Key words Weakly measurable, consistency of a measurable cardinal, least weakly compact cardinal, first failure of the GCH, surgery method, Silver iteration. MSC (2010) 03E35, 03E45, 03E55In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection A containing at most κ + many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring all sets in A. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add(κ, η) for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent.In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar notion of a measurable cardinal, and we show some of the features that can be exhibited by it but not by measurable cardinals. We know that a cardinal κ is measurable if there exists a nonprincipal κ-complete (ultra)filter on κ measuring all of its subsets. We propose a weakening of this notion by insisting only that for any A ⊆ P(κ) of size at most κ + , we can find a nonprincipal κ-complete filter on κ measuring all subsets in A. Of course, if 2 κ = κ + , then these large cardinal concepts are equivalent assertions for κ because we can set A = P(κ). However, our analysis reveals the fact that a measurable cardinal κ can become nonmeasurable and yet still be weakly measurable in a forcing extension (where the GCH fails first at κ).Main Definition A cardinal κ is weakly measurable if for every collection A ⊆ P(κ) containing at most κ + many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring every set in A. (i.e., for every subset A ∈ A, either A or κ \ A is in the filter.)
Main Theorem(1) If κ is measurable, then there exists a forcing extension where κ is weakly measurable, but not measurable.(2) In fact, if κ is measurable, then there exists a forcing extension where κ is weakly measurable (but not measurable), and the GCH fails first at κ.(3) Indeed, if κ is measurable then there exists a forcing extension where the GCH holds, the measurability of κ is preserved, and the weak measurability of κ is indestructible by further forcing Add(κ, η) for any η.(4) Finally, if κ is weakly measurable, then κ is measurable in an inner model, and so the two concepts of measurability are equiconsistent.While the consistency strength of weakly measurable cardinals is settled by statement (4) of the Main Theorem, many open questions remain. If we had only required such filters for A containing at most κ many subsets of κ, then the ...
