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Between strong and superstrong

Published online by Cambridge University Press:  12 March 2014

Stewart Baldwin
Stewart Baldwin*
Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849
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Extract

Definition. A cardinal κ is strong iff for every x there is an elementary embedding j:V → M with critical point κ such that x ∈ M.

κ is superstrong iff ∃j:V → M with critical point κ such that Vj(κ) ∈ M.

These definitions are natural weakenings of supercompactness and hugeness respectively and display some of the same relations. For example, if κ is superstrong then Vκ ⊨ “∃ proper class of strong cardinals”, but the smallest superstrong cardinal is less than the smallest strong cardinal (if both types exist). (See [SRK] and [Mo] for the arguments involving supercompact and huge, which translate routinely to strong and superstrong.)

Given any two types of large cardinals, a typical vague question which is often asked is “How large is the gap in consistency strength?” In one sense the gap might be considered relatively small, since the “higher degree” strong cardinals described below (a standard trick that is nearly always available) and the Shelah and Woodin hierarchies of cardinals (see [St] for a definition of these) seem to be (at least at this point in time) the only “natural” large cardinal properties lying between strong cardinals and superstrong cardinals in consistency strength.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 3 , September 1986 , pp. 547 - 559
Copyright
Copyright © Association for Symbolic Logic 1986

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References

REFERENCES

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