This paper is a systematic exploration of non-wellfounded mereology. Motivations and applications suggested in the literature are considered. Some are exotic like Borges’ Aleph, and the Trinity; other examples are less so, like time traveling bricks, and even Geach’s Tibbles the Cat. The authors point out that the transitivity of non-wellfounded parthood is inconsistent with extensionality. A non-wellfounded mereology is developed with careful consideration paid to rival notions of supplementation and fusion. Two equivalent axiomatizations are given, and (...) are compared to classical mereology. We provide a class of models with respect to which the non-wellfounded mereology is sound and complete. (shrink)

In this paper we present and study a categorical formulation of the W-types of Martin-Löf. These are essentially free term algebras where the operations may have finite or infinite arity. It is shown that W-types are preserved under the construction of sheaves and Artin gluing. In the proofs we avoid using impredicative or nonconstructive principles.

If there is a good wellordering of the reals, then there is a wellfounded relation for which the comparison relation is not projective.

We show that it is relatively consistent with ZFC that there is a projective wellfounded relation with rank higher than all projective prewellorderings.

We study the non-wellfounded sets as fixed points of substitution. For example, we show that ZFA implies that every function has a fixed point. As a corollary we determine for which functions f there is a function g such that . We also present a classification of non-wellfounded sets according to their branching structure.

The paper explores the proof theory of non-wellfounded mereology with binary fusions and provides a cut-free sequent calculus equivalent to the standard axiomatic system.

We prove that if I is a partially ordered set in a countable transitive model M of ZFC then M can be extended by a generic sequence of reals a i , i ∈ I, such that ℵ M 1 is preserved and every a i is Sacks generic over $\mathfrak{M}[\langle \mathbf{a}_j: j . The structure of the degrees of M-constructibility of reals in the extension is investigated. As applications of the methods involved, we define a cardinal invariant to distinguish (...) product and iterated Sacks extensions, and give a short proof of a theorem (by Budinas) that in ω 2 -iterated Sacks extension of L the Burgess selection principle for analytic equivalence relations holds. (shrink)

In a previous article on cosmological arguments, I have put forward a few examples of complete infinite and circular explanations, and argued that complete non-wellfounded explanations such as these might explain the present state of the world better than their well-founded theistic counterparts (Billon, 2021). Although my aim was broader, the examples I gave there implied merely causal explanations. In this article, I would like to do three things: • Specify some general informative conditions for complete and incomplete non- (...) class='Hi'>wellfounded causal explanations that can be used to assess candidate explanations and to generate new examples of complete non-wellfounded explanations. • Show that these conditions, which concern chains of causal explanations, easily generalize to chains of metaphysical, grounding explanations and even to chains involving other “determination relations” such as supervenience. • Apply these general conditions to the recent debates against the existence of nonwellfounded chains of grounds and show, with a couple of precise examples, that the latter can be complete, and that just like in the case of causal explanations, non-wellfoundedness can in fact be an aset rather than a liability. (shrink)

If there is a good $\Delta^1_3$ wellordering of the reals, then there is a $\Pi^1_1$ wellfounded relation for which the comparison relation is not projective.

Let ZFB be ZF + "every set is the same size as a wellfounded set". Then the following are true. Every sentence true in every (Rieger-Bernays) permutation model of a model of ZF is a theorem of ZFB. (i.e.. ZFB is the theory of Rieger-Bernays permutation models of models of ZF) ZF and ZFAFA are both extensions of ZFB conservative for stratified formulæ. The class of models of ZFB is closed under creation of Rieger-Bernays permutation models.

This is a review of Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena, written by Jon Barwise and Lawrence Moss and published by CSLI Publications in 1996.

We approach the virtual reality phenomenon by studying its relationship to set theory. This approach offers a characterization of virtual reality in set theoretic terms, and we investigate the case where this is done using the wellfoundedness property. Our hypothesis is that non-wellfounded sets (so-called hypersets) give rise to a different quality of virtual reality than do familiar wellfounded sets. To elaborate this hypothesis, we describe virtual reality through Sommerhoff’s categories of first- and second-order self-awareness; introduced as necessary (...) conditions for consciousness in terms of higher cognitive functions. We then propose a representation of first- and second-order self-awareness through sets, and assume that these sets, which we call events, originally form a collection of wellfounded sets. Strong virtual reality characterizes virtual reality environments which have the limited capacity to create only events associated with wellfounded sets. In contrast, the logically weaker and more general concept of weak virtual reality characterizes collections of virtual reality mediated events altogether forming an entirety larger than any collection of wellfounded sets. By giving reference to Aczel’s hyperset theory we indicate that this definition is not empty because hypersets encompass wellfounded sets already. Moreover, we argue that weak virtual reality could be realized in human history through continued progress in computer technology. Finally, within a more general framework, we use Baltag’s structural theory of sets (STS) to show that within this hyperset theory Sommerhoff’s first- and second-order self-awareness as well as both concepts of virtual reality admit a consistent mathematical representation. To illustrate our ideas, several examples and heuristic arguments are discussed. (shrink)

The iterative conception of set commonly is regarded as supporting the axioms of Zermelo-Fraenkel set theory (ZF). This paper presents a modified version of the iterative conception of set and explores the consequences of that modified version for set theory. The modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterative conception of set supports the axioms of Quine's set theory NF.

The hypothetical notion of consequence is normally understood as the transmission of a categorical notion from premisses to conclusion. In model-theoretic semantics this categorical notion is 'truth', in standard proof-theoretic semantics it is 'canonical provability'. Three underlying dogmas, (I) the priority of the categorical over the hypothetical, (II) the transmission view of consequence, and (III) the identification of consequence and correctness of inference are criticized from an alternative view of proof-theoretic semantics. It is argued that consequence is a basic semantical (...) concept which is directly governed by elementary reasoning principles such as definitional closure and definitional reflection, and not reduced to a categorical concept. This understanding of consequence allows in particular to deal with non-wellfounded phenomena as they arise from circular definitions. (shrink)

We analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form$\Sigma _{1 + \alpha + 3}^0 $, forα<ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to requireα+ 1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of weak reflection principles, Π1-RAPα, whose consistency strength corresponds exactly to the logical strength of${\rm{\Sigma }}_{1 + \alpha + (...) 3}^0 $determinacy, for$\alpha < \omega _1^{CK} $. This yields a characterization of the levels ofLby or at which winning strategies in these games must be constructed. Whenα= 0, we have the following concise result: The leastθso that all winning strategies in${\rm{\Sigma }}_4^0 $games belong toLθ+1is the least so that$L_\theta \models {\rm{``}}{\cal P}\left$exists, and all wellfounded trees are ranked”. (shrink)

Ross Cameron (2022) argues that metaphysical infinitists should reject the generally accepted idea that metaphysical determination relations back metaphysical explanations. Otherwise it won’t be possible for them to come up with successful explanations for the existence of dependent entities in non-wellfounded chains of dependence. I argue that his argument suffers from what he calls the finitist dogma, although indirectly so. However, there is a better way of motivating Cameron’s conclusion. Assuming Cameron’s principle of Essence, explanations for the existence of (...) dependent entities turn out to be circular if determination relations back explanations. This latter argument provides a stronger case as it puts the foundationalist under significant pressure, besides putting the infinitist under some pressure, to deny the idea that determination relations back explanations. (shrink)

A semantics is presented for belief-revision in the face of common announcements to a group of agents that have beliefs about each other's beliefs. The semantics is based on the idea that possible worlds can be viewed as having an internal structure, representing the belief independent features of the world, and the respective belief states of the agents in a modular fashion. Modularity guarantees that changing one aspect of the world (a belief independent feature or a belief state) has no (...) effect on any other aspect of the world. This allows us to employ an AGM-style selection function to represent revision. The semantics is given a complete axiomatisation (identical to the axiomatisation found by Gerbrandy and Groeneveld for a semantics based on non-wellfounded set theory) for the special case of expansion. (shrink)

We introduce a system of simply typed lambda terms and show that a rather comprehensive class of recursion equations on streams or non-wellfounded trees can be solved in our system. Moreover certain conditions are presented which guarantee that the defined functionals are primitive recursive. As a major example we give a co-recursive treatment of Mints’ continuous cut-elimination operator.

The main claim of this paper is that the boundary between scientific and non scientific knowledge does exist -- which means several things. First, it's not the case that anything goes: some irrationalists have been mistaken into acceptance of that wrong conclusion because they have remarked that, however the boundary might be drawn, some important scientific developments would fall afoul of the standards entitling a research practice to count as scientific. Second, the boundary is not an imaginary one, that is (...) to say besides what is scientific and what is unscientific there also is what lies at the boundary, certain research practices which are neither wholly scientific nor fully unscientific. Third, studying what is science is itself a kind of research belonging to the boundary, since the methods available in that research are not as strictly rigorous as those used in science proper; in fact, all of philosophy is included in the boundary in question. Fourth, the boundary (and in fact science itself) displays a characteristic structure pertaining to what are by now usually called «non wellfounded sets» -- sets, that is, which are somehow or other involved in themselves, whether as members, or as members of members or so on; the significance of the last thesis is that the best way of approaching philosophy of science is not standard set theory, but theories allowing non wellfounded sets are preferable. Fifth, and last, admission of the boundary's existence compels us to go beyond standard classical logic and to look for a more suitable logic, as for instance some kind of fuzzy paraconsistent logic. (shrink)

A semantics is presented for belief revision in the face of common announcements to a group of agents that have beliefs about each other’s beliefs. The semantics is based on the idea that possible worlds can be viewed as having an internal-structure, representing the belief independent features of the world, and the respective belief states of the agents in a modular fashion. Modularity guarantees that changing one aspect of the world (a belief independent feature or a belief state) has no (...) effect on any other aspect of the world. This allows us to employ an AGM-style selection function to represent revision. The semantics is given a complete axiomatisation (identical to the axiomatisation found by Gerbrandy and Groeneveld for a semantics based on non-wellfounded set theory) for the special case of expansion. (shrink)

We show that Mitchell order on downward closed extenders below rank-to-rank type is wellfounded.

Section 1 is devoted to the study of countable recursively saturated models with an automorphism moving every non-algebraic point. We show that every countable theory has such a model and exhibit necessary and sufficient conditions for the existence of automorphisms moving all non-algebraic points. Furthermore we show that there are many complete theories with the property that every countable recursively saturated model has such an automorphism. In Section 2 we apply our main theorem from Section 1 to models of Quine's (...) set theory New Foundations (NF) to answer an old consistency question. If NF is consistent, then it has a model in which the standard natural numbers are a definable subclass N of the model's set of internal natural numbers Nn. In addition, in this model the class of wellfounded sets is exactly $\bigcup_{n\in \mathbb{N}}\mathscr{P}^n(\varnothing)$. (shrink)

The satisfiability problem for the two-variable fragment of first-order logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wellfounded relations. It is shown that FO 2 over ordered, respectively wellordered, domains or in the presence of one well-founded relation, is decidable for satisfiability as well as for finite satisfiability. Actually the complexity of these decision problems is (...) essentially the same as for plain unconstrained FO 2 , namely non-deterministic exponential time. In contrast FO 2 becomes undecidable for satisfiability and for finite satisfiability, if a sufficiently large number of predicates are required to be interpreted as orderings, wellorderings, or as arbitrary wellfounded relations. This undecidability result also entails the undecidability of the natural common extension of FO 2 and computation tree logic CTL. (shrink)

It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the T-function which is peculiar to NF turn out to be equivalent to the truth-in-certain-permutation-models of assertions which have perfectly sensible ZF-style meanings, such as: the existence of wellfounded sets of great size or rank, or the nonexistence of small counterexamples to the wellfoundedness of ∈. Everything here holds also for NFU if the permutations are taken to fix all urelemente.

A theory of two-sided containers, denoted ZF2, is introduced. This theory is then shown to be synonymous to ZF in the sense of Visser (2006), via an interpretation involving Quine pairs. Several subtheories of ZF2, and their relationships with ZF, are also examined. We include a short discussion of permutation models (in the sense of Rieger–Bernays) over ZF2. We close with highlighting some areas for future research, mostly motivated by the need to understand non-wellfounded games.

In [10] Friedman showed that is a conservative extension of <ε0for-sentences wherei= min, i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with instead of. Feferman and Sieg established in [9] the conservativeness of over <ε0for-sentences for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem (...) functionals.In this paper we study the same choice principles in the presence of theBar Rule, which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result characterizes + as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences. Forn= 0 this has been proved by Feferman in the form that + is a conservative extension of <Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ + and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below. (shrink)

A foundational algebra ( , f, ) consists of a hemimorphism f on a Boolean algebra with a greatest solution to the condition f(x). The quasi-variety of foundational algebras has a decidable equational theory, and generates the same variety as the complex algebras of structures (X, R), where f is given by R-images and is the non-wellfounded part of binary relation R.The corresponding results hold for algebras satisfying =0, with respect to complex algebras of wellfounded binary relations. These (...) algebras, however, generate the variety of all ( ,f) with f a hemimorphism on ). (shrink)

The paper addresses the widely held position that the Third Man regress in the Parmenides is caused at least in part by the self-predicational aspect of Plato's Ideas. I offer a critique of the logic behind this type of interpretation, and argue that if the Ideas are construed as genuinely applying to themselves, then the regress is dissolved. Furthermore, such an interpretation can be made technically precise by modeling Platonic Universals as non-wellfounded sets. This provides a solution to the (...) Third Man regress, and allows a consistent reading of both self-predication and the singularity of the respective Forms. (shrink)

We study a partial order on countably complete ultrafilters introduced by Ketonen [2] as a generalization of the Mitchell order. The following are our main results: the order is wellfounded; its linearity is equivalent to the Ultrapower Axiom, a principle introduced in the author’s dissertation [1]; finally, assuming the Ultrapower Axiom, the Ketonen order coincides with Lipschitz reducibility in the sense of generalized descriptive set theory.

With less than 0# two generic extensions ofL are identified: one in which ${\aleph_1}$ , and the other ${\aleph_2}$ , is almost precipitous. This improves the consistency strength upper bound of almost precipitousness obtained in Gitik M, Magidor M (On partialy wellfounded generic ultrapowers, in Pillars of Computer Science, 2010), and answers some questions raised there. Also, main results of Gitik (On normal precipitous ideals, 2010), are generalized—assumptions on precipitousness are replaced by those on ∞-semi precipitousness. As an application (...) it is shown that if δ is a Woodin cardinal and there is an ${f:\omega_1 \to \omega_1}$ with ${\|f\|=\omega_2}$ , then after ${Col(\aleph_2,\delta)}$ there is a normal precipitous ideal over ${\aleph_1}$ . The existence of a pseudo-precipitous ideal over a successor cardinal is shown to give an inner model with a strong cardinal. (shrink)

We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. As an application, we introduce a new method to force (weak) measurability of all definable sets with respect to a certain (non-ccc) ideal.

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 consider foundational questions related to the definition of functions by corecursion. This method is especially suited to functions into the greatest fixed point of some monotone operator, and it is most applicable in the context of non-wellfounded sets. We review the work on the Special Final Coalgebra Theorem of Aczel [1] and the Corecursion Theorem of Barwise and Moss [4]. We offer a condition weaker than Aczel's condition of uniformity on maps, and then we prove a result relating (...) the operators satisfying the new condition to the smooth operators of [4]. (shrink)

We present sound and complete sequent calculi for the modal mu-calculus with converse modalities, aka two-way modal mu-calculus. Notably, we introduce a cyclic proof system wherein proofs can be represented as finite trees with back-edges, i.e., finite graphs. The sequent calculi incorporate ordinal annotations and structural rules for managing them. Soundness is proved with relative ease as is the case for the modal mu-calculus with explicit ordinals. The main ingredients in the proof of completeness are isolating a class of non- (...) class='Hi'>wellfounded proofs with sequents of bounded size, called slim proofs, and a counter-model construction that shows slimness suffices to capture all validities. Slim proofs are further transformed into cyclic proofs by means of re-assigning ordinal annotations. (shrink)

Building on previous work by Mints, Buchholz and Schwichtenberg, a simplified version of continuous normalization for the untyped λ-calculus and Gödel’s is presented and analysed in the coalgebraic framework of non-wellfounded terms with so-called repetition constructors.The primitive recursive normalization function is uniformly continuous w.r.t. the natural metric on non-wellfounded terms. Furthermore, the number of necessary repetition constructors is locally related to the number of reduction steps needed to reach the normal form and its size.It is also shown how (...) continuous normal forms relate to derivations of strong normalizability in the typed λ-calculus and how this leads to new bounds for the sum of the height of the reduction tree and the size of the normal form.Finally, the methods are extended to an infinitary λ-calculus with ω-rule and permutative conversions and this is used to derive a strong form of normalization for an iterative version of Gödel’s system , leading to a value table semantics for number-theoretic functions. (shrink)

A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of a greatest fixpoint formula of the modal μ-calculus. This generalizes the standard result in the literature where a finitary modal characterization is provided only for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and non-well-founded sets.

This paper establishes a conjecture of Steel [7] regarding the structure of elementary embeddings from a level of the cumulative hierarchy into itself. Steel’s question is related to the Mitchell order on these embeddings, studied in [5] and [7]. Although this order is known to be illfounded, Steel conjectured that it has certain large wellfounded suborders, which is what we establish. The proof relies on a simple and general analysis of the much broader class of extender embeddings and a (...) variant of the Mitchell order called the internal relation. (shrink)

In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially different universes of such sets. The unisets of the first universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the sets of the AFA-universe. We will have a closer look into the connection of the iterated circular multisets and arbitrary trees. RID=""ID="" Mathematics Subject Classification (2000): 03B45, 03E65, 03E70, 18A15, 18A22, (...) 18B05, 68Q85 RID=""ID="" Key words or phrases: Multiset – Non-wellfounded set – Scott-universe – AFA – Coalgebra – Modal logic – Graded modalities. (shrink)

A complete list of Finsler, Scott and Boffa sets whose transitive closures contain 1, 2 and 3 elements is given. An algorithm for deciding the identity of hereditarily finite Scott sets is presented. Anti-well-founded sets, i. e., non-well-founded sets whose all maximal ∈-paths are circular, are studied. For example they form transitive inner models of ZFC minus foundation and empty set, and they include uncountably many hereditarily finite awf sets. A complete list of Finsler and Boffa awf sets with 2 (...) and 3 elements in their transitive closure is given. Next the existence of infinite descending ∈-sequences in Aczel universes is shown. Finally a theorem of Ballard and Hrbáček concerning nonstandard Boffa universes of sets is considerably extended. (shrink)

Frege’s Basic Law V and its successor, Boolos’s New V, are axioms postulating abstraction operators: mappings from the power set of the domain into the domain. Basic Law V proved inconsistent. New V, however, naturally interprets large parts of second-order ZFC via a construction discovered by Boolos in 1989. This paper situates these classic findings about abstraction operators within the general theory of F-algebras and coalgebras. In particular, we show how Boolos’s construction amounts to identifying an initial F-algebra in a (...) certain category, we identify a natural coalgebraic dual to Boolos’s axiom which naturally interprets large parts of Aczel’s non-well-founded set theory via the construction of a certain terminal F-coalgebra, and we suggest a coalgebraic way forward for an abstraction-theoretic axiomatization of the real numbers. (shrink)