This paper concerns late 1920 s attempts to construct unitary theories of gravity and electromagnetism. A first attempt using a non-standard connection—with torsion and zero-curvature—was carried out by Albert Einstein in a number of publications that appeared between 1928 and 1931. In 1929, Tullio Levi-Civita discussed Einstein’s geometric structure and deduced a new system of differential equations in a Riemannian manifold endowed with what is nowadays known as Levi-Civita connection. He attained an important result: Maxwell’s electromagnetic equations and the (...) gravitational equations were obtained exactly, while Einstein had deduced them only as a first order approximation. A main feature of Levi-Civita’s theory is the essential use of the Ricci’s rotation coefficients, introduced by Gregorio Ricci Curbastro many years before. We trace the history of Ricci’s coefficients that are still used today, and highlight their geometric and mechanical meaning. (shrink)

Levada, William Cardinal; Ladaria, Luis F The supreme law of the Church is the salvation of souls. As such, throughout its history, the Church has always found the pastoral and juridical means to care for the good of the faithful. With the Apostolic Construction Anglicanorum coetibus, promulgated on 4 November 2009, the Holy Father, Pope Benedict XVI, provided for the establishment of Personal Ordinariates through which Anglican faithful may enter, even in a corporate manner, into full communion with the (...) Catholic Church. On that same date, the Congregation for the Doctrine of the Faith published Complementary Norms relating to such Ordinariates. In conformity with what is established in Art, I and 1 and 2 of the Apostolic Coinstitution Anglicanorum coetibus, having received requests from a considerable number of Anglican faithful, and having consulted with the Australian Catholic Bishops Conference, the Congregation for the Doctrine of the Faith. (shrink)

An overview of what Frege accomplishes in Part II of Grundgesetze, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, and the relationship between these notions. One might think of this paper as an extremely compressed form of Part II of my book Reading Frege's Grundgesetze.

This paper presents a model-theoretic semantics for discourse "about" natural numbers, one that captures what I call "the mathematical-object picture", but avoids what I can "the mathematical-object theory".

We discuss the concept of a cardinal number and its history, focussing on Cantor's work and its reception. J'ay fait icy peu pres comme Euclide, qui ne pouvant pas bien >faire< entendre absolument ce que c'est que raison prise dans le sens des Geometres, definit bien ce que c'est que memes raisons. (Leibniz) 1.

Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of (...) class='Hi'>number turns out to be valid on its own terms, even though it depends on two epistemological principles logicist philosophers of mathematics may find too ‘constructivist’. (shrink)

The Axiom of Choice implies the Partition Principle and the existence, uniqueness, and monotonicity of (possibly infinite) sums of cardinal numbers. We establish several deductive relations among those principles and their variants: the monotonicity follows from the existence plus uniqueness; the uniqueness implies the Partition Principle; the Weak Partition Principle is strictly stronger than the Well-Ordered Choice.

A frege-Russell cardinal number is a maximal class of equinumerous classes. Since anything can be numbered, A frege-Russell cardinal should contain classes whose members are cardinal numbers. This is not possible in standard set theories, Since it entails that some classes are members of members of themselves. However, A consistent set theory can be constructed in which such membership circles are allowed and in which, Consequently, Genuine frege-Russell cardinals can be defined.

I catalog several concepts associated with finite cardinals, and then invoke them to interpret and evaluate several passages in Rips et al.'s target article. Like the literature it discusses, the article seems overly quick to ascribe the possession of certain concepts to children (and of set-theoretic concepts to non-mathematicians).

In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal (...) axiom that (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects. (shrink)

This paper investigates a certain puzzling argument concerning number expressions and their meanings, the Easy Argument for Numbers. After finding faults with previous views, I offer a new take on what’s ultimately wrong with the Argument: it equivocates. I develop a semantics for number expressions which relates various of their uses, including those relevant to the Easy Argument, via type-shifting. By marrying Romero ’s :687–737, 2005) analysis of specificational clauses with Scontras ’ semantics for Degree Nouns, I show (...) how to extend Landman ’s Adjectival Theory to numerical specificational clauses. The resulting semantics can explain various contrasts observed by Moltmann, but only if Scontras’ contention that degrees and numbers are sortally distinct is correct. At the same time, the Easy Argument can establish its intended conclusion only if numbers and degrees are mistakenly assumed to be identical. (shrink)

We consider minimal ranks of extenders associated with Shelah cardinals by introducing witnessing numbers. Using these numbers we shall investigate effects of Shelah cardinals above themselves. MSC: 03E55.

Against Clarke and Beck's proposal that the approximate number system represents natural and rational numbers, I suggest that the experimental evidence is better accommodated by the thesis that the ANS represents cardinality comparisons. Cardinality comparisons do not stand in arithmetical relations and being able to apply them does not involve basic arithmetical concepts and operations.

We study a generalization of the splitting number s to uncountable cardinals. We prove that $\mathfrak{s}(\kappa) > \kappa^+$ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption $\mathfrak{s}(\aleph_\omega) > \aleph_{\omega + 1}$ has a considerable large cardinal strength as well.

We investigate several ideal versions of the pseudointersection number \(\mathfrak {p}\), ideal slalom numbers, and associated topological spaces with the focus on selection principles. However, it turns out that well-known pseudointersection invariant \(\mathtt {cov}^*({\mathcal I})\) has a crucial influence on the studied notions. For an invariant \(\mathfrak {p}_\mathrm {K}({\mathcal J})\) introduced by Borodulin-Nadzieja and Farkas (Arch. Math. Logic 51:187–202, 2012), and an invariant \(\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J})\) introduced by Repický (Real Anal. Exchange 46:367–394, 2021), we have $$\begin{aligned} \min (...) \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathtt {cov}^*({\mathcal I})\}=\mathfrak {p},\qquad \min \{\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}\le \mathtt {cov}^*({\mathcal I}), \end{aligned}$$ respectively. In addition to the first inequality, for a slalom invariant \(\mathfrak {sl_e}({\mathcal I},{\mathcal J})\) introduced in Šupina (J. Math. Anal. Appl. 434:477–491, 2016), we show that $$\begin{aligned} \min \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathfrak {sl_e}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}=\mathfrak {p}. \end{aligned}$$ Finally, we obtain a consistency that ideal versions of the Fréchet–Urysohn property and the strictly Fréchet–Urysohn property are distinguished. (shrink)

It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact cardinal that there is a uniform ultrafilter on \ which is generated by fewer than \ sets.

According to Rips et al., numerical cognition develops out of two independent sets of cognitive primitives – one that supports enumeration, and one that supports arithmetic and the concepts of natural numbers. I argue against this proposal because it incorrectly predicts that natural number concepts could develop without prior knowledge of enumeration.

This article sets out to analyse some of the most basic elements of our number concept - of our awareness of the one and the many in their coherence with multiplicity, succession and equinumerosity. On the basis of the definition given by Cantor and the set theoretical definition of cardinal numbers and ordinal numbers provided by Ebbinghaus, a critical appraisal is given of Frege’s objection that abstraction and noticing (or disregarding) differences between entities do not produce the concept (...) of number. By introducing the notion of subject functions, an account is advanced of the (nominalistic) reason why Frege accepted physical, kinematic and spatial properties (subject functions) of entities, but denied the ontic status of their quantitative properties (their quantitative subject function). With reference to intuitionistic mathematics (Brouwer, Weyl, Troelstra, Kreisel, Van Dalen) the primitive meaning of succession is acknowledged and connected to an analysis of what is entailed in the term ‘Gleichzahligkeit’ (‘equinumerosity’). This expression enables an analysis of the connections between ordinality and cardinality on the one hand and succession and wholeness (totality) on the other. The conceptions of mathematicians such as Frege, Cantor, Dedekind, Zermelo, Brouwer, Skolem, Fraenkel, Von Neumann, Hilbert, Bernays and Weyl, as well as the views of the philosopher Cassirer, are discussed in order to arrive at an assessment of the relation between ordinality and cardinality (also taking into account the relation between logic and arithmetic) - and on the basis of this evaluation, attention is briefly given to the definition of an ordered pair in axiomatic set theory (with reference to the set theory of Zermelo-Fraenkel) and to the defmition of an ordered pair advanced by Wiener and Kuratowski. (shrink)

We examine the number of normal measures a successor cardinal can carry, in universes in which the Axiom of Choice is false. When considering successors of singular cardinals, we establish relative consistency results assuming instances of supercompactness, together with the Ultrapower Axiom (introduced by Goldberg in [12]). When considering successors of regular cardinals, we establish relative consistency results only assuming the existence of one measurable cardinal. This allows for equiconsistencies.

Letκ, λ be regular uncountable cardinals such that λ >κ+is not a successor of a singular cardinal of low cofinality. We construct a generic extension withs = λ starting from a ground model in whicho = λ and prove that assuming ¬0¶,s = λ implies thato ≥ λ in the core model.

Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one-to-one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, (...) 2008) linked children's understanding of cardinality to their understanding of succession for the numbers five and six. This study investigates the link between cardinality and equinumerosity for these numbers, finding that children either understand both cardinality and equinumerosity or they understand neither. This suggests that cardinality and equinumerosity (along with succession) are interrelated facets of the concepts five and six, the acquisition of which is an important conceptual achievement of early childhood. (shrink)

Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, (...) however, that Frege was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many. (shrink)

This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

The theorem that the arithmetic mean is greater than or equal to the geometric mean is investigated for cardinal and ordinal numbers. It is shown that whereas the theorem of the means can be proved for n pairwise comparable cardinal numbers without the axiom of choice, the inequality a2 + b2 ≥ 2ab is equivalent to the axiom of choice. For ordinal numbers, the inequality α2 + β2 ≥ 2αβ is established and the conditions for equality are derived; (...) stronger inequalities are obtained for finite and infinite sequences of ordinals under suitable monotonicity hypotheses. MSC: 03E10, 04A10, 03E25, 04A25. (shrink)

The consistency of a second-order version of Morley’s Theorem on the number of countable models was proved in [EHMT23] with the aid of large cardinals. We here dispense with them.

Cardinality arguments against regular probability measures aim to show that no matter which ordered field ℍ we select as the measures for probability, we can find some event space F of sufficiently large cardinality such that there can be no regular probability measure from F into ℍ. In particular, taking ℍ to be hyperreal numbers won't help to guarantee that probability measures can always be regular. I argue that such cardinality arguments fail, since they rely on the wrong conception of (...) the role of numbers as measures of probability. With the proper conception of their role we can see that for any event space F, of any cardinality, there are regular hyperreal-valued probability measures. (shrink)

§1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2ℵα for singular ℵα's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals (...) problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory. The most striking result to date states that if 2ℵn < ℵω for every n = 0, 1, 2, …, then 2ℵω < ℵω4.In this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory.§2. The arithmetic of cardinal numbers. Cardinal numbers were introduced by Cantor in the late 19th century and problems arising from investigations of rules of arithmetic of cardinal numbers led to the birth of set theory. The operations of addition, multiplication and exponentiation of infinite cardinal numbers are a natural generalization of such operations on integers. Addition and multiplication of infinite cardinals turns out to be simple: when at least one of the numbers κ, λ is infinite then both κ + λ and κ·λ are equal to max {κ, λ}. In contrast with + and ·, exponentiation presents fundamental problems. In the simplest nontrivial case, 2κ represents the cardinal number of the power set P, the set of all subsets of κ. (Here we adopt the usual convention of set theory that the number κ is identified with a set of cardinality κ, namely the set of all ordinal numbers smaller than κ. (shrink)

For each natural number n, let C(n) be the closed and unbounded proper class of ordinals α such that Vα is a Σn elementary substructure of V. We say that κ is a C(n)-cardinal if it is the critical point of an elementary embedding j : V → M, M transitive, with j(κ) in C(n). By analyzing the notion of C(n)-cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at (...) the level of superstrong cardinals and up to the level of rank-into-rank embeddings, C(n)-cardinals form a much finer hierarchy. The naturalness of the notion of C(n)-cardinal is exemplified by showing that the existence of C(n)-extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of Bagaria et al. (2010), we give new characterizations of Vopeňka’s Principle in terms of C(n)-extendible cardinals. (shrink)

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal axioms consistent with ZFC, since it appears to be `possible' to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we (...) argue that there are several conceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. (...) First, we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others. This paper addresses two related issues. First, we review some (...) of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege's foundationalist epistemology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally ‘legitimate’. (shrink)

We explore the connection between combinatorial principles on uncountable cardinals, like stick and club, on the one hand, and the combinatorics of sets of reals and, in particular, cardinal invariants of the continuum, on the other hand. For example, we prove that additivity of measure implies that Martin’s axiom holds for any Cohen algebra. We construct a model in which club holds, yet the covering number of the null ideal is large. We show that for uncountable cardinals κ≤λ (...) and , if all subsets of λ either contain, or are disjoint from, a member of , then has size at least etc. As an application, we solve the Gross space problem under by showing that there is such a space over any countable field. In two appendices, we solve problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively. (shrink)

Unfoldable cardinals are preserved by fast function forcing and the Laver-like preparations that fast functions support. These iterations show, by set-forcing over any model of ZFC, that any given unfoldable cardinal κ can be made indestructible by the forcing to add any number of Cohen subsets to κ.

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.

In 2002, Yorioka introduced the σ-ideal ${{\mathcal {I}}_f}$ for strictly increasing functions f from ω into ω to analyze the cofinality of the strong measure zero ideal. For each f, we study the cardinal coefficients (the additivity, covering number, uniformity and cofinality) of ${{\mathcal {I}}_f}$.

Unfoldable cardinals are preserved by fast function forcing and the Laver-like preparations that fast functions support. These iterations show, by set-forcing over any model of ZFC, that any given unfoldable cardinal $\kappa$ can be made indestructible by the forcing to add any number of Cohen subsets to $\kappa$.

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory.§0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + (...) ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture:The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: ;Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture. (shrink)

A classic result of Baumgartner-Harrington-Kleinberg [1] implies that assuming CH a stationary subset of ω1 has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω1. Therefore, assuming that $\omega_2^L$ is countable: { $X \in L \mid X \subseteq \omega_1^L$ and X has a CUB subset in a cardinal -preserving extension of L} is constructible, as it equals the set of constructible subsets of $\omega_1^L$ which in L are stationary. Is there a (...) similar such result for subsets of $\ omega_2^L$ ? Building on work of M. Stanley [9], we show that there is not. We shall also consider a number of related problems, examining the extent to which they are "solvable" in the above sense, as well as defining a notion of reduction between them. (shrink)

We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new [Formula: see text]-sequences (for some regular [Formula: see text]). As an application, we show that consistently the following cardinal characteristics can be different: The (“independent”) characteristics in Cichoń’s diagram, plus [Formula: see text]. (So we get thirteen different values, including [Formula: see text] and continuum). We also give constructions to alternatively separate other MA-numbers (instead of [Formula: see text]), namely: MA for (...) [Formula: see text]-Knaster from MA for [Formula: see text]-Knaster; and MA for the union of all [Formula: see text]-Knaster forcings from MA for precaliber. (shrink)