Supercompact cardinal

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In set theory, a supercompact cardinal a type of large cardinal. They display a variety of reflection properties.

[edit] Formal definition

If λ is any ordinal, κ is λ-supercompact means that there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ and

{ }^\lambda M\subseteq M

That is, M contains all of its λ-sequences. Then κ is supercompact means that it is λ-supercompact for all ordinals λ.

[edit] References

  • Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0444105352.
  • Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, 2nd ed, Springer. ISBN 3540003843.
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