Cofinality of the partial ordering of functions from ω1 into ω under eventual domination

Thomas Jech; Karel Prikry

doi:10.1017/S0305004100061272

Cofinality of the partial ordering of functions from ω₁ into ω under eventual domination

Thomas Jech
, Karel Prikry

School of Mathematics

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

It is unknown whether there can exist a family of functions from ω₁ into ω of size less than 2^N₁that dominates all functions fromω1 Intoω. We show that there is no such family if the continuum is real-valued measurable, and that the existence of such a family has consequences for cardinal arithmetic, and is related to large cardinal axioms.

Original language	English (US)
Pages (from-to)	25-32
Number of pages	8
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Volume	95
Issue number	1
DOIs	https://doi.org/10.1017/S0305004100061272
State	Published - Jan 1984

Bibliographical note

Funding Information:
THEOREM 3*9 (Jensen's Covering Theorem [5]). / / L[U] does not exist then for every set X of ordinals of size Ki there exists a set YeK of sizefc^such that X £ Y. Combining Theorem 3-2, 3-9 and Lemma 3-8 we get: THEOREM 3-10. J/Cof (wwi) < 2Ni and if 2Ki is greater than the least possible value for 2K» then L[U] exists. This is Theorem C with L[U] in place of'a model with large cardinals'. The general theory of transitive models for large cardinals axioms along with corresponding covering theorems, as developed by Mitchell [8], provides the framework for the appropriate generalization of Theorem 3-10: 'L[U] exists' can be replaced by much stronger large cardinal axioms. Research supported by NSF grants MCS 78-01525 and MCS 80-20467.

Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

Publisher link

10.1017/S0305004100061272

Find It

Find It at U of M Libraries

Cite this

@article{9c5f8531bf6c40f1ab6ea567019bdc02,

title = "Cofinality of the partial ordering of functions from ω1 into ω under eventual domination",

abstract = "It is unknown whether there can exist a family of functions from ω1 into ω of size less than 2N1 that dominates all functions fromω1 Intoω. We show that there is no such family if the continuum is real-valued measurable, and that the existence of such a family has consequences for cardinal arithmetic, and is related to large cardinal axioms.",

author = "Thomas Jech and Karel Prikry",

note = "Funding Information: THEOREM 3*9 (Jensen's Covering Theorem [5]). / / L[U] does not exist then for every set X of ordinals of size Ki there exists a set YeK of sizefc\textasciicircum{}such that X £ Y. Combining Theorem 3-2, 3-9 and Lemma 3-8 we get: THEOREM 3-10. J/Cof (wwi) < 2Ni and if 2Ki is greater than the least possible value for 2K» then L[U] exists. This is Theorem C with L[U] in place of'a model with large cardinals'. The general theory of transitive models for large cardinals axioms along with corresponding covering theorems, as developed by Mitchell [8], provides the framework for the appropriate generalization of Theorem 3-10: 'L[U] exists' can be replaced by much stronger large cardinal axioms. Research supported by NSF grants MCS 78-01525 and MCS 80-20467. Copyright: Copyright 2016 Elsevier B.V., All rights reserved.",

year = "1984",

month = jan,

doi = "10.1017/S0305004100061272",

language = "English (US)",

volume = "95",

pages = "25--32",

journal = "Mathematical Proceedings of the Cambridge Philosophical Society",

issn = "0305-0041",

publisher = "Cambridge University Press",

number = "1",

}

TY - JOUR

T1 - Cofinality of the partial ordering of functions from ω1 into ω under eventual domination

AU - Jech, Thomas

AU - Prikry, Karel

N1 - Funding Information: THEOREM 3*9 (Jensen's Covering Theorem [5]). / / L[U] does not exist then for every set X of ordinals of size Ki there exists a set YeK of sizefc^such that X £ Y. Combining Theorem 3-2, 3-9 and Lemma 3-8 we get: THEOREM 3-10. J/Cof (wwi) < 2Ni and if 2Ki is greater than the least possible value for 2K» then L[U] exists. This is Theorem C with L[U] in place of'a model with large cardinals'. The general theory of transitive models for large cardinals axioms along with corresponding covering theorems, as developed by Mitchell [8], provides the framework for the appropriate generalization of Theorem 3-10: 'L[U] exists' can be replaced by much stronger large cardinal axioms. Research supported by NSF grants MCS 78-01525 and MCS 80-20467. Copyright: Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1984/1

Y1 - 1984/1

N2 - It is unknown whether there can exist a family of functions from ω1 into ω of size less than 2N1 that dominates all functions fromω1 Intoω. We show that there is no such family if the continuum is real-valued measurable, and that the existence of such a family has consequences for cardinal arithmetic, and is related to large cardinal axioms.

AB - It is unknown whether there can exist a family of functions from ω1 into ω of size less than 2N1 that dominates all functions fromω1 Intoω. We show that there is no such family if the continuum is real-valued measurable, and that the existence of such a family has consequences for cardinal arithmetic, and is related to large cardinal axioms.

UR - https://www.scopus.com/pages/publications/0039089270

UR - https://www.scopus.com/pages/publications/0039089270#tab=citedBy

U2 - 10.1017/S0305004100061272

DO - 10.1017/S0305004100061272

M3 - Article

AN - SCOPUS:0039089270

SN - 0305-0041

VL - 95

SP - 25

EP - 32

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

IS - 1

ER -