Abstract
In this survey paper, we will summarise some of the more and less known results on the generalisation of the Easton theorem in the context of large cardinals. In particular, we will consider inaccessible, Mahlo, weakly compact, Ramsey, measurable, strong, Woodin, and supercompact cardinals. The paper concludes with a result from the opposite end of the spectrum: namely, how to kill all large cardinals in the universe.
Highlights
One of the questions which stood at the birth of set theory as a mathematical discipline concerns the size of real numbers R
Can we find a generic extension of V which realises F and preserves the largeness of a fixed large cardinal κ? Clearly, a necessary condition on F is that it should keep κ strong limit
Considering the variety of large cardinal concepts, it is no surprise that many of them have not been studied from the point of their compatibility with patterns of the continuum function
Summary
One of the questions which stood at the birth of set theory as a mathematical discipline concerns the size of real numbers R. We can view (*) as a statement about consistency of a theory, in which case κ should either be a parameter or should be definable in ZFC, or (*) can be taken as a statement about pairs of models of ZFC It is the latter approach which is more useful and general: Theorem 1.1 (Cohen, Solovay) Let κ be a cardinal with uncountable cofinality in V , and assume κω = κ in V. There are more general statements of Easton’s theorem which remove the restriction of definability of F Such generalisations usually require additional assumptions above ZFC: one can for instance start with an inaccessible cardinal κ and GCH below κ, and set M = H(κ). Easton’s theorem solves the problem of the possible behaviours of the continuum function on regular cardinals in models of ZFC in full generality. Particular, in Lemma 1.17, the set A is required to be in a normal measure, not just stationary, as in Silver’s theorem
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