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Peano Arithmetic and \( \epsilon _0\)
We bound the growth rate of computable functions provably total in Peano Arithmetic. This is applied to show the independence of Goodstein’s number...
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On the Performance of Axiom Systems
One of the aims of proof theory is to calibrate the strength of axiom systems by invariants. According to Gödel’s discoveries these invariants will...
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The Two Sides of Modern Axiomatics: Dedekind and Peano, Hilbert and Bourbaki
This chapter focuses on two different facets of axiomatics: 1. the formal-logical side, linked to careful, rigorous establishing of the inferential...
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Provability logic: models within models in Peano Arithmetic
In 1994 Jech gave a model-theoretic proof of Gödel’s second incompleteness theorem for Zermelo–Fraenkel set theory in the following form:
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Intuitionistic sets and numbers: small set theory and Heyting arithmetic
It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set...
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Reverse Mathematics
Reverse mathematics is a new take on an old idea: asking which axioms are necessary to prove a given theorem. This question was first asked about the...
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Deduktive Systeme und Unvollständigkeit
In der Antike entwickelte sich der wissenschaftliche Diskurs rasant. Arithmetische und geometrische Theoreme wurden unter Annahme von Axiomen präzise...
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Two Paths to Logical Consequence: Pieri and the Peano School
This chapter1 has two main goals. First, it will explore the “negative” avenue leading from the concepts of independence and consistency to that of...
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So, What Does It All Mean?
Shelah–Soifer’s results we have discussed in this book seem surprising and even strange. How can the presence of the Axiom of Choice or its version...
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A Modern Rigorous Approach to Stratification in
NF /NFU The main feature of
NF /NFU is the notion of stratification, which sets it apart from other set theories. We define stratification and prove... -
Natural, Integral, and Rational Numbers
In this chapter we present a very detailed and slow-paced arithmetic exposition of the natural, integral, and rational number systems. Natural...
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Natürliche Zahlen und vollständige Induktion
Ein mathematischer Beweis ist eine Argumentationskette, durch welche die zu beweisende Aussage (der zu beweisende Satz) in mehr oder weniger...
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H.S.M. Coxeter’s Theory of Accessibility: From Mario Pieri to Marvin Greenberg
In the 1960s, H. S. M. Coxeter adopted a set of incidence axioms similar to one O. Veblen and J. W. Young proposed in 1910, to study projective...
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Completeness: From Husserl to Carnap
In his Doppelvortrag (1901), Edmund Husserl introduced two concepts of “definiteness” which have been interpreted as a vindication of his role in the...
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Logic in the History and Philosophy of Mathematical Practice
Mathematical logic is the study of reasoning about mathematical objects and the degree to which mathematical and scientific reasoning can be...
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Reverse Mathematics
Reverse mathematics is a new take on an old idea: asking which axioms are necessary to prove a given theorem. This question was first asked about the...
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Implicitly Defining Mathematical Terms
This chapter expounds the view that mathematical implicit definitions, i.e., systems of axioms as well as abstraction principles, underpin some...
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Eine kurze Einführung in die mathematische Logik
Um die bedeutenden Beiträge von Kurt Gödel zur mathematischen Logik und Mengenlehre zu verstehen und zu würdigen, sollten die Leser zumindest eine...
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Logic in the History and Philosophy of Mathematical Practice
Mathematical logic is the study of reasoning about mathematical objects and the degree to which mathematical and scientific reasoning can be...
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The typewritten manuscripts
If one builds on top of the Peano axioms the logic of the Principia mathematica1 (natural numbers as individuals), with the axiom of choice (for all...