Supercompact cardinal

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

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 over [A]< κ, in the following sense.

[A]< κ is defined as follows:

An ultrafilter U over [A]< κ is fine if it is κ-complete and , for every . A normal measure over [A]< κ is a fine ultrafilter U over [A]< κ with the additional property that every function such that is constant on a set in . Here "constant on a set in U" means that there is such that .

Properties

Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-huge cardinal) that is witnessed by a structure of limited rank exists above a supercompact cardinal κ, then a cardinal with that property exists below κ. For example, if κ is supercompact and the Generalized Continuum Hypothesis holds below κ then it holds everywhere because a bijection between the powerset of ν and a cardinal at least ν++ would be a witness of limited rank for the failure of GCH at ν so it would also have to exist below κ.

Finding a canonical inner model for supercompact cardinals is one of the major problems of inner model theory.

gollark: As planned. Now to compile ttyd, for purposes.
gollark: Wait, is that *my* blog post?
gollark: Apparently it uses 100% CPU when none are connected, somehow.
gollark: What an exciting session of HBMud.
gollark: ```< ======================= hbmud=======================welcome to hbmud (< 0 users)what is your name?> bee> look< there is a wall next to you.there is a sandbag herethere is also a sign here.there is also a poster posted on the wallto the north is a room< present in this room are: bee> f< what?> look< there is a wall next to you.there is a sandbag herethere is also a sign here.there is also a poster posted on the wallto the north is a room< present in this room are: bee> north> look< you feel that this room is bee.there is a sign herethere is another sandbag here. you feel like you could attack itto the south is a room< present in this room are: bee< also, a(n)< bee< also, a(n)< sandbagalso, a(n) death> kill death< you are attacked by the< deathyou attack the death dealing 3 damageyou are dead. goodbye```

See also

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.
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