This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most (...) striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic. (shrink)

Using hitherto unpublished manuscripts from the Zermelo Nachlass, I describe the development of the notion of definiteness and the discussion about it, giving a conclusive picture of Zermelo's thoughts up to the late thirties. As it turns out, Zermelo's considerations about definiteness are intimately related to his concept of a Cantorian universe of categorically definable sets that may be considered an inner model of set theory in an ideationally given universe of classes.

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz” in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world of objects that could only be grasped adequately by (...) infinitary means. So the refutation might serve as a final clue to his epistemological credo.Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic. However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”. (shrink)

Towards the end of his 1930 paper on boundary numbers and domains of sets Zermelo briefly discusses the questions of consistency and of the existence of an unbounded sequence of strongly inaccessible cardinals, deferring a detailed discussion to a later paper which never appeared. In a report to the Emergency Community of German Science from December 1930 about investigations in progress he mentions that some of the intended extensions of these topics had been worked out and were nearly ready for (...) publication. Using manuscripts from his Nachlass, we exhibit essential features of these extensions. (shrink)

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz”(“Relativism in Set Theory and the So-Called Theorem of Skolem”) in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world (...) of objects that could only be grasped adequately by infinitary means. So the refutation might serve as a final clue to his epistemological credo.Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic (cf. [8]). However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, (cf. [1]) he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”. (shrink)

Colonization of public education—the process by which schools are overwhelmed and penetrated by non-educational imperatives—is usually believed to be caused by capitalism and the hegemonic ideological structures it produces. In this paper I argue that in the case of the United States an additional mechanism produces strong colonizing effects: the institution of local control. In the context of contemporary institutional conditions, local control is the lynch-pin for the production of socio-economic segregation, cumulative disadvantages, and the mythology of popular control disguising (...) the growing control of public schooling through unaccountable bureaucracies and private corporations. (shrink)

This book presents a range of essays on the conceptual foundations of physicalism, mental causation and human agency, written by established and leading authors ...

Epiphenomenalism (with respect to the intentional) may be construed as the claim that intentional events as such are causally inert. If property-dualism is true, that is if intentional properties aren't identical with or reducible to physical properties, then epiphenomenalism is inevitable. Some philosophers (most notably Fodor) have argued that their irreducibility notwithstanding intentional properties are causally responsible properties (or at least supervenient upon causally responsible properties). The arguments supporting these promising allegations aren't convincing. Given the tmth of property-dualism, there is (...) no (Fodorian or any other) escape-route from epiphenomenalism. (shrink)

Epiphenomenalism (with respect to the intentional) may be construed as the claim that intentional events as such are causally inert. If property-dualism is true, that is if intentional properties aren't identical with or reducible to physical properties, then epiphenomenalism is inevitable. Some philosophers (most notably Fodor) have argued that their irreducibility notwithstanding intentional properties are causally responsible properties (or at least supervenient upon causally responsible properties). The arguments supporting these promising allegations aren't convincing. Given the tmth of property-dualism, there is (...) no (Fodorian or any other) escape-route from epiphenomenalism. (shrink)

Für die Sinnesdatenphilosophie sind zwei Behauptungen zentral: (1) Der sinnliche Gehalt des sinnlich-perzeptiven Bewußtseins ist gegenständlich oder objektual aufzufassen, d.h. es gibt Sinnesdaten oder phänomenale Gegenstände mit phänomenalen Beschaffenheiten; (2) Das sinnlich-perzeptive Bewußtsein ist nach dem Modell: Subjekt — zweistelliges Prädikat — Objekt zu analysieren, d.h. Sinnesdaten sind uns als Objekte unseres sinnlich-perzeptiven Bewußtseins unmittelbar gegeben. Die Annahmen (l)und (2) führen in philosophische Aporien. Es ist jedoch möglich, den sinnlichen Gehalt des sinnlich-perzeptiven Bewußtseins nicht-gegenständlich aufzufassen und nach einem Modell zu (...) analysieren, welches eine solche Vergegenständlichung und die damit verbundenen Probleme vermeidet. Die sich gegen ein solches nicht-objektuales oder adverbiales Analysemodell erhebenden phänomenologischen und semantischen Einwände lassen sich entkräften. (shrink)

Für die Sinnesdatenphilosophie sind zwei Behauptungen zentral: (1) Der sinnliche Gehalt des sinnlich-perzeptiven Bewußtseins ist gegenständlich oder objektual aufzufassen, d.h. es gibt Sinnesdaten oder phänomenale Gegenstände mit phänomenalen Beschaffenheiten; (2) Das sinnlich-perzeptive Bewußtsein ist nach dem Modell: Subjekt — zweistelliges Prädikat — Objekt zu analysieren, d.h. Sinnesdaten sind uns als Objekte unseres sinnlich-perzeptiven Bewußtseins unmittelbar gegeben. Die Annahmen (l)und (2) führen in philosophische Aporien. Es ist jedoch möglich, den sinnlichen Gehalt des sinnlich-perzeptiven Bewußtseins nicht-gegenständlich aufzufassen und nach einem Modell zu (...) analysieren, welches eine solche Vergegenständlichung und die damit verbundenen Probleme vermeidet. Die sich gegen ein solches nicht-objektuales oder adverbiales Analysemodell erhebenden phänomenologischen und semantischen Einwände lassen sich entkräften. (shrink)

Die Struktur des unmittelbaren epistemischen Selbstbewußtseins läßt sich am besten unter Zugrundelegung und Entfaltung folgen der Annahmen explizieren: (i) Es gibt mentale Substanzen, (ii) Es gibt Wahrheitswerttransparente Cogitopropositionen oder CogitoSachverhalte, (iii) Jedes Ich kann seine und nur seine es betreffenden Cogitopropositionen fassen, womit es ipso facto ein unmittelbares Wissen um sich selbst hat. Dieses unmittelbare selbstreflexive Wissen kann in assertorischen Sätzen seinen Ausdruck finden, (iv) Die andere Iche betreffenden Cogitopropositionen können diesen Ichen nur mit Hilfe quasiindikatorischer Mittel zugeschrieben werden. Jedes (...) Ich kann wenigstens eine Entität per se individuieren, nämlich sich selbst. Per se Individuation anderer Iche ist nur dann möglich, wenn jedem Ich nicht nur Cogitopropositionen, sondern auch es und nur es betreffende Propositionen zugeordnet werden, die von ihm selbst, aber auch von anderen gefaßt bzw. gewußt werden können. (shrink)

Die Struktur des unmittelbaren epistemischen Selbstbewußtseins läßt sich am besten unter Zugrundelegung und Entfaltung folgen der Annahmen explizieren: (i) Es gibt mentale Substanzen, (ii) Es gibt Wahrheitswerttransparente Cogitopropositionen oder CogitoSachverhalte, (iii) Jedes Ich kann seine und nur seine es betreffenden Cogitopropositionen fassen, womit es ipso facto ein unmittelbares Wissen um sich selbst hat. Dieses unmittelbare selbstreflexive Wissen kann in assertorischen Sätzen seinen Ausdruck finden, (iv) Die andere Iche betreffenden Cogitopropositionen können diesen Ichen nur mit Hilfe quasiindikatorischer Mittel zugeschrieben werden. Jedes (...) Ich kann wenigstens eine Entität per se individuieren, nämlich sich selbst. Per se Individuation anderer Iche ist nur dann möglich, wenn jedem Ich nicht nur Cogitopropositionen, sondern auch es und nur es betreffende Propositionen zugeordnet werden, die von ihm selbst, aber auch von anderen gefaßt bzw. gewußt werden können. (shrink)