The prevalent interpretation of Gödel’s Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of the underlying mathematical theorem depends on several arbitrary formalisation choices. In this paper I examine the theorem’s dependency regarding Gödel numberings. I introducedeviantnumberings, yielding provability predicates satisfying Löb’s conditions, which result in provable consistency sentences. According to the main result of this paper however, these (...) “counterexamples” do not refute the theorem’s prevalent interpretation, since once a natural class ofadmissiblenumberings is singled out, invariance is maintained. (shrink)

In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study (...) of certain principles of self-referential truth. (shrink)

There is a well-known gap between metamathematical theorems and their philosophical interpretations. Take Tarski’s Theorem. According to its prevalent interpretation, the collection of all arithmetical truths is not arithmetically definable. However, the underlying metamathematical theorem merely establishes the arithmetical undefinability of a set of specific Gödel codes of certain artefactual entities, such as infix strings, which are true in the standard model. That is, as opposed to its philosophical reading, the metamathematical theorem is formulated (and proved) relative to a specific (...) choice of the Gödel numbering and the notation system. A similar observation applies to Gödel’s and Church’s theorems, which are commonly taken to impose severe limitations on what can be proved and computed using the resources of certain formalisms. The philosophical force of these limitative results heavily relies on the belief that these theorems do not depend on contingencies regarding the underlying formalisation choices. The main aim of this paper is to provide metamathematical facts which support this belief. While employing a fixed notation system, I showed in previous work (_Review of Symbolic Logic_, 2021, 14(1):51–84) how to abstract away from the choice of the Gödel numbering. In the present paper, I extend this work by establishing versions of Tarski’s, Gödel’s and Church’s theorems which are invariant regarding _both_ the notation system and the numbering. This paper thus provides a further step towards absolute versions of metamathematical results which do not rely on contingent formalisation choices. (shrink)

This paper investigates the conditions under which diagonal sentences can be taken to constitute paradigmatic cases of self-reference. We put forward well-motivated constraints on the diagonal operator and the coding apparatus which separate paradigmatic self-referential sentences, for instance obtained via Gödel’s diagonalization method, from accidental diagonal sentences. In particular, we show that these constraints successfully exclude refutable Henkin sentences, as constructed by Kreisel.

Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of (...) arithmetic. In doing so, we provide set-theoretical facts which, we believe, are crucial for informed assessment of reductionism. (shrink)

On business ethics of Papuan peoples in Indonesia.

Ce livre est la brillante synthèse d'une œuvre qui figure au premier plan de la pensée catholique au XXe siècle. L'élan qui anime tous les écrits de Hans Urs von Balthasar ne s'affirme dans aucun livre mieux que dans cette Théologie de l'histoire. Celle-ci est parfaitement centrée sur le temps du Christ : Dieu toujours présent parmi les hommes. Depuis l'incarnation, l'histoire se déroule dans ce temps nouveau ; il est la norme de l'évolution humaine ; c'est lui qui (...) mesure le cours de l'existence, le progrès vertical et horizontal de l'humanité. Une fresque extraordinaire se déploie sous les yeux du lecteur l'histoire s'intériorise en un destin personnel, la vocation de chacun se situe dans un destin historique universel. Cette vision eschatologique, d'un réalisme éblouissant et pathétique, nous montre que la fin des temps et les temps nouveaux sont déjà commencés. (shrink)

v. 1. Truth of the world -- v. 2 Truth of God -- v. 3. The spirit of truth.

In this paper, I set out to argue in favour of a philosophical posthumanist and Nietzschean reading of Ira Levin’s This Perfect Day while demonstrating how transhumanism isunbefitting of being called a Nietzschean theory. I will do this by establishing Chip, the protagonist, as a posthuman and being on the path of the ‘Overhuman’ whereas Wei, the antagonist, will be illustrated as Chip’s intended counterpart the Last Human. Through explaining ‘Transhumansim’ and connecting the field to Wei, I will showcase a (...) new way of reading transhumanist ideology, namely ‘Weiism’. The polar opposite in characters of ‘Overhuman’ and ‘Last Human’ will argue in favour of the Nietzschean paradoxical, as part of philosophical posthumanism. Amor fati and the eternal recurrence become posthumanist narrative tools through their weaving into the posthuman narrative of Chip. My argument is that by making philosophical posthumanism more existentialist in nature, it becomes simpler to argue against nihilistic and totalitarian tendencies of worshipping technology. Finally, it is the spiralling motion forward as a mix between the Nietzschean, eternally cyclical re-inventing of oneself and the posthumanist consideration for the future which makes for a literary combination of the two fields of study. (shrink)