as constructions of the mind, it is clear that the axiom is clearly not a logical axiom and in fact makes an implausible claim. I look at two other ways of understanding propositional functions, a nominalist interpretation along the lines of Landini and a realist interpretation along the lines of Linsky and Mares. I argue that while on either of these interpretations it is not easy to see the axiom as a non-logical claim about the world, there are also appear to be difficulties in accepting it as a purely logical axiom. (shrink)

We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.

There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some (...) important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down. (shrink)

Ce texte a déjà paru dans la mineure « Rythmanalyses » de la revue Multitudes, n° 46, 2011. Nous remercions Yves Citton et la revue Multitudes de nous avoir autorisé à le reproduire ici. Les propositions rythmologiques qui suivent sont formulées de manière dogmatique, en une série d'axiomes et de règles. Elles prolongent les deux axiomes articulés par Frédéric Bisson dans « Ainsi marche Anna Cruz ». Une telle axiomatique, ouverte à discussion, a valeur d'orientation générale - Pour une éthique (...) et une politique du rythme – Nouvel article. (shrink)

In this paper I argue for the view that the axioms of ZF are analytic truths of a particular concept of set. By this I mean that these axioms are true by virtue only of the meaning attached to this concept, and, moreover, can be derived from it. Although I assume that the object of ZF is a concept of set, I refrain from asserting either its independent existence, or its dependence on subjectivity. All I presuppose is that (...) this concept is given to us with a certain sense as the objective focus of a phenomenologically reduced intentional experience. The concept of set that ZF describes, I claim, is that of a multiplicity of coexisting elements that can, as a consequence, be a member of another multiplicity. A set is conceived as a quantitatively determined collection of objects that is, by necessity, ontologically dependent on its elements, which, on the other hand, must exist independently of it. A close scrutiny of the essential characters of this conception seems to be sufficient to ground the set-theoretic hierarchy and the axioms of ZF. (shrink)

In this paper I argue for the view that the axioms of ZF are analytic truths of a particular concept of set. By this I mean that these axioms are true by virtue only of the meaning attached to this concept, and, moreover, can be derived from it. Although I assume that the object of ZF is a concept of set, I refrain from asserting either its independent existence, or its dependence on subjectivity. All I presuppose is that (...) this concept is given to us with a certain sense as the objective focus of a ”phenomenologically reduced“ intentional experience.The concept of set that ZF describes, I claim, is that of a multiplicity of coexisting elements that can, as a consequence, be a member of another multiplicity. A set is conceived as a quantitatively determined collection of objects that is, by necessity, ontologically dependent on its elements, which, on the other hand, must exist independently of it. A close scrutiny of the essential characters of this conception seems to be sufficient to ground the set-theoretic hierarchy and the axioms of ZF. (shrink)

The method of morphisms is a well-known application of Dialectica categories to set theory. 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 of either the Axiom of Choice or Partition Principle —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. (shrink)

A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.

On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. (...) The distinctions introduced aim at clarifying discussions in philosophy of mathematics and contributing towards a more refined view of mathematical practice. (shrink)

We present a new axiomatization of classical mereology in which the three components of the theory—ordering, composition, and decomposition prin-ciples—are neatly separated. The equivalence of our axiom system with other, more familiar systems is established by purely deductive methods, along with additional results on the relative strengths of the composition and decomposition axioms of each theory.

In [10], we have shown that the statement that all ∑ 1 1 partitions are Ramsey is deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition,but the reversal needs П 1 1 - CA 0 rather than ATR 0 . By contrast, we show in this paper that the statement that all ∑ 0 2 games are determinate is also deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition, but the (...) reversal is provable even in ACA 0 . These results illuminate the substantial differences among lightface theorems which can not be observed in boldface. (shrink)

We give a new characterization of the class of completely representable cylindric algebras of dimension 2 #lt; n w via special neat embeddings. We prove an independence result connecting cylindric algebra to Martin''s axiom. Finally we apply our results to finite-variable first order logic showing that Henkin and Orey''s omitting types theorem fails for L n, the first order logic restricted to the first n variables when 2 #lt; n#lt;w. L n has been recently (and quite extensively) studied as a (...) many-dimensional modal logic. (shrink)

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a (...) new account of the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments. (shrink)

This paper discusses a sequence of extensions ofNFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].The original theoryNFof Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] thatNFdisproves Choice (and so proves Infinity). Even if one assumes the consistency ofNF, one is hampered by the lack of powerful methods for proofs of consistency and (...) independence such as are available for use withZFC; very clever work has been done with permutation methods, starting with [18] and [5], and exemplified more recently by [14], but permutation methods can only be applied to show the consistency or independence of unstratified sentences (see the definition ofNFUbelow for a definition of stratification). For example, there is no method available to determine whether the assertion “the continuum can be well-ordered” is consistent with or independent ofNF. There is one substantial independence result for an assertion with nontrivial stratified consequences, using metamathematical methods: this is Orey's proof of the independence of the Axiom of Counting fromNF(see below for a statement of this axiom).We mention these difficulties only to reassure the reader of their irrelevance to the present work. Jensen's modification of “New Foundations” (in [13]), which was to restrict extensionality to sets, allowing many non-sets (urelements) with no elements, has almost magical effects. (shrink)