We study the intersection number of families of tall ideals. We show that the intersection number of the class of analytic P-ideals is equal to the bounding number ${\mathfrak{b}}$ , the intersection number of the class of all meager ideals is equal to ${\mathfrak{h}}$ and the intersection number of the class of all F σ ideals is between ${\mathfrak{h}}$ and ${\mathfrak{b}}$ , consistently different from both.

Let $\mathcal {I}$ be an ideal on $\omega $. For $f,\,g\in \omega ^{\omega }$ we write $f \leq _{\mathcal {I}} g$ if $f(n) \leq g(n)$ for all $n\in \omega \setminus A$ with some $A\in \mathcal {I}$. Moreover, we denote $\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$ (in particular, $\mathcal {D}_{\mathrm {Fin}}$ denotes the family of all finite-to-one functions).We examine cardinal numbers $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times (...) \mathcal {D}_{\mathcal {I}}))$ and $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}}\times \mathcal {D}_{\mathrm {Fin}}))$ describing the smallest sizes of unbounded from below with respect to the order $\leq _{\mathcal {I}}$ sets in $\mathcal {D}_{\mathrm {Fin}}$ and $\mathcal {D}_{\mathcal {I}}$, respectively. For a maximal ideal $\mathcal {I}$, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers.We show that $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}} \times \mathcal {D}_{\mathrm {Fin}})) =\mathfrak {b}$ for all ideals $\mathcal {I}$ with the Baire property and that $\aleph _1 \leq \mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) \leq \mathfrak {b}$ for all coanalytic weak P-ideals (this class contains all $\bf {\Pi ^0_4}$ ideals). What is more, we give examples of Borel (even $\bf {\Sigma ^0_2}$ ) ideals $\mathcal {I}$ with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\mathfrak {b}$ as well as with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) =\aleph _1$.We also study cardinals $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {J}} \times \mathcal {D}_{\mathcal {K}}))$ describing the smallest sizes of sets in $\mathcal {D}_{\mathcal {K}}$ not bounded from below with respect to the preorder $\leq _{\mathcal {I}}$ by any member of $\mathcal {D}_{\mathcal {J}}\!$. Our research is partially motivated by the study of ideal-QN-spaces: those cardinals describe the smallest size of a space which is not ideal-QN. (shrink)

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)

SummaryAlthough desired family size is often different from actual family size, the dynamics of this difference are not well understood. This paper examines the patterns and determinants of the difference between desired and actual number of children among women aged 15–49 years using pooled data from the 1990, 1999 and 2003 Nigeria Demographic and Health Surveys. The results show that more than two-thirds of the sample have unmet fertility desires. It was found that early and late childbearing increased the odds (...) of unmet fertility desires. Also, women with low levels of education, from poor households, rural residents as well as those who had experienced child death were at a higher risk of unmet fertility desires in the multivariate context. The study highlights the policy and programme implications of the findings. (shrink)

This is a continuation of the paper [J. Symb. Log. 87 (2022), 1065–1092]. For an ideal $\mathcal {I}$ on $\omega $ we denote $\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$ and write $f\leq _{\mathcal {I}} g$ if $\{n\in \omega :f(n)>g(n)\}\in \mathcal {I}$, where $f,g\in \omega ^{\omega }$. We study the cardinal numbers $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$ describing the smallest sizes of subsets (...) of $\mathcal {D}_{\mathcal {I}}$ that are unbounded from below with respect to $\leq _{\mathcal {I}}$. In particular, we examine the relationships of $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$ with the dominating number $\mathfrak {d}$. We show that, consistently, $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))>\mathfrak {d}$ for some ideal $\mathcal {I}$, however $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))\leq \mathfrak {d}$ for all analytic ideals $\mathcal {I}$. Moreover, we give example of a Borel ideal with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\operatorname {\mathrm {add}}(\mathcal {M})$. (shrink)

The Mycielski ideal M k is defined to consist of all sets $A \subseteq ^{\mathbb{N}}k$ such that $\{f \upharpoonright X: f \in A\} \neq ^Xk$ for all X ∈ [N] ℵ 0 . It will be shown that the covering numbers for these ideals are all equal. However, the covering numbers of the closely associated Roslanowski ideals will be shown to be consistently different.

We investigate the covering numbers of some ideals on $${^{\omega }}{2}{}$$ ω 2 associated with tree forcings. We prove that the covering of the Sacks ideal remains small in the Silver and uniform Sacks model, respectively, and that the coverings of the uniform Sacks ideal and the Mycielski ideal, $${\mathfrak {C}_{2}}$$ C 2, remain small in the Sacks model.

For ${b \in {^{\omega}}{\omega}}$ , let ${\mathfrak{c}^{\exists}_{b, 1}}$ be the minimal number of functions (or slaloms with width 1) to catch every functions below b in infinitely many positions. In this paper, by using the technique of forcing, we construct a generic model in which there are many coefficients ${\mathfrak{c}^{\exists}_{{b_\alpha}, 1}}$ with pairwise different values. In particular, under the assumption that a weakly inaccessible cardinal exists, we can construct a generic model in which there are continuum many coefficients ${\mathfrak{c}^{\exists}_{{b_\alpha}, 1}}$ (...) with pairwise different values. In conjunction with these results, we give a generic model in which there are many Yorioka’s ideals ${\mathcal{I}_{f_\alpha}}$ with pairwise different covering numbers. (shrink)

We propose a reformulation of the ideal $\mathcal {N}$ of Lebesgue measure zero sets of reals modulo an ideal J on $\omega $, which we denote by $\mathcal {N}_J$. In the same way, we reformulate the ideal $\mathcal {E}$ generated by $F_\sigma $ measure zero sets of reals modulo J, which we denote by $\mathcal {N}^*_J$. We show that these are $\sigma $ -ideals and that $\mathcal {N}_J=\mathcal {N}$ iff J has the Baire property, which in turn (...) is equivalent to $\mathcal {N}^*_J=\mathcal {E}$. Moreover, we prove that $\mathcal {N}_J$ does not contain co-meager sets and $\mathcal {N}^*_J$ contains non-meager sets when J does not have the Baire property. We also prove a deep connection between these ideals modulo J and the notion of nearly coherence of filters (or ideals). We also study the cardinal characteristics associated with $\mathcal {N}_J$ and $\mathcal {N}^*_J$. We show their position with respect to Cichoń’s diagram and prove consistency results in connection with other very classical cardinal characteristics of the continuum, leaving just very few open questions. To achieve this, we discovered a new characterization of $\mathrm {add}(\mathcal {N})$ and $\mathrm {cof}(\mathcal {N})$. We also show that, in Cohen model, we can obtain many different values to the cardinal characteristics associated with our new ideals. (shrink)

The main result of this paper is an improvement of the upper bound on the cardinal invariant $cov^*(L_0)$ that was discovered in [11]. Here $L_0$ is the ideal of subsets of the set of natural numbers that have asymptotic density zero. This improved upper bound is also dualized to get a better lower bound on the cardinal $non^*(L_0)$. En route some variations on the splitting number are introduced and several relationships between these variants are proved.

We say that $\mathcal {I}$ is an ideal independent family if no element of ${\mathcal {I}}$ is a subset mod finite of a union of finitely many other elements of ${\mathcal {I}}.$ We will show that the minimum size of a maximal ideal independent family is consistently bigger than both $\mathfrak {d}$ and $\mathfrak {u},$ this answers a question of Donald Monk.

Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals’ uniformity numbers can be pairwise different. In addition we show that, in the same forcing extension, for two other types of simple cardinal characteristics parametrised by reals, for uncountably many parameters the corresponding cardinals are pairwise different.

Ideal epistemologists investigate the nature of pure epistemic rationality, abstracting away from human cognitive limitations. Non-ideal epistemologists investigate epistemic norms that are satisfiable by most humans, most of the time. Ideal epistemology faces a number of challenges, aimed at both its substantive commitments and its philosophical worth. This paper explains the relation between ideal and non-ideal epistemology, with the aim of justifying ideal epistemology. Its approach is meta-epistemological, focusing on the meaning and purpose of (...) epistemic evaluations. I provide an account on which the fundamental difference between ideal and non-ideal epistemic evaluations is that only the non-ideal epistemic ‘ought’ implies any substantive ‘can’. I argue that only ideal epistemic evaluations are ‘normatively robust’: they are neither conventional nor seriously context-sensitive. Non-ideal epistemic evaluations are normatively non-robust, exhibiting both conventionality and serious context-sensitivity from an interesting variety of distinct sources. For this reason, non-ideal epistemic evaluations won’t characterize the fundamental nature of epistemic rationality. Non-ideal epistemic rationality depends, not merely on what’s epistemically valuable, but also on modally contingent epistemic conventions and contextually contingent constraints on epistemic options. If we want a normatively robust theory of epistemic rationality, ideal epistemology is the only game in town. (shrink)

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal \. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that \<\mathrm {cov}<\mathrm {cof}\), which is the first consistency result where more than two cardinal (...) invariants associated with \ are pairwise different. Another consequence is that \ in ZFC where \ denotes Marczewski’s ideal. (shrink)

This study focuses on determining the number of children considered ideal by Pakistani husbands and identifying the factors associated with this, with a special emphasis on family type. A cross-sectional survey was conducted among married males residing in four areas of Khairpur district. An equal number of study participants were selected systematically from each field site to achieve the required sample size of 500. Interviews were conducted by trained fieldworkers using a structured questionnaire to obtain information on background socioeconomic (...) characteristics, family structure and reproductive health knowledge and practices, in particular family planning. Multivariate logistic regression analyses were used to assess the hypothesis that family type has an independent association with husbands opinions regarding ideal number of children can help strengthen population control efforts in Pakistan. (shrink)

Science is the study of our world, as it is in its messy reality. Nonetheless, science requires idealization to function—if we are to attempt to understand the world, we have to find ways to reduce its complexity. Idealization and the Aims of Science shows just how crucial idealization is to science and why it matters. Beginning with the acknowledgment of our status as limited human agents trying to make sense of an exceedingly complex world, Angela Potochnik moves on to explain (...) how science aims to depict and make use of causal patterns—a project that makes essential use of idealization. She offers case studies from a number of branches of science to demonstrate the ubiquity of idealization, shows how causal patterns are used to develop scientific explanations, and describes how the necessarily imperfect connection between science and truth leads to researchers’ values influencing their findings. The resulting book is a tour de force, a synthesis of the study of idealization that also offers countless new insights and avenues for future exploration. (shrink)

We prove that the covering number of the Marczewski ideal is equal to ℵ1 in the extension with the iteration of Hechler forcing.

A number of social theorists have contended that the essence of historical analysis is the employment of ideal types to comprehend past goings-on. But, while acknowledging that the study of history through ideal types can yield genuine insight, we may still ask if it represents the full emancipation of historical understanding from other modes of conceiving the past. This paper follows Michael Oakeshott's work on the philosophy of history in arguing that explaining the historical past by means of (...) ideal types, even though offering a coherent and fruitful enterprise, nevertheless falls short of fully embodying the characteristics that differentiate historical understanding. -/- This dispute involves fundamental issues regarding the nature of the social sciences. Furthermore, I suggest that it is of more than purely theoretical interest, in that social theorists who have accepted the Weberian view of historical inquiry will tend to be unaware of the defects inherent in all attempts to capture the nature of past events in a net of generalizations, and thus may be lead to ignore the inherent partiality of the understanding of historical happenings provided by ideal typification. (shrink)

Formal epistemologists often claim that our credences should be representable by a probability function. Complete probabilistic coherence, however, is only possible for ideal agents, raising the question of how this requirement relates to our everyday judgments concerning rationality. One possible answer is that being rational is a contextual matter, that the standards for rationality change along with the situation. Just like who counts as tall changes depending on whether we are considering toddlers or basketball players, perhaps what counts as (...) rational shifts according to whether we are considering ideal agents or creatures more like ourselves. Even though a number of formal epistemologists have endorsed this type of solution, I will argue that there is no way to spell out this contextual account that can make sense of our everyday judgments about rationality. Those who defend probabilistic coherence requirements will need an alternative account of the relationship between real and ideal rationality. (shrink)

Let I be an F σ -ideal on natural numbers. We characterize the ultrafilters which are generic over the model L for the poset of I -positive sets of natural numbers ordered by inclusion.

§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters. There is also a substantial interest in nicely definable ideals—these by old results of Sierpiński are very far from being maximal— and the structure of such ideals will concern us in this announcement. In addition to being interesting in their own right, Borel and analytic ideals occur naturally (...) in the investigations of compact subsets of the space of all Baire class 1 functions on a Polish space, see [12, 18]. Also, certain objects associated with such ideals are of considerable interest and were quite extensively studied by several authors. Let us list here three examples; in all three of them I stands for an analytic or Borel ideal.1. The partial order induced by I on P: X ≥I Y iff X \ Y ϵ I and the partial order.2. Boolean algebras of the form P/I and their automorphisms.3. The equivalence relation associated with I: XEI Y iff X Δ ϵ I.In Section 4, we will have an opportunity to state some consequences of our results for equivalence relations as in 3. (shrink)

Hilbert took to using ideal elements in the 1890's, in both algebraic number theory and geometry. His Zahlbericht of 1897 popularized the concept of the ideal introduced by Dedekind in 1871 (which in turn formalized the concept of "ideal number" introduced by Kummer in the 1840's). His geometric work likewise followed a long history of ideal elements, some that originated in geometry and others that originated elsewhere and were applied to geometry. Important examples were:Piero della Francesca's (...) Ideal City.Points at infinity in projective geometry, originating in the 1430s from the "vanishing points" in perspective drawing.Imaginary points in algebraic geometry, originating from the imaginary numbers used by .. (shrink)

In this paper, I first outline the view developed in my recent book on the role of idealization in scientific understanding. I discuss how this view leads to the recognition of a number of kinds of variability among scientific representations, including variability introduced by the many different aims of scientific projects. I then argue that the role of idealization in securing understanding distances understanding from truth, but that this understanding nonetheless gives rise to scientific knowledge. This discussion will clarify how (...) my view relates to three other recent books on understanding by Henk de Regt, Catherine Elgin, and Kareem Khalifa. (shrink)

Ancient Greek Philosophy routinely relied upon concepts of number to explain the tangible order of the universe. Plotinus' contribution to this tradition, however, has been often omitted, if not ignored. The main reason for this, at first glance, is the Plotinus does not treat the subject of number in the Enneads as pervasively as the Neopythagoreans or even his own successors Lamblichus, Syrianus, and Proclus. Nevertheless, a close examination of the Enneads reveals that Plotinus systematically discusses number in relation to (...) each of his underlying principles of existence--the One, Intellect, and Soul. Plotinus on Number offers the first comprehensive analysis of Plotinus' concept of number, beginning with its origins in Plato and the Neopythagoreans and ending with its influence on Porphyry's arrangement of the Enneads. It's main argument is that Plotinus adapts Plato's and the Neopythagoreans' cosmology to place number in the foundation of the intelligible realm and in the construction of the universe. Through Plotinus' defense of Plato's Ideal Numbers from Aristotle's criticism, Svetla Slaveva-Griffin reveals the founder of Neoplatonism as the first post-Platonic philosopher who purposefully and systematically develops what we may call a theory of number, distinguishing between number in the intelligible realm and number in the quantitative, mathematical realm. Finally, the book draws attention to Plotinus' concept as a necesscary and fundamental linke between Platonic and late Neoplatonic schools of philosophy. (shrink)

'Numbers and Proofs' presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational, complex and integers). An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might have (...) before any attempt proof is made. Established proofs which the student is in a better position to follow then follow. Presented in the author's entertaining and informal style, and written to reflect the changing profile of students entering universities, this book will prove essential reading for all seeking an introduction to the notion of proof as well as giving a definitive guide to the more common forms. Stressing the importance of backing up "truths" found through experimentation, with logically sound and watertight arguments, it provides an ideal bridge to more complex undergraduate maths. (shrink)

This article analyses a hitherto neglected problem at the transition from ideal to non‐ideal theory: the problem of knowledge. Ideal theories often make idealising assumptions about the availability of knowledge, for example knowledge of social scientific facts. This can lead to problems when this knowledge turns out not to be available at the non‐ideal level. Knowledge can be unavailable in a number of ways: in principle, for practical reasons, or because there are normative reasons not to (...) use it. This can make it necessary to revise ideal theories, because the principle of ‘ought implies can’ rules out certain theories, at least insofar as they are understood as action‐guiding. I discuss a number of examples and argue that there are two tendencies that will increase the relevance of this problem in the future: the availability of large amounts of sensitive data whose use is problematic from a normative point of view, and the increasing complexity of an interrelated world that makes it harder to predict the effects of institutional changes. To address these issues, philosophers need to cooperate with social scientists and philosophers of the social sciences. Normative theorising can then be understood as one step in a long process that includes thinkers from different disciplines. Ideal theory can respond to many of the charges raised against it if it is understood along these lines and if it takes the problem of knowledge and its implications seriously. (shrink)

This article provides a conceptual map of the debate on ideal and non‐ideal theory. It argues that this debate encompasses a number of different questions, which have not been kept sufficiently separate in the literature. In particular, the article distinguishes between the following three interpretations of the ‘ideal vs. non‐ideal theory’ contrast: (i) full compliance vs. partial compliance theory; (ii) utopian vs. realistic theory; (iii) end‐state vs. transitional theory. The article advances critical reflections on each of (...) these sub‐debates, and highlights areas for future research in the field. (shrink)

Husserl describes arithmetic as a branch of formal ontology. It is an ontology because its goal is to lay out the essential truths about a region of objects, and it is formal because the determinate region of number deals with a characteristic of every possible object. The mathematical experience proper requires something more than the constitution of "concrete numbers" in acts of collecting and counting, for its objects are "ideal numbers" that emerge from eidetic variation over corresponding (...) concrete numbers. With remarkable clarity, Miller coordinates Husserl's various writings on the nature of number by stressing the key theme of identity in presence and absence. Concrete numbers are "determinate multitudes" which are not presented in sensuous intuition, but rather in sensuously founded categorial operations, i.e., explication, in which the parts of an indiscriminately given whole are explicitly articulated as parts, and comparison, in which the multitude generated through such explication is explicitly related to other multitudes. These operations account for the original presence of number. But we can intend numbers in their absence as well as in their presence. For example we may intend a determinate multitude in the absence of the sensuous group in question, or we may experience a different mode of absence when an intended number is frustrated by subsequent intuitive registration, or we may deploy the distinctive absence generated by signs in the process of calculating without authentic articulation of the relevant determinate multitudes. The presence of ideal numbers requires the free play of fantasy, for we must imagine collections of items and also imagine ourselves counting the items in order to disengage their purely formal character. We must discern an identical presence of "four" in a cluster of persons, a collection of apples, and even in some arbitrarily contrived group such as a "feeling, an angel, the moon, Italy." The absence of generic community in the latter example helps us to appreciate what Husserl means by the purely formal nature of arithmetic. Formal ontology is not restricted in scope to some particular region, but is concerned with properties that belong to every possible object precisely as such. Thus, whereas Euclidean geometry is a regional ontology, set-theory and number-theory must be considered as branches of formal ontology, for the basic concepts of these disciplines are strictly formal. Anything that can be thought of can be counted. (shrink)

Aristotle’s classification of ideal number in Metaphysics M 6 has often been considered an unfair presentation of Plato’s actual views. I take another look at the passage and argue that Aristotle is a more careful critic than has been usually recognised. In particular, I argue that much of the scholarly discussion on the passage has failed to take account of Aristotle’s deeper concern, namely, the conditions necessary for numbers to be ordinal. I then set Aristotle’s critique within the (...) broader context of his concern to separate Form from Number, and thus to separate metaphysical questions from mathematical ones. The latter half of the article shows some of the consequences of mixing mathematics with metaphysics from a systematic and historical point of view. (shrink)

It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:If${\cal I}$is a normal ideal on$\omega _2 $which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal;For any$n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal${\cal I}$on$\omega _n $which satisfiesprojective antichain catching, yet${\cal I}$is not saturated. This provides (...) a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory. (shrink)

This essay poses the question of how many rulers are envisaged in Plato’s Statesman. After pointing out that this is a crucial question for issues concerning non-ideal as well as ideal approaches to political rule, the essay focuses on three relevant aspects of rule in the Statesman: the notion of kingly rule, the limitations posed by human nature, and the importance of self-rule. It is shown how each of these dimensions of Plato’s discussion demonstrates the complexity of the (...) question. Particular attention is then given to features inherent to political rule: the need for subordinate functions and a distribution of offices, seen in light of the ends of political rule as helping citizens obtain their potential. It is argued that while the Statesman does not lead to any certain conclusion concerning the number of rulers, and some of its considerations conflict with each other, the text as a whole allows for a fairly broad basis of political rule. (shrink)

Wittgenstein's treatment of number words and arithmetic in the Tractatus reflects central features of his early conception of philosophy. In rejecting Frege's and Russell's analyses of number, Wittgenstein rejects their respective conceptions of function, object, logical form, generality, sentence, and thought. He, thereby, surrenders their shared ideal of the clarity a Begriffsschrift could bring to philosophy. The development of early analytic philosophy thus evinces far less continuity than some readers of Wittgenstein, from Russell and the Vienna positivists to many (...) contemporary readers of the Tractatus, have supposed. (shrink)

The author discusses the concept of an ideal object. The statement of O.I. Genisaretsky is quoted and problematized, stating that the obligatory feature that has been preserved for the object and the terms "object" and "ideal object" is, apparently, its representability or visibility. The author shows that ideal objects began to be created during the formation of ancient philosophy and thinking. Faced with contradictions, ancient thinkers dealt with this situation in different ways. If Protogoras recognized the right (...) of the reasoners to receive contradictions and, accordingly, to consider the world contradictory, then Parmenides considered the world consistent, arguing that the criterion for the correctness (truth) of knowledge is not a reasoning person, but this world, existence, which still needs to be reached with the help of correct thought. It was within the framework of the implementation of the Parmenides program that ideal objects began to be created, which, as the author shows, performed three main functions: they allowed us to think consistently, to know the world and its phenomena, to comprehend and interpret empirical phenomena (facts) in a different way. Based on the case of Plato's "Feast", the semiotic schemes on the basis of which Plato creates ideal objects are analyzed. It is argued that it is impossible to build ideal objects, bypassing schemes, since their inventors (creators) are people. By solving problems, they synthesize meanings belonging to different areas of consciousness. The analysis of Galileo's "Conversations" allowed the author to suggest that the construction of ideal objects is due to the historical type of philosophical or scientific thinking. The author considers it necessary, before claiming to create a modern doctrine of ideal objects, to analyze the main typical cases of the construction of these objects in philosophy and science, and such an analysis can make significant adjustments to the understanding of the nature and features of this concept. Nevertheless, he suggests, it is already clear that the construction of ideal objects is an important link in philosophical, scientific and a number of other types of thinking. (shrink)

We introduce a notion of ideal type such that any two ideals with the same ideal type are isomorphic. From this we infer, under the axiom t = h, that each ideal which consists of all nowhere Ramsey sets contained in some family of infinite subsets of natural numbers is isomorphic with the ideal of all nowhere Ramsey sets.

This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. However, such (...) later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism. (shrink)

We study the structure of analytic ideals of subsets of the natural numbers. For example, we prove that for an analytic ideal I, either the ideal {X (Ω × Ω: En X ({0, 1,…,n} × Ω } is Rudin-Keisler below I, or I is very simply induced by a lower semicontinuous submeasure. Also, we show that the class of ideals induced in this manner by lsc submeasures coincides with Polishable ideals as well as analytic P-ideals. We study (...) this class of ideals and characterize, for example, when the ideals in it are Fσ or when they carry a locally compact group topology. We apply these results to Borel partial orders to rederive a theorem of Todorcevic and to Borel equivalence relations to answer a question of Kechris and Louveau. As another application we give a characterization of σ-ideals of μ-zero sets for Maharam submeasures μ on the Cantor set which is to a large extent analogous to a characterization of the meager ideal due to Kechris and the author. (shrink)

In this paper we inquire into the fundamental assumptions that underpin the ideal of the Bio-Based Economy as it is currently developed . By interpreting the BBE from the philosophical perspective on economy developed by Georges Bataille, we demonstrate how the BBE is fully premised on a thinking of scarcity. As a result, the BBE exclusively frames economic problems in terms of efficient production, endeavoring to exclude a thinking of abundance and wastefulness. Our hypothesis is that this not only (...) entails a number of internal tensions and inconsistencies with regard to the ideal of BBE, but ultimately undermines the ideal itself, by pushing purported regenerativity into a cataclysmic and terminal discharge. We here point to the strategies that the BBE employs in this exclusion, the fundamental assumptions regarding the relation between energy and economy that underpin this endeavor, as well as to the resulting inconsistencies and their catastrophic consequences. We finally argue for the introduction of the presently excluded question of abundance and wastefulness and explore the implications of such a question for the ideal of a zero-waste humanity. (shrink)

It's standard in epistemology to approach questions about knowledge and rational belief using idealized, simplified models. But while the practice of constructing idealized models in epistemology is old, metaepistemological reflection on that practice is not. Greco argues that the fact that epistemologists build idealized models isn't merely a metaepistemological observation that can leave first-order epistemological debates untouched. Rather, once we view epistemology through the lens of idealization and model-building, the landscape looks quite different. Constructing idealized models is likely the best (...) epistemologists can do. Once one starts using epistemological categories like belief, knowledge, and confidence, the realm of idealization and model-building is entered. We can object to a model of knowledge by pointing to a better model, but in the absence of a better model, the fact that a framework for epistemologizing theorizing involves simplifications, approximations, and other inaccuracies-the fact of its status as an idealized model-is not in itself objectionable. Once we accept that theorizing in epistemological terms is inescapably idealized, a number of intriguing possibilities open up. Greco defends a package of epistemological views that might otherwise have looked indefensibly dismissive of our cognitive limitations-a package according to which we know a wide variety of facts with certainty, including what our evidence is, what we know and don't know, and what follows from our knowledge. (shrink)

Bayesian epistemology provides a popular and powerful framework for modeling rational norms on credences, including how rational agents should respond to evidence. The framework is built on the assumption that ideally rational agents have credences, or degrees of belief, that are representable by numbers that obey the axioms of probability. From there, further constraints are proposed regarding which credence assignments are rationally permissible, and how rational agents’ credences should change upon learning new evidence. While the details are hotly disputed, (...) all flavors of Bayesianism purport to give us norms of ideal rationality. This raises the question of how exactly these norms apply to you and me, since perfect compliance with those ideal norms is out of reach for human thinkers. A common response is that Bayesian norms are ideals that human reasoners are supposed to approximate – the closer they come to being ideally rational, the better. To make this claim plausible, we need to make it more precise. In what sense is it better to be closer to ideally rational, and what is an appropriate measure of such closeness? This article sketches some possible answers to these questions. (shrink)

A number of formal epistemologists have argued that perfect rationality requires probabilistic coherence, a requirement that they often claim applies only to ideal agents. However, in “Rationality as an Absolute Concept,” Roy Sorensen contends that ‘rational’ is an absolute term. Just as Peter Unger argued that being flat requires that a surface be completely free of bumps and blemishes, Sorensen claims that being rational requires being perfectly rational. When we combine these two views, though, they lead to counterintuitive results. (...) If being rational requires being perfectly rational, and only the probabilistically coherent are perfectly rational, then this indicts all ordinary agents as irrational. In this paper, I will attempt to resolve this conflict by arguing that Sorensen is only partly correct. One important sense of ‘rational’, the sanctioning sense of ‘rational’, is an absolute term, but another important sense of ‘rational’, the sense in which someone can have rational capacities, is not. I will, then, show that this distinction has important consequences for theorizing about ideal rationality, developing an account of the relationship between ordinary and ideal rationality. Because the sanctioning sense of ‘rational’ is absolute, it is rationally required to adopt the most rational attitude available, but which attitude is most rational can change depending on whether we are dealing with ideal agents or people more like ourselves. (shrink)

To any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that any set of reals of size ℵ 1 is meagre yet there are (...) ℵ 1 rectifiable curves in R 3 whose union is not meagre. The consistency of this statement when the phrase "rectifiable curves" is replaced by "straight lines" remains open. (shrink)

My paper sets out to demonstrate that Weber's ideal-typical theory of concept formation, subject to certain modifications, is compatible with the principles of philosophical hermeneutics and is therefore a valuable strategy of concept formation for interpretive historical inquiry. The essay begins with a brief recapitulation of the philosophical-hermeneutic approach to the human sciences. I then chart out the affinities as well as the discrepancies between philosophical hermeneutics and Weber's theory of the ideal type. Against this backdrop, I proceed (...) to offer a number of correctives and additions to Weber's theory so as to tighten its fit with philosophical hermeneutics. First, I argue that the ideal type's proper logic of concept formation is a logic of significance rather than a logic of commonality. Second, I claim that the relationship between the various empirical cases to which a given ideal type is (or can be) applied is a Wittgensteinian relationship of family resemblance. Finally, I present two main kinds of epistemological functions that ideal types can fulfill within the framework of historical inquiry: argumentative and orientational-clarificatory. (shrink)

Abstraction and idealization are the two notions that are most often discussed in the context of assumptions employed in the process of model building. These notions are also routinely used in philosophical debates such as that on the mechanistic account of explanation. Indeed, an objection to the mechanistic account has recently been formulated precisely on these grounds: mechanists cannot account for the common practice of idealizing difference-making factors in models in molecular biology. In this paper I revisit the debate and (...) I argue that the objection does not stand up to scrutiny. This is because it is riddled with a number of conceptual inconsistencies. By attempting to resolve the tensions, I also draw several general lessons regarding the difficulties of applying abstraction and idealization in scientific practice. Finally, I argue that more care is needed only when speaking of abstraction and idealization in a context in which these concepts play an important role in an argument, such as that on mechanistic explanation. (shrink)

Ordinary language and scientific language enable us to speak about, in a singular way, what we recognize not to exist: fictions, the contents of our hallucinations, abstract objects, and various idealized but nonexistent objects that our scientific theories are often couched in terms of. Indeed, references to such nonexistent items-especially in the case of the application of mathematics to the sciences-are indispensable. We cannot avoid talking about such things. Scientific and ordinary languages thus enable us to say things about Pegasus (...) or about hallucinated objects that are true, such as "Pegasus was believed by the ancient Greeks to be a flying horse," or "That elf I'm now hallucinating over there is wearing blue shoes." Standard contemporary metaphysical views and semantic analyses of singular idioms on offer in contemporary philosophy of language have not successfully accommodated these routine practices of saying true and false things about the nonexistent while simultaneously honoring the insight that such things do not exist in any way at all. That is, philosophers often feel driven to claim that such objects do exist, or they claim that all our talk isn't genuine truth-apt talk, but only pretence. This book reconfigures metaphysics in radical ways that allow the accommodation of our ordinary ways of speaking of what does not exist while retaining the absolutely crucial presupposition that such objects exist in no way at all, have no properties, and so are not the truth-makers for the truths and falsities that are about them. (shrink)