Silver cardinal is the virtual version of an inconsistent notion of what a Silver indiscernible would be if zero sharp "exists". If κ is the cardinal, then it says that there is (in a forcing extension of V) a club of ordinals in Vκ of order type κ which are indiscernible. That is, there are elementary embeddings between each pair of distinct indiscernibles.
Limits of Silver ordinals are also Silver.
Let an ordinal, κ, be called α-Silver, if it is Silver and for every β < α the set of β-Silver ordinals below it has order type κ. Which particular ordinals are Silver and α-Silver cannot be determined in L itself, but if Ord is the least Silver ordinal strictly greater than some ordinal, then they can be identified in a V only slightly larger than L (that is, having the same initial ordinals). In this case, the set of Silver ordinals is countable and no 1-Silver ordinals exist. A Silver cardinal is the virtual version of a 1-Silver ordinal, that is, it is 1-Silver in a forcing extension of V. In that forcing extension, the Silver ordinals below the cardinal all have cofinality ω even though they may have different cofinalities in V (reality).
Let an ordinal be called weakly Silver, if it is Silver in some multiply-iterated forcing extension of V (without added ordinals below existing ordinals). What prevents us from forcing another weakly Silver ordinal below the least weakly Silver cardinal?
If α is any ordinal, then the least weakly Silver ordinal strictly greater than α has cofinality ω in V. If β has cofinality 0, 1, or ω and α is any ordinal, then the least β-Silver ordinal strictly greater than α has cofinality ω in V. In L, they are all regular ordinals.
Strength relative to other large cardinals
Silver cardinal is stronger than any other large cardinals consistent with V=L including: the least Mahlo cardinal, the least weakly compact cardinal, the least unfoldable cardinal, the least ineffable cardinal, the least remarkable cardinal, the least virtually extendible cardinal, the least ω-iterable cardinal, the least virtually rank-into-rank cardinal, the least ω-Erdős cardinal, and the least λ-iterable cardinal, and the least λ-Erdős cardinal (for ω+1≤λ<ω1).
But Silver cardinal is weaker than zero sharp, ω1-iterable cardinal, ω1-Erdos cardinal, Ramsey cardinal, and measurable cardinal. If zero sharp exists, then all stronger large cardinals (and indeed all uncountable initial ordinals of cardinals) are Silver L-indiscernibles and thus Silver cardinals. Zero sharp "exists" is equivalent to the existence of a ω1-Silver cardinal, and the least ω1-Silver cardinal would be ω1 itself.
Forcing
One possible way of doing the forcing would be: Let the forcing conditions be constructible functions from n×κ to Ord where n<ω≤κ and the functions are strictly increasing in both arguments. Condition p is stronger than q iff q is a subset of p. Once you have the generic function f from ω×κ to Ord, the α-th member of the club will be the supremum of { f(n,α) : n<ω }. This succeeds in showing that κ is Silver if κ is the supremum of the club.
See also
References
- Gitman, Victoria; Schindler, Ralf (2018-12-01). "Virtual large cardinals". Annals of Pure and Applied Logic. Logic Colloquium 2015. 169 (12): 1317–1334. doi:10.1016/j.apal.2018.08.005. ISSN 0168-0072.