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, j(κ)>λ and

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

Alternatively, an uncountable cardinal κ is supercompact if for every A such that |A| ≥ κ there exists a normal measure on [A]^{< κ}.

[A]^{< κ} is defined as follows:

.

[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 0-444-10535-2. 

• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2. 

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, 2nd ed, Springer. ISBN 3-540-00384-3.