Partition Relations

Hajnal, A.; Larson, Jean A.

doi:10.1007/978-1-4020-5764-9_3

Cited by 17 publications

(8 citation statements)

References 45 publications

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“…The delay between the thesis and the paper is indicative of the difficulty of the proof and the process of checking its correctness. A sketch of Schipperus' proof, divided in seven subsections, is given on pages 188-209 of the excellent survey article [17] by András Hajnal and Larson. The reason that Schipperus focused on ordinals of the type ω ω β is that if the ordinal α is not a power of ω then it cannot satisfy α −→ (α, 3), as shown in Observation 2.2.…”

Section: Ordinal Partition Relationsmentioning

confidence: 99%

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Formalizing Ordinal Partition Relations Using Isabelle/HOL

Džamonja

Koutsoukou-Argyraki

Paulson

2021

Experimental Mathematics

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This is an overview of a formalization project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erd ős-Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all m ∈ N, ω ω −→ (ω ω , m). This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalization process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic.

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Section: Ordinal Partition Relationsmentioning

confidence: 99%

“…2.2 In this section, we have mostly concentrated on the result we formalized and Erdős's problem. Information on some additional instances of α −→ (α, m) for m > 3 can be found in the Hajnal-Larson paper [17].…”

Section: Ordinal Partition Relationsmentioning

confidence: 99%

Formalizing Ordinal Partition Relations Using Isabelle/HOL

Džamonja

Koutsoukou-Argyraki

Paulson

2021

Experimental Mathematics

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“…The delay between the thesis and the paper is indicative of the difficulty of the proof and the process of checking its correctness. A sketch of Schipperus' proof, divided in seven subsections, is given on pages 188-209 of the excellent survey article [19] by András Hajnal and Larson. The reason that Schipperus focused on ordinals of the type ω ω β is that if the ordinal α is not a power of ω then it cannot satisfy α −→ (α, 3), as shown in the following Observation 2.2.…”

Section: Ordinal Partition Relationsmentioning

confidence: 99%

“…In this section, we have mostly concentrated on the result we formalised and Erdős's problem. Information on some additional instances of α −→ (α, m) for m > 3 can be found in the Hajnal-Larson paper [19].…”

Section: Ordinal Partition Relationsmentioning

confidence: 99%

Formalising Ordinal Partition Relations Using Isabelle/HOL

Džamonja,

Koutsoukou-Argyraki,

Paulson

2020

Preprint

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This is an overview of a formalisation project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erdős-Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all m ∈ N, ω ω −→ (ω ω , m). This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalisation process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic.

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“…104 The results of Erdős-Hajnal-Rado [1965] were extended in Byzantine detail to the general situation without GCH by the book Erdős-Hajnal-Máté-Rado [1984]. See Hajnal-Larson [2008] for recent work on partition relations.…”

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confidence: 99%

Set Theory from Cantor to Cohen

Kanamori¹

2012

Handbook of the History of Logic

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Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The whole transfinite landscape can be viewed as having been articulated by Cantor in significant part to solve the Continuum Problem. Zermelo's axioms can be construed as clarifying the set existence commitments of a single proof, of his Well-Ordering Theorem. Set theory is a particular case of a field of mathematics in which seminal proofs and pivotal problems actually shaped the basic concepts and forged axiomatizations, these transmuting the very notion of set. There were two main junctures, the first being when Zermelo through his axiomatization shifted the notion of set from Cantor's range of inherently structured sets to sets solely structured by membership and governed and generated by axioms. The second juncture was when the Replacement and Foundation Axioms were adjoined and a first-order setting was established; thus transfinite recursion was incorporated and results about all sets could established through these means, including results about definability and inner models. With the emergence of the cumulative hierarchy picture, set theory can be regarded as becoming a theory of well-foundedness, later to expand to a study of consistency strength. Throughout, the subject has not only been sustained by the axiomatic tradition through Gödel and Cohen but also fueled by Cantor's two legacies, the extension of number into the transfinite as transmuted into the theory of large cardinals and the investigation of definable sets of reals as transmuted into descriptive set theory. All this can be regarded as having a historical and mathematical logic internal to set theory, one that is often misrepresented at critical junctures in textbooks (as will be pointed out). This view, from inside set theory and about itself, serves to shift the focus to Akihiro K...

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Partition Relations

Cited by 17 publications

References 45 publications

Formalizing Ordinal Partition Relations Using Isabelle/HOL

Formalizing Ordinal Partition Relations Using Isabelle/HOL

Formalising Ordinal Partition Relations Using Isabelle/HOL

Set Theory from Cantor to Cohen

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