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A combinatorial proof of a combinatorial theorem

A combinatorial proof of a combinatorial theorem Acta Mathematicae Academia Scientiarum Hungaricae Tomus 26 (1--2), (1975), 3--7. A COMBINATORIAL PROOF OF A COMBINATORIAL THEOREM By J. M. HENLE and E. M. KLEINBERG (Cambridge) w 1. Many theorems of pure (combinatorial) set theory, such as those con- cerned with the relative sizes of so called "large cardinals", were first proved using techniques which were primarily "logical" rather than "set theoretic". By this we mean that although the theorem at hand was concerned only with basic properties about sets, its proof strayed from using only properties of sets and rather made strong use of the theory associated with logical formulas, models, satisfaction, and so forth. For example, a standard "model theoretic" format for showing the least "type x large cardinal" to exceed the least "type y large cardinal" is to show that type x large cardinals are "indescribable" with respect to a certain class of logical formulas (i.e., that if ~p is a formula of the appropriate class which is true at the level of some type x cardinal z, ~hen q~ is true at the level of some cardinal less than z), and then to show that type y cardinals are describable by a formula over which http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

A combinatorial proof of a combinatorial theorem

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References (4)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01895944
Publisher site
See Article on Publisher Site

Abstract

Acta Mathematicae Academia Scientiarum Hungaricae Tomus 26 (1--2), (1975), 3--7. A COMBINATORIAL PROOF OF A COMBINATORIAL THEOREM By J. M. HENLE and E. M. KLEINBERG (Cambridge) w 1. Many theorems of pure (combinatorial) set theory, such as those con- cerned with the relative sizes of so called "large cardinals", were first proved using techniques which were primarily "logical" rather than "set theoretic". By this we mean that although the theorem at hand was concerned only with basic properties about sets, its proof strayed from using only properties of sets and rather made strong use of the theory associated with logical formulas, models, satisfaction, and so forth. For example, a standard "model theoretic" format for showing the least "type x large cardinal" to exceed the least "type y large cardinal" is to show that type x large cardinals are "indescribable" with respect to a certain class of logical formulas (i.e., that if ~p is a formula of the appropriate class which is true at the level of some type x cardinal z, ~hen q~ is true at the level of some cardinal less than z), and then to show that type y cardinals are describable by a formula over which

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Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: May 21, 2016

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