The set-theoretic multiverse

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.Comment: 35 page

An order-theoretic analysis of interpretations among propositional deductive systems

In this paper we study interpretations and equivalences of propositional deductive systems by using a quantale-theoretic approach introduced by Galatos and Tsinakis. Our aim is to provide a general order-theoretic framework which is able to describe and characterize both strong and weak forms of interpretations among propositional deductive systems also in the cases where the systems have different underlying languages

Buddhist Philosophy of Logic

Logic in Buddhist Philosophy concerns the systematic study of anumāna (often translated as inference) as developed by Dignāga (480-540 c.e.) and Dharmakīti (600-660 c.e.). Buddhist logicians think of inference as an instrument of knowledge (pramāṇa) and, thus, logic is considered to constitute part of epistemology in the Buddhist tradition. According to the prevalent 20th and early 21st century ‘Western’ conception of logic, however, logical study is the formal study of arguments. If we understand the nature of logic to be formal, it is difficult to see what bearing logic has on knowledge. In this paper, by weaving together the main threads of thought that are salient in Dignāga’s and Dharmakīti’s texts, I shall re-conceive the nature of logic in the context of epistemology and demarcate the logical part of epistemology which can be recognised as logic. I shall demonstrate that we can recognise the logical significance of inference as understood by Buddhist logicians despite the fact that its logical significance lies within the context of knowledge

Formation of adenosine 5′-tetraphosphate from the acyl phosphate intermediate: a difference between the MurC and MurD synthetases of Escherichia coli

AbstractThe mechanism of the Mur synthetases of peptidoglycan biosynthesis is thought to involve in each case the successive formation of an acyl phosphate and a tetrahedral intermediate. The existence of the acyl phosphates for the MurC and MurD enzymes from Escherichia coli was firmly established by their in situ reduction by sodium borohydride followed by acid hydrolysis, yielding the corresponding amino alcohols. Furthermore, it was found that MurD, but not MurC, catalyses the synthesis of adenosine 5′-tetraphosphate from the acyl phosphate, thereby substantiating its existence and pointing out a difference between the two enzymes

When logic meets engineering: introduction to logical issues in the history and philosophy of computer science

Introduction to a Journal Special issue on Logical Issues in the History and Philosophy of Computer Scienc

Periplasmic phosphorylation of lipid A is linked to the synthesis of undecaprenyl phosphate

One-third of the lipid A found in the Escherichia coli outer membrane contains an unsubstituted diphosphate unit at position 1 (lipid A 1-diphosphate). We now report an inner membrane enzyme, LpxT (YeiU), which specifically transfers a phosphate group to lipid A, forming the 1-diphosphate species. 32P-labelled lipid A obtained from lpxT mutants do not produce lipid A 1-diphosphate. In vitro assays with Kdo2-[4′-32P]lipid A as the acceptor shows that LpxT uses undecaprenyl pyrophosphate as the substrate donor. Inhibition of lipid A 1-diphosphate formation in wild-type bacteria was demonstrated by sequestering undecaprenyl pyrophosphate with the cyclic polypeptide antibiotic bacitracin, providing evidence that undecaprenyl pyrophosphate serves as the donor substrate within whole bacteria. LpxT-catalysed phosphorylation is dependent upon transport of lipid A across the inner membrane by MsbA, a lipid A flippase, indicating a periplasmic active site. In conclusion, we demonstrate a novel pathway in the periplasmic modification of lipid A that is directly linked to the synthesis of undecaprenyl phosphate, an essential carrier lipid required for the synthesis of various bacterial polymers, such as peptidoglycan

Computability, G\"odel's Incompleteness Theorem, and an inherent limit on the predictability of evolution

The process of evolutionary diversification unfolds in a vast genotypic space of potential outcomes. During the past century there have been remarkable advances in the development of theory for this diversification, and the theory's success rests, in part, on the scope of its applicability. A great deal of this theory focuses on a relatively small subset of the space of potential genotypes, chosen largely based on historical or contemporary patterns, and then predicts the evolutionary dynamics within this pre-defined set. To what extent can such an approach be pushed to a broader perspective that accounts for the potential open-endedness of evolutionary diversification? There have been a number of significant theoretical developments along these lines but the question of how far such theory can be pushed has not been addressed. Here a theorem is proven demonstrating that, because of the digital nature of inheritance, there are inherent limits on the kinds of questions that can be answered using such an approach. In particular, even in extremely simple evolutionary systems a complete theory accounting for the potential open-endedness of evolution is unattainable unless evolution is progressive. The theorem is closely related to G\"odel's Incompleteness Theorem and to the Halting Problem from computability theory.Comment: Journal of the Royal Society, Interface 201