The questions:
- Are there any cardinals $\kappa$ such that $\mathrm{V}\setminus\{\kappa\}\prec\mathrm{V}$? If so, I think a neat name for them would be "ghost cardinals", because their existance leaves no impact on $\mathrm{V}$ and it's properties.
- Given a theory $\mathrm{T}$, what is the minimum cardinality of $\mathcal{M}$ such that $\mathcal{M}$ is nonempty and $\mathcal{M}\models\mathrm{T}$? If so, what is it respective to Peano Arithmetic, Z, ZF, ZFC, KM, etc.? (SOLVED)
- Are there any cardinals $\kappa$ larger than the smallest correct cardinal such that $\forall\lambda<\kappa(\mathrm{V}_\lambda\prec\mathrm{V}\rightarrow\mathrm{V}_{\lambda}\prec\mathrm{V}_\kappa)$? (In other words, every rank of a correct cardinal is a substructure of the rank of $\kappa$, making $\mathrm{V}_\kappa$ very similar to $\mathrm{V}$, a good name for these is hypercorrect)
- Given a structure $\mathcal{M}$, are there any non-singleton chains of substructures of $\mathcal{M}$ ordered by $\prec$? What is the largest suprememum of these chains' order types when $\mathcal{M}=\mathrm{V}$? What about when $\mathcal{M}=L$?
Where I have gotten to so far on them:
- If they do exist, they are not "definable" (i.e. there is no formula that is true for them and only them.) Every $\aleph_\alpha$ and $\beth_\alpha$ for finite $\alpha$ is not a ghost cardinal, and GCH implies every ghost cardinal is a limit cardinal.
- For Peano Arithmetic, the answer is clearly $\aleph_0$. For all the others, there are no finite models of Z, ZF, ZFC, or KM, and since they are all $L(\omega,\omega)$-theories, there is a countable model (if any). Thus, for all of these, $\aleph_0$.
- Every one of these cardinals are correct. Since not much is known on correct cardinals, this one seems hard to work on.
- Yes, there is an $\mathcal{M}$ where this property is true. In fact, that $\mathcal{M}$ is $\mathrm{V}$. Clearly, the existance of a correct cardinal implies that this supremum is at least $3$ (as $\{\mathrm{V}_\kappa,\mathrm{V}\}$ is well-ordered by $\prec$). It is also true that the existence of a hypercorrect cardinal implies it is at least $4$ (as $\{\mathrm{V}_\lambda,\mathrm{V}_\kappa,\mathrm{V}\}$ is well-ordered by $\prec$)