Let M be a model of ZFAC (ZFC modified to allow a set of atoms), and let N be an inner model with the same set of atoms and the same pure sets (sets with no atoms in their transitive closure) as M. We show that N is a permutation submodel of M if and only if N satisfies the principle SVC (Small Violations of Choice), a weak form of the axiom of choice which says that in some sense, (...) all violations of choice are localized in a set. A special case is considered in which there exists an SVC witness which satisfies a certain homogeneity condition. (shrink)

We show that if N and M are transitive models of ZFA such that N M, N and M have the same kernel and same set of atoms, and M AC, then N is a Fraenkel-Mostowski-Specker (FMS) submodel of M if and only if M is a generic extension of N by some almost homogeneous notion of forcing. We also develop a slightly modified notion of FMS submodels to characterize the case where M is a generic extension of N (...) not necessarily by an almost homogeneous notion of forcing. (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.

This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Restall, J Philos Logic 22(5):481–511, 1993 ) concerning the modelling conditions for the axioms of assertion A → (( A → B ) → B ) (there called c 6) and permutation ( A → ( B → C )) → ( B → ( A → C )) (there called c 7). We show that the modelling conditions for assertion and (...) class='Hi'>permutation proposed in ‘Simplified Semantics’ overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the “base point” is inconsistent. This problem is not restricted to ‘Simplified Semantics.’ The techniques of that paper are used in Graham Priest’s textbook An Introduction to Non-Classical Logic (Priest, 2001 ), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose two different corrections. (shrink)

The purpose of this article is to give a general overview of permutations in physics, particularly the symmetry of theories under permutations. Particular attention is paid to classical mechanics, classical statistical mechanics and quantum mechanics. There are two recurring themes: (i) the metaphysical dispute between haecceitism and anti-haecceitism, and the extent to which this dispute may be settled empirically; and relatedly, (ii) the way in which elementary systems are individuated in a theory's formalism, either primitively or in terms of the (...) properties and relations those systems are represented as bearing. Section 1 introduces permutations and provides a brief outline of the symmetric and braid groups. Section 2 discusses permutations in the general setting provided by model theory, in particular providing some definitions and elementary results regarding the permutability and indiscernibility of objects. Section 3 lays some philosophical groundwork for later sections, in particular articulated the distinction between haecceitism and anti-haecceitism and the distinction between transcendental and qualitative individuation. Section 4 addresses classical mechanics and introduces the procedure of quotienting, under which permutable states are identified. Section 5 addresses classical statistical mechanics, and outlines a number of equivalent ways to implement permutation invariance. I also briefly outline how particles may be qualitatively individuated in this framework. Section 6 addresses quantum mechanics. This contains an outline of: the representation theory of the symmetric groups; the topological approach to quantum statistics, in which the braid groups become relevant; and a brief proposal for qualitatively individuating quantum particles, and its implications for entanglement. Section 7 concludes with a discussion of equilibrium ensembles in the classical and quantum theories under permutation invariance. A (much) shorter version of this paper was published as a chapter in E. Knox & A. Wilson (eds), the Routledge Companion to Philosophy of Physics (Routledge, 2021), pp. 578-594. (shrink)

We give a derivation of exclusion principles for the elementary particles of the standard model, using simple mathematical principles arising from a set theory of identical particles. We apply the theory of permutation group actions, stating some theorems which are proven elsewhere, and interpreting the results as a heuristic derivation of Pauli's Exclusion Principle (PEP) which dictates the formation of elements in the periodic table and the stability of matter, and also a derivation of quark confinement. We arrive at (...) these properties by using a symmetry property of collections of the particles themselves as compared for example to the symmetry property of their wave function under interchange of two particles. (shrink)

We extend Shelah's first many model result to show that an unstable theory has 2 κ many non-permutation group isomorphic models of size κ, where κ is an uncountable regular cardinal.

Adopting fair value measurement may bring more earnings fluctuations and induce irrational psychology and radical financing behavior of managers. Based on behavioral corporate governance theory, using the sample of Chinese A-share nonfinancial listed companies during 2007–2017, this paper empirically examines the regulatory effect of fair value measurement, that is, whether fair value measurement affects the company's financing decisions when managers have irrational psychological characteristics, i.e., overconfidence. The study found that overconfident managers of the company that have fair value measurement assets (...) will be more aggressive for debt decisions, indicating that fair value measurement has a positively regulatory effect on overconfident managers. (shrink)

We consider the possible cardinalities of the following three cardinal invariants which are related to the permutation group on the set of natural numbers: the least cardinal number of maximal cofinitary permutation groups; the least cardinal number of maximal almost disjoint permutation families; the cofinality of the permutation group on the set of natural numbers.We show that it is consistent with that ; in fact we show that in the Miller model.

The notions of permutable and weak-permutable convergence of a series|$\sum _{n=1}^{\infty }a_{n}$|of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD-|$\mathbb {N}$|implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation (...) theorem for series holds but BD-|$\mathbb{N}$|does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD-|$\mathbb {N}$|. We show that this is the case when the property is weak-permutable convergence. (shrink)

The idea that the world is made of particles — little discrete, interacting objects that compose the material bodies of everyday experience — is a durable one. Following the advent of quantum theory, the idea was revised but not abandoned. It remains manifest in the explanatory language of physics, chemistry, and molecular biology. Aside from its durability, there is good reason for the scientific realist to embrace the particle interpretation: such a view can account for the prominent epistemic fact that (...) only limited knowledge of a portion of the material universe is needed in order to make reliable predictions about that portion. Thus, particle interpretations can support an abductive argument from the epistemic facts in favor of a realist reading of physical theory. However, any particle interpretation with this property is untenable. The empirical adequacy of modern particle theories requires adoption of a postulate known as permutation invariance — the claim that interchanging the role of two particles of the same kind in a dynamical state description results in a description of the identical state. It is the central claim of this essay that PI is incompatible with any particle interpretation strong enough to account for the epistemic facts. This incompatibility extends across all physical theories. To frame and motivate the inconsistency argument, I begin by fixing the relevant notion of particle. To single out those accounts of greatest appeal to the realist, I develop the logically weakest particle ontology that entails the epistemic fact that the world is piecewise predictable, an ontology I call ‘minimal atomism’. The entire series of scientific conceptions of the particle, from Newton’s mechanically interacting corpuscles to the ‘centers of force’ in classical field theories, all comport with MA. As long as PI is left out, even quantum mechanics can be viewed this way. To assess the impact of PI on this picture, I present a framework for rigorously connecting interpretations to physical theories. In particular, I represent MA as a set of formal conditions on the models of physical theories, the mathematical structures taken to represent states of the world. I also formulate PI — originally introduced as a postulate of non-relativistic quantum mechanics — in theory independent terms. With all of these pieces in hand, I am then able to present a proof of the inconsistency of PI and MA. In the second part of the essay, I survey responses to the inconsistency result open to the scientific realist. The two most plausible approaches involve abandoning particles in one way or another. The first alternative interpretation considered takes the property bearing objects of the world to be regions of space rather than particles. In this view, the properties once attributed to particles in quantum states are attributed instead to one or more regions of space. PI no longer obtains in this case, at least not as a statement about the permutation symmetry of property bearers. Rather, the new interpretation naturally imposes an analogous constraint on quantum states. The second major approach to evading the inconsistency result is to dispense with objects altogether. This is the recommendation of so-called ‘Ontic Structural Realism’. The central OSR thesis is that structure rather than entities are the basic ontological components of the world. OSR is intended to embrace the ‘miracle’ argument in favor of scientific realism while avoiding the pessimistic meta-induction. I demonstrate that one principal motivation for OSR based on the under-determination of interpretations in QM is actually dissolved by the incompatibility result. At the same time, I suggest reasons to think that OSR fares no better with respect to the pessimistic meta-induction than traditional realism does. Thus, while PI and MA may be incompatible, object ontologies remain the best option for the realist. (shrink)

By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued,pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos E of M is said to represent M if it is complete and its cumulative hierarchy, as (...) defined by Fourman and Hayashi, coincides with the usual cumulative hierarchy of M. We show that, although M need not be a complete topos, it has a smallest complete representing subtopos, and we describe this subtopos in terms of definability in M. We characterize, again in terms of definability, those models M whose smallest representing topos is a Grothendieck topos. Finally, we discuss the extent to which a model can be reconstructed when its smallest representing topos is given. (shrink)

Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical (...) challenges. (shrink)

Due to the NP-hard nature, the permutation flowshop scheduling problem is a fundamental issue for Industry 4.0, especially under higher productivity, efficiency, and self-managing systems. This paper proposes an improved genetic-shuffled frog-leaping algorithm to solve the permutation flowshop scheduling problem. In the proposed IGSFLA, the optimal initial frog in the initialized group is generated according to the heuristic optimal-insert method with fitness constrain. The crossover mechanism is applied to both the subgroup and the global group to avoid the (...) local optimal solutions and accelerate the evolution. To evolve the frogs with the same optimal fitness more outstanding, the disturbance mechanism is applied to obtain the optimal frog of the whole group at the initialization step and the optimal frog of the subgroup at the searching step. The mathematical model of PFSSP is established with the minimum production cycle as the objective function, the fitness of frog is given, and the IGSFLA-based PFSSP is proposed. Experimental results have been given and analyzed, showing that IGSFLA not only provides the optimal scheduling performance but also converges effectively. (shrink)

A varietyV satisfies a strong Malcev condition ∃f1,…, ∃fnθ where θ is a conjunction of equations in the function variablesf1, …,fnand the individual variablesx1, …,xm, if there are polynomial symbolsp1, …,pnin the language ofVsuch that ∀x1, …,xmθ is a law ofV. Thus a strong Malcev condition involves restricted second order quantification of a strange sort. The quantification is restricted to functions which are “polynomially definable”. This notion was introduced by Malcev [6] who used it to describe those varieties all of (...) whose members have permutable congruence relations. The general formal definition of Malcev conditions is due to Grätzer [1]. Since then and especially since Jónsson's [3] characterization of varieties with distributive congruences there has been extensive study of strong Malcev conditions and the related concepts: Malcev conditions and weak Malcev conditions.In [9], Taylor gives necessary and sufficient semantic conditions for a class of varieties to be defined by a Malcev condition. A key to the proof is the translation of the restricted second order concepts into first order concepts in a certain many sorted language. In this paper we show that, given this translation, Taylor's theorem is an easy consequence of a result of Tarski [8] and the standard preservation theorems of first order logic. (shrink)

Applying Weglorz' mode s of set theory without the axiom of choice, we investigate Arrow-type social we fare functions for infinite societies with restricted coalition algebras. We show that there is a reasonable, nondictatorial social welfare function satisfying “finite discrimination”, if and only if in Weglorz' mode there is a free ultrafilter on a set representing the individuals.

This chapter concentrates on the version of Putnam's argument set forth in his Reason, Truth and History. It explains how, in general terms, that argument is best conceived as working. Cursory inspection of Putnam's overall dialectic reveals it to incorporate three sub‐arguments, collectively designed to show that the metaphysical realist confronts an insuperable problem over explaining how our words may possess determinate reference. The chapter considers Putnam's version of the Permutation Argument, aimed at showing that reference cannot be determined (...) by fixing the truth‐conditions of whole sentences. It then reviews Putnam's 'just more theory' argument, designed to show that the metaphysical realist cannot rescue the situation by appeal to causal or other natural connections between our words and the world. The chapter considers how the considerations adduced by Hilary Putnam might be seen as an argument telling selectively against metaphysical realism. (shrink)

This chapter sets out the relevant core of Putnam’s case. Section 3.1 extracts three arguments from Putnam’s writings: the Arguments from Cardinality, Completeness, and Permutation. Of these, section 3.2 argues that only the second is of direct relevance. Section 3.3 examines attempts to frame constraints based on causal and psycho-behavioural reductions of reference. Section 3.4 investigates the Translational Reference Constraint, a constraint on reference which does not rely on a reduction of reference but makes essential use of translation to (...) sort out the models which get reference right. The claims made in this section, however, require foundation in a theory of translation, sufficient to sustain the assumptions it makes about that controversial and opaque notion. This foundation is supplied in section 3.5, whose general tenor is Davidsonian, its key notion being that of a ‘hermeneutic theory’, i.e., a Davidsonian theory of interpretation cast into model-theoretic terms. With Translational Truth Constraint now identified as the most fundamental constraint on intendedness, it remains to see if it will suffice to rule out as unintended all the models of ideal theory whose existence the Completeness Theorem guarantees. The issue is examined in section 3.6. (shrink)

The main features of the tetron model of elementary particles are discussed in the light of recent developments, in particular the formation of strong and electroweak vector bosons and a microscopic understanding of how the observed tetrahedral symmetry of the fermion spectrum may arise.

Hilary Putnam’s BIV argument first occurred to him when ‘thinking about a theorem in modern logic, the “Skolem–Löwenheim Theorem”’ (Putnam 1981: 7). One of my aims in this paper is to explore the connection between the argument and the Theorem. But I also want to draw some further connections. In particular, I think that Putnam’s BIV argument provides us with an impressively versatile template for dealing with sceptical challenges. Indeed, this template allows us to unify some of Putnam’s most enduring (...) contributions to the realism/antirealism debate: his discussions of brains-in-vats, of Skolem’s Paradox, and of permutations. In all three cases, we have an argument which does not merely defeat the sceptic; it also shows us that we must reject some prima facie plausible philosophical picture. (shrink)

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that ifMis a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2be countable arithmetically saturated (...) class='Hi'>models of Peano Arithmetic such that Aut(M1)≅ Aut(M2).ThenSSy(M1) = SSy(M2).We show that ifMis a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1,M2be countable arithmetically saturated models of Peano Arithmetic such thatAut(M1) ≅ Aut(M2).Then for every n<ωHere RT2nis Infinite Ramsey's Theorem stating that every 2-coloring of [ω]nhas an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic. (shrink)

For \(n\in \omega \), the weak choice principle \(\textrm{RC}_n\) is defined as follows: _For every infinite set_ _X_ _there is an infinite subset_ \(Y\subseteq X\) _with a choice function on_ \([Y]^n:=\{z\subseteq Y:|z|=n\}\). The choice principle \(\textrm{C}_n^-\) states the following: _For every infinite family of_ _n_-_element sets, there is an infinite subfamily_ \({\mathcal {G}}\subseteq {\mathcal {F}}\) _with a choice function._ The choice principles \(\textrm{LOC}_n^-\) and \(\textrm{WOC}_n^-\) are the same as \(\textrm{C}_n^-\), but we assume that the family \({\mathcal {F}}\) is linearly orderable (...) (for \(\textrm{LOC}_n^-\) ) or well-orderable (for \(\textrm{WOC}_n^-\) ). In the first part of this paper, for \(m,n\in \omega \) we will give a full characterization of when the implication \(\textrm{RC}_m\Rightarrow \textrm{WOC}_n^-\) holds in \({\textsf {ZF}}\). We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that \(\textrm{RC}_5\Rightarrow \textrm{LOC}_5^-\) and that \(\textrm{RC}_6\Rightarrow \textrm{C}_3^-\), answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that \(\textrm{RC}_6\Rightarrow \textrm{C}_9^-\) and that \(\textrm{RC}_7\Rightarrow \textrm{LOC}_7^-\). (shrink)

For a set x, let S(x) be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF: (1) There is an infinite set x such that |p(x)|<|S(x)|<|seq^1-1(x)|<|seq(x)|, where p(x) is the powerset of x, seq(x) is the set of all finite sequences of elements of x, and seq^1-1(x) is the set of all finite sequences of elements of x without repetition. (2) There is a Dedekind (...) infinite set x such that |S(x)|<|[x]^3| and such that there exists a surjection from x onto S(x). (3) There is an infinite set x such that there is a finite-to-one function from S(x) into x. (shrink)

We study properties of certain subclasses of the Dedekind finite sets in set theory without the axiom of choice with respect to the comparability of their elements and to the boundedness of such classes, and we answer related open problems from Herrlich’s “The Finite and the Infinite.” The main results are as follows: 1. It is relatively consistent with ZF that the class of all finite sets is not the only finiteness class such that any two of its elements are (...) comparable. 2. The principle “Small Violations of Choice” —introduced by A. Blass—implies that the class of all Dedekind finite sets is bounded above. 3. “The class of all Dedekind finite sets is bounded above” is true in every permutation model of ZFA in which the class of atoms is a set, and in every symmetric model of ZF. 4. There exists a model of ZFA set theory in which the class of all atoms is a proper class and in which the class of all infinite Dedekind finite sets is not bounded above. 5. There exists a model of ZF in which the class of all infinite Dedekind finite sets is not bounded above. (shrink)

Let ${\mathfrak{M}}$ be a nonstandard model of Peano Arithmetic with domain M and let ${n \in M}$ be nonstandard. We study the symmetric and alternating groups S n and A n of permutations of the set ${\{0,1,\ldots,n-1\}}$ internal to ${\mathfrak{M}}$ , and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that A n and S n are not split extensions by these normal subgroups, by showing that any such complement if (...) it exists, cannot be a limit of definable sets. We conclude by identifying an ${\mathbb{R}}$ -valued metric on ${\tilde{S}_n = S_n /B_S}$ and ${\tilde{A}_n = A_n /B_A}$ (where B S , B A are the maximal normal subgroups of S n and A n identified earlier) making these groups into topological groups, and by showing that if ${\mathfrak{M}}$ is ${\mathfrak\aleph_1}$ -saturated then ${\tilde{S}_n}$ and ${\tilde{A}_n}$ are complete with respect to this metric. (shrink)

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 (...) class='Hi'>permutation models. (shrink)

Quasi-set theory????\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}$$\end{document} is a first order theory which allows us to cope with certain collections of objects where the usual notion of identity is not applicable, in the sense that x = x is not a formula, if x is an arbitrary term. The terms of????\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}$$\end{document} are either collections or atoms (empty terms who are not collections), in a precise sense. (...) Within this context,????\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}$$\end{document} is supposed to be some sort of generalization of ZFU (ZF with atoms). Strangely enough, no one published any permutation model of quasi-set theory until now, although this theory is already 30 years old. In this paper I introduce a class of permutation models of????′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}'$$\end{document} defined as????−{Axiom of Choice and all axioms involving quasi-cardinality}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}-\{\text{Axiom of Choice and all axioms involving quasi-cardinality}\}$$\end{document}. I show, for example, that all permutation models of????′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {Q}'$$\end{document} are models of ZFU−{Axiom of Choice}. (shrink)

If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable (...) without using the axiom of choice. (shrink)

It is a long standing open problem whether or not the Axiom of Countable Choice implies the fragment of Martin's Axiom either in or in. In this direction, we provide a partial answer by establishing that the Boolean Prime Ideal Theorem in conjunction with the Countable Union Theorem does not imply restricted to complete Boolean algebras in. Furthermore, we prove that the latter (formally) weaker form of and the Δ‐system Lemma are independent of each other in.We also answer open questions (...) from Tachtsis [16] which concern the status of restricted to complete Boolean algebras in certain Fraenkel–Mostowski permutation models of and we strengthen some results from the above paper. (shrink)

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.

The ways in which space-time points and elementary particles are modeled share a curious feature: neither seems to specify which basic object has which properties. This chapter sketches the motivation for this claim and searches for an explanation for it. After reviewing several proposals, it argues for a view according to which objects occupy their place in a given relational structure essentially. This view, which is termed minimal structural essentialism, provides a metaphysical grounding for the physical equivalence of models (...) related by permutation. An interesting consequence of this position is that space-time points and elemental particles turn out to be individuals, albeit of a rather different sort than has traditionally been considered. (shrink)

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order and Zermelo’s quasi-categoricity theorem (...) for second-order —these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

In this paper, the connections between model theory and the theory of infinite permutation groups are used to study the n-existence and the n-uniqueness for n-amalgamation problems of stable theories. We show that, for any n ⩾ 2, there exists a stable theory having -existence and k-uniqueness, for every k ⩽ n, but has neither -existence nor -uniqueness. In particular, this generalizes the example, for n = 2, due to Hrushovski given in 3. © 2011 WILEY-VCH Verlag GmbH & (...) Co. KGaA, Weinheim. (shrink)

In set theory without the Axiom of Choice ( $\mathsf {AC}$ ), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinite-dimensional subspaces of a given infinite-dimensional vector space, as well as the extension of almost disjoint families in infinite-dimensional vector spaces to MAD families.

In the realm of metric spaces the role of choice principles is investigated.

In set theory without the axiom of choice (AC), we study the relative strength of the principle “No decreasing sequence of cardinals,” that is, “There is no function f on ω such that |f(n+1)|<|f(n)| for all n∈ω” (NDS) with regard to its position in the hierarchy of weak choice principles. We establish the following results: (1) The Boolean prime ideal theorem plus countable choice does not imply NDS in ZF; (2) “Every non-well-orderable set has a well-orderable partition into denumerable sets” (...) (Dℵ0) plus the axiom of choice for well-ordered families of nonempty finite sets does not imply NDS ∨ “The axiom of choice for countable families of nonempty countable sets” in ZFA; and (3) The axiom of choice for well-ordered families of nonempty sets, each of cardinality not greater than or equal to 2ℵ0, does not imply NDS ∨ “countable union theorem” in ZFA. The above results answer corresponding questions left open in works by Howard and Rubin (1998) and Howard and Tachtsis (2016). We also show the following: (4) “Every non-well-orderable set has a well-orderable partition into infinite well-orderable sets” (Dw) is strictly weaker than D ℵ0 in ZFA. This resolves an open problem from Keremedis and Tachtsis (2003). (shrink)

The dynamics-from-permutations of classical Ising spins is generalized here for an arbitrarily long chain. This serves as an ontological model with discrete dynamics generated by pairwise exchange interactions defining the unitary update operator. The model incorporates a finite signal velocity and resembles in many aspects a discrete free field theory. We deduce the corresponding Hamiltonian operator and show that it generates an exact terminating Baker–Campbell–Hausdorff formula. Motivation for this study is provided by the Cellular Automaton Interpretation of Quantum Mechanics. We (...) find that our ontological model, which is classical and deterministic, appears as if of quantum mechanical kind in an appropriate formal description. However, it is striking that small deformations of the model turn it into a genuine quantum theory. This supports the view that quantum mechanics stems from an epistemic approach handling physical phenomena. (shrink)

We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1] (...) I which consists of functions with finite support is compact, is not provable in ZF set theory. 3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰ (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ₀-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰. (shrink)

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, (...) from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators. (shrink)

For every$n\in \omega \setminus \{0,1\}$we introduce the following weak choice principle:$\operatorname {nC}_{<\aleph _0}^-:$For every infinite family$\mathcal {F}$of finite sets of size at least n there is an infinite subfamily$\mathcal {G}\subseteq \mathcal {F}$with a selection function$f:\mathcal {G}\to \left [\bigcup \mathcal {G}\right ]^n$such that$f(F)\in [F]^n$for all$F\in \mathcal {G}$.Moreover, we consider the following choice principle:$\operatorname {KWF}^-:$For every infinite family$\mathcal {F}$of finite sets of size at least$2$there is an infinite subfamily$\mathcal {G}\subseteq \mathcal {F}$with a Kinna–Wagner selection function. That is, there is a function$g\colon \mathcal (...) {G}\to \mathcal {P}\left (\bigcup \mathcal {G}\right )$with$\emptyset \not =f(F)\subsetneq F$for every$F\in \mathcal {G}$.We will discuss the relations between these two choice principles and their relations to other well-known weak choice principles. Moreover, we will discuss what happens when we replace$\mathcal {F}$by a linearly ordered or a well-ordered family. (shrink)

A partition is finitary if all its members are finite. For a set A, $\mathscr {B}(A)$ denotes the set of all finitary partitions of A. It is shown consistent with $\mathsf {ZF}$ (without the axiom of choice) that there exist an infinite set A and a surjection from A onto $\mathscr {B}(A)$. On the other hand, we prove in $\mathsf {ZF}$ some theorems concerning $\mathscr {B}(A)$ for infinite sets A, among which are the following: (1) If there is a finitary (...) partition of A without singleton blocks, then there are no surjections from A onto $\mathscr {B}(A)$ and no finite-to-one functions from $\mathscr {B}(A)$ to A. (2) For all $n\in \omega $, $|A^n|<|\mathscr {B}(A)|$. (3) $|\mathscr {B}(A)|\neq |\mathrm {seq}(A)|$, where $\mathrm {seq}(A)$ is the set of all finite sequences of elements of A. (shrink)