Conceptions of infinity and set in Lorenzen’s operationist system

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity

Universism and extensions of V

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist

Modern Class Forcing

We survey recent developments in the theory of class forcing for- malized in the second-order set-theoretic setting

A general framework for a Second Philosophy analysis of set-theoretic methodology

Penelope Maddy’s Second Philosophy is one of the most well-known ap- proaches in recent philosophy of mathematics. She applies her second-philosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of means-ends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows us to clarify the procedure and aims of the Second Philosopher’s investigation into set-theoretic methodology; pro- vides a platform to analyze the Second Philosopher’s methods themselves; and can be applied to further questions in the philosophy of set theory

An AP2/ERF transcription factor ERF139 coordinates xylem cell expansion and secondary cell wall deposition

Abstract Differentiation of xylem elements involves cell expansion, secondary cell wall deposition and programmed cell death. Transitions between these phases require strict spatiotemporal control. The function of Populus ERF139 (Potri.013G101100) in xylem differentiation was characterized in transgenic overexpression and dominant repressor lines of ERF139 in hybrid aspen (Populus tremula x tremuloides). Xylem properties, secondary cell wall (SCW) chemistry and downstream targets were analyzed in both types of transgenic trees using microscopy techniques, FT-IR, pyrolysis-GC/MS, wet chemistry methods and RNA sequencing. Opposite phenotypes were observed in the secondary xylem vessel sizes and SCW chemistry in the two different types of transgenic trees, supporting the function of ERF139 in suppressing the radial expansion of vessel elements and stimulating accumulation of guaiacyl-type lignin and possibly also xylan. Comparative transcriptomics identified genes related to SCW biosynthesis (LAC5, LBD15, MYB86) and salt and drought stress responsive genes (ANAC002, ABA1) as potential direct targets of ERF139. The phenotypes of the transgenic trees and the stem expression profiles of ERF139 potential target genes support the role of ERF139 as a transcriptional regulator of xylem cell expansion and SCW formation, possibly in response to osmotic changes of the cells. This article is protected by copyright. All rights reserved.Peer reviewe