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Diagonal Prikry extensions

Published online by Cambridge University Press:  12 March 2014

James Cummings  and
Matthew Foreman
James Cummings
Affiliation:
Mathematical Sciences Department, Carnegie Mellon University, Pittsburgh PA 15215, USA. E-mail: jcumming@andrew.cmu.edu
Matthew Foreman
Affiliation:
Department of Mathematics, University of California, Irvine, Irvine CA 92697, USA. E-mail: mforeman@math.uci.edu
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Extract

§1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then

1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W.

2. W weakly covers V if W computes the successor of every V-singular cardinal correctly.

Strong covering implies weak covering.

In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include

1. X is “0# exists”, Kx is L, Y is “strongly”.

2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”.

3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 4 , December 2010 , pp. 1383 - 1402
Copyright
Copyright © Association for Symbolic Logic 2010

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References

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