Peter Koellner

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Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general(More)
The ActA protein of the gram-positive pathogen Listeria monocytogenes is a 90-kDa polypeptide required for interaction of the bacteria with components of the host cell microfilament system to generate intra- and intercellular movement. To study the localization, distribution, and expression of the ActA polypeptide in L. monocytogenes grown either in broth(More)
The discovery of non-Euclidean geometries (in the 19 century) undermined the claim that Euclidean geometry is the one true geometry and instead led to a plurality of geometries no one of which could be said (without qualification) to be “truer” than the others. In a similar spirit many have claimed that the discovery of independence results for arithmetic(More)
The incompleteness theorems show that for every sufficiently strong consistent formal system of mathematics there are mathematical statements undecided relative to the system. A natural and intriguing question is whether there are mathematical statements that are in some sense absolutely undecidable, that is, undecidable relative to any set of axioms that(More)
In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger(More)
The continuum hypotheses (CH) is one of the most central open problems in set theory, one that is important for both mathematical and philosophical reasons. The problem actually arose with the birth of set theory; indeed, in many respects it stimulated the birth of set theory. In 1874 Cantor had shown that there is a one-to-one correspondence between the(More)
The relative-dose-response (RDR) assay first proposed by Underwood has proved to be a useful indicator of marginal vitamin A status. We suggested that 3,4-didehydroretinol might be used in a simpler assay that requires only one blood sample for analysis. In the present study 24 healthy children aged 3.7-7.1 y were given 100 micrograms 3,4-didehydroretinyl(More)
In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and V B1 and V B2 are generic extensions of V satisfying CH then V B1 and V B2 agree on all Σ1-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for Σ1.(More)
Ever since the rise of non-standard models and the proliferation of the independence results there have been two conflicting positions in the foundations of mathematics. The first position—which we shall call pluralism—maintains that certain statements of mathematics do not have determinate truth-values. On this view, although it is admitted that there are(More)
The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measurable?”), cardinal arithmetic (“Does Cantor’s Continuum(More)