Confused about inner models and On

[nomadreid]

I do not understand how On, the class of ordinals, can be included in the universe of an inner model of ZFC such as < V_{[tex]\kappa[/tex]}, epsilon>, where [tex]\kappa[/tex] is the first inaccessible ordinal. My confused attempts to do so have led me to believe that I am getting off on the wrong foot in my analysis, that there is something very basic that I am missing here. I list four of my dead-ends, to show what sort of confusion I am asking someone to lead me out of.

One, it would seem that On was simply too big. For example, On would contain the ordinal that is associated with the cardinal number [tex]\alpha[/tex], where a is a measurable cardinal, there are at least a Ramsey cardinals below [tex]\alpha[/tex]. [tex]\alpha[/tex] is much larger than the first inaccessible [tex]\kappa[/tex], so how would so many ordinals fit into V[tex]\kappa[/tex]?

Two, V_{[tex]\kappa[/tex]} can be, in a higher order, be construed as a set, whereas On can never be anything but a class. So how can On be part of V_{[tex]\kappa[/tex]}?

Three: I thought of the collapsing of V by using an appropriate ultrafilter over a measurable cardinal, where also the ordinals would be collapsed, but since there are a lot of ordinals which are skipped by the elementary embedding, having the measurable cardinal as a critical point, then it would seem that On doesn't all get into the universe of the collapsed model. Or does it somehow?

Fourth, I was convinced me that we were not just talking about V_{[tex]\kappa[/tex]} intersect On, because the condition for the supercompact cardinal is that it be lambda-supercompact for all ordinal lambda greater than or equal to the supercompact cardinal, which means that somehow they all have to be in the model.

Any help in the right direction would be appreciated.

PS For some reason, the kappas and alphas got superscripted rather than subscripted, even though it clearly shows subscripting. Sorry. Maybe the two functions, Greek letters and subscripting, don't go together.

 

[Citan Uzuki]

Briefly, because inner models are proper classes, not sets. The model <V_κ, ∈> is not an inner model of V, because it (obviously) does not contain all ordinals. It is, however, a transitive model.

I think you are somewhat confused by the definition of supercompact cardinal. If κ is λ-supercompact, that means that there exists a transitive inner model M with ^λM⊆M and an elementary embedding j:V→M with critical point κ and j(k)>λ. It does not, however, mean that M is V_κ -- M is a much larger class than V_κ.

 

[nomadreid]

Thank you, Citan Uzuki. You found the major points of my confusion and cleared them up. Very helpful.