Constructive set theory and Brouwerian principles

The paper furnishes realizability models of constructive Zermelo-Fraenkel set theory, CZF, which also validate Brouwerian principles such as the axiom of continuous choice (CC), the fan theorem (FT), and monotone bar induction (BIM), and thereby determines the proof-theoretic strength of CZF augmented by these principles. The upshot is that CZF+CC+FT possesses the same strength as CZF, or more precisely, that CZF+CC+FTis conservative over CZF for 02 statements of arithmetic, whereas the addition of a restricted version of bar induction to CZF (called decidable bar induction, BID) leads to greater proof-theoretic strength in that CZF+BID proves the consistency of CZF

A system to enrich for primitive streak-derivatives, definitive endoderm and mesoderm, from pluripotent cells in culture

Two lineages of endoderm develop during mammalian embryogenesis, the primitive endoderm in the pre-implantation blastocyst and the definitive endoderm at gastrulation. This complexity of endoderm cell populations is mirrored during pluripotent cell differentiation in vitro and has hindered the identification and purification of the definitive endoderm for use as a substrate for further differentiation. The aggregation and differentiation of early primitive ectoderm-like (EPL) cells, resulting in the formation of EPL-cell derived embryoid bodies (EPLEBs), is a model of gastrulation that progresses through the sequential formation of primitive streak-like intermediates to nascent mesoderm and more differentiated mesoderm populations. EPL cell-derived EBs have been further analysed for the formation of definitive endoderm by detailed morphological studies, gene expression and a protein uptake assay. In comparison to embryoid bodies derived from ES cells, which form primitive and definitive endoderm, the endoderm compartment of embryoid bodies formed from EPL cells was comprised almost exclusively of definitive endoderm. Definitive endoderm was defined as a population of squamous cells that expressed Sox17, CXCR4 and Trh, which formed without the prior formation of primitive endoderm and was unable to endocytose horseradish peroxidase from the medium. Definitive endoderm formed in EPLEBs provides a substrate for further differentiation into specific endoderm lineages; these lineages can be used as research tools for understanding the mechanisms controlling lineage establishment and the nature of the transient intermediates formed. The similarity between mouse EPL cells and human ES cells suggests EPLEBs can be used as a model system for the development of technologies to enrich for the formation of human ES cell-derived definitive endoderm in the future.Sveltana Vassilieva, Hweee Ngee Goh, Kevin X. Lau, James N. Hughes, Mary Familari, Peter D. Rathjen and Joy Rathje

Investigations of subsystems of second order arithmetic and set theory in strength between Pi-1-1-CA and delta-1-2-CA+BI: part I

This paper is the rst of a series of two. It contains proof{theoretic investigations on subtheories of second order arithmetic and set theory. Among the principles on which these theories are based one nds autonomously iterated positive and monotone inductive de ni- tions, 1 1 trans nite recursion, 1 2 trans nite recursion, trans nitely iterated 1 1 dependent choices, extended Bar rules for provably de nable well-orderings as well as their set-theoretic counterparts which are based on extensions of Kripke-Platek set theory. This rst part intro- duces all the principles and theories. It provides lower bounds for their strength measured in terms of the amount of trans nite induction they achieve to prove. In other words, it determines lower bounds for their proof-theoretic ordinals which are expressed by means of ordinal representation systems. The second part of the paper will be concerned with ordinal analysis. It will show that the lower bounds established in the present paper are indeed sharp, thereby providing the proof-theoretic ordinals. All the results were obtained more then 20 years ago (in German) in the author's PhD thesis [43] but have never been published before, though the thesis received a review (MR 91m#03062). I think it is high time it got published

Ordinal Analysis of Intuitionistic Power and Exponentiation Kripke Platek Set Theory

Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive definitions, subsystems of analysis and finite type systems. With the pioneering work of Gerhard Jäger in the late 1970 s and early 1980s, the focus switched to set theories, furnishing ordinal-theoretic proof theory with a uniform and elegant framework. More recently it was shown that these tools can even sometimes be adapted to the context of strong axioms such as the powerset axiom, where one does not attain complete cut elimination but can nevertheless extract witnessing information and characterize the strength of the theory in terms of provable heights of the cumulative hierarchy. Here this technology is applied to intuitionistic Kripke-Platek set theories IKP(P) and IKP(E), where the operation of powerset and exponentiation, respectively, is allowed as a primitive in the separation and collection schemata. In particular, IKP(P) proves the powerset axiom whereas IKP(E) proves the exponentiation axiom. The latter expresses that given any sets A and B, the collection of all functions from A to B is a set, too. While IKP(P) can be dealt with in a similar vein as its classical cousin, the treatment of IKP(E) posed considerable obstacles. One of them was that in the infinitary system the levels of terms become a moving target as they cannot be assigned a fixed level in the formal cumulative hierarchy solely based on their syntactic structure. As adumbrated in an earlier paper, the results of this paper are an important tool in showing that several intuitionistic set theories with the collection axiom possess the existence property, i.e., if they prove an existential theorem then a witness can be provably described in the theory, one example being intuitionistic Zermelo-Fraenkel set theory with bounded separation

Preservation of choice principles under realizability

Especially nice models of intuitionistic set theories are realizability models V(A)V(A)⁠, where AA is an applicative structure or partial combinatory algebra. This paper is concerned with the preservation of various choice principles in V(A)V(A) if assumed in the underlying universe VV⁠, adopting Constructive Zermelo–Fraenkel as background theory for all of these investigations. Examples of choice principles are the axiom schemes of countable choice, dependent choice, relativized dependent choice and the presentation axiom. It is shown that any of these axioms holds in V(A)V(A) for every applicative structure AA if it holds in the background universe.1 It is also shown that a weak form of the countable axiom of choice, ACω,ωACω,ω⁠, is rendered true in any V(A)V(A) regardless of whether it holds in the background universe. The paper extends work by McCarty (1984, Realizability and Recursive Mathematics, PhD Thesis) and Rathjen (2006, Realizability for constructive Zermelo–Fraenkel set theory

Strategies for wheat stripe rust pathogenicity identified by transcriptome sequencing

Stripe rust caused by the fungus Puccinia striiformis f.sp. tritici (Pst) is a major constraint to wheat production worldwide. The molecular events that underlie Pst pathogenicity are largely unknown. Like all rusts, Pst creates a specialized cellular structure within host cells called the haustorium to obtain nutrients from wheat, and to secrete pathogenicity factors called effector proteins. We purified Pst haustoria and used next-generation sequencing platforms to assemble the haustorial transcriptome as well as the transcriptome of germinated spores. 12,282 transcripts were assembled from 454-pyrosequencing data and used as reference for digital gene expression analysis to compare the germinated uredinospores and haustoria transcriptomes based on Illumina RNAseq data. More than 400 genes encoding secreted proteins which constitute candidate effectors were identified from the haustorial transcriptome, with two thirds of these up-regulated in this tissue compared to germinated spores. RT-PCR analysis confirmed the expression patterns of 94 effector candidates. The analysis also revealed that spores rely mainly on stored energy reserves for growth and development, while haustoria take up host nutrients for massive energy production for biosynthetic pathways and the ultimate production of spores. Together, these studies substantially increase our knowledge of potential Pst effectors and provide new insights into the pathogenic strategies of this important organism.J.P.R. is an ARC Future Fellow (FT0992129). This project has been supported by Bioplatforms Australia through funding from the Commonwealth Government NCRIS and Education Investment Fund Super Science programs

Formation of a primitive ectoderm like cell population, EPL cells, from ES cells in response to biologically derived factors

The primitive ectoderm of the mouse embryo arises from the inner cell mass between 4.75 and 5.25 days post coitum, around the time of implantation. Positioned at a pivotal time in development, just prior to formation of the three germ layers of the embryo proper, the primitive ectoderm responds directly to the signals generated during gastrulation. We have identified a conditioned medium, MEDII, which caused the homogeneous conversion of ES cells to a morphologically distinct cell population, termed early primitive ectoderm-like (EPL) cells. EPL cells expressed the pluripotent cell markers Oct4, SSEA1 and alkaline phosphatase. However, the formation of EPL cells was accompanied by alterations in Fgf5, Gbx2 and Rex1 expression, a loss in chimaera forming ability, changes in factor responsiveness and modified differentiation capabilities, all consistent with the identification of EPL cells as equivalent to the primitive ectoderm population of the 5.5 to 6.0 days post coitum embryo. EPL cell formation could be reversed in the presence of LIF and withdrawal of MEDII, which suggested that EPL cell formation was not a terminal differentiation event but reflected the ability of pluripotent cells to adopt distinct cell states in response to specific factors. Partial purification of MEDII revealed the presence of two separable biological activities, both of which were required for the induction and maintenance of EPL cells. We show here the first demonstration of uniform differentiation of ES cells in response to biological factors. The formation of primitive ectoderm, both in vivo and in vitro, appears to be an obligatory step in the differentiation of the inner cell mass or ES cells into cell lineages of the embryonic germ layers. EPL cells potentially represent a model for the development of lineage specific differentiation protocols and analysis of gastrulation at a molecular level. An understanding of the active components of MEDII may provide a route for the identification of factors which induce primitive ectoderm formation in vivo.Joy Rathjen, Julie-Anne Lake, Michael D. Bettess, Jenny M. Washington, Gavin Chapman and Peter D. Rathje

Directed differentiation of pluripotent cells to neural lineages: homogeneous formation and differentiation of a neurectoderm population

During embryogenesis the central and peripheral nervous systems arise from a neural precursor population, neurectoderm, formed during gastrulation. We demonstrate the differentiation of mouse embryonic stem cells to neurectoderm in culture, in a manner which recapitulates embryogenesis, with the sequential and homogeneous formation of primitive ectoderm, neural plate and neural tube. Formation of neurectoderm occurs in the absence of extraembryonic endoderm or mesoderm and results in a stratified epithelium of cells with morphology, gene expression and differentiation potential consistent with positionally unspecified neural tube. Differentiation of this population to homogeneous populations of neural crest or glia was also achieved. Neurectoderm formation in culture allows elucidation of signals involved in neural specification and generation of implantable cell populations for therapeutic use