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0# and some forcing principles

Published online by Cambridge University Press:  12 March 2014

Matthew Foreman
Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Menachem Magidor
Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Saharon Shelah
Institute of Mathematics, The Hebrew University, Jerusalem, Israel


It has been considered desirable by many set theorists to find maximality properties which state that the universe has in some sense “many sets”. The properties isolated thus far have tended to be consistent with each other (as far as we know). For example it is a widely held view that the existence of a supercompact cardinal is consistent with the axiom of determinacy holding in L(R). This consistency has been held to be evidence for the truth of these properties. It is with this in mind that the first author suggested the following:

Maximality Principle If P is a partial ordering and GP is a V-generic ultrafilter then either

  • a) there is a real number rV [G] with rV, or

  • b) there is an ordinal α such that α is a cardinal in V but not in V[G].

This maximality principle applied to garden variety partial orderings has startling results for the structure of V.

For example, if for some , then P = 〈{p: pκ, ∣p∣ < κ}, ⊆〉 neither adds a real nor collapses a cardinal. Thus from the maximality principle we can deduce that the G. C. H. fails everywhere and there are no inaccessible cardinals. (Hence this principle contradicts large cardinals.) Similarly one can show that there are no Suslin trees on any cardinal κ. These consequences help justify the title “maximality principle”.

Since the maximality principle implies that the G. C. H. fails at strong singular limit cardinals it has consistency strength at least that of “many large cardinals”. (See [M].) On the other hand it is not known to be consistent, relative to any assumptions.

Research Article
Copyright © Association for Symbolic Logic 1986

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[FW]Foreman, M. and Woodin, H., The G.C.H. can fail everywhere (to appear).Google Scholar
[J]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[M]Mitchell, W., The core model for sequences of ultrafilters (manuscript).Google Scholar
[S]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar