(Weak) diamond can fail at the least inaccessible cardinal

Abstract

Starting from suitable large cardinals, we force the failure of (weak) diamond at the least inaccessible cardinal. The result improves an unpublished theorem of Woodin and a recent result of Ben-Neria, Garti and Hayut.1. Introduction. We study the combinatorial principles diamond, introduced by Jensen [9], and weak diamond, introduced by Devlin-Shelah [5], and prove the consistency of their failure at the least inaccessible cardinal.Suppose κ is an uncountable regular cardinal. Recall that diamond at κ, denoted ♦ κ , asserts the existence of a sequence S α | α < κ such that for each α < κ, S α ⊆ α, and if X ⊆ κ, then {α < κ | X ∩ α = S α } is stationary in κ.Also, weak diamond at κ, denoted Φ κ , is the assertion "For every c : 2 <κ → 2, there exists g : κ → 2 such that for all f : κ → 2, the setIt is easily seen that ♦ κ + implies 2 κ = κ + , and in fact by a celebrated theorem of Shelah [12], for all uncountable cardinals κ, ♦ κ + is equivalent to 2 κ = κ + . It follows that it is easy to force the failure of diamond at successor cardinals.By [5], 2 κ < 2 κ + implies Φ κ + . It was later observed by Abraham and Baumgartner that Φ κ + implies 2 κ < 2 κ + : see for example [7], [11] where the proof is attributed to Abraham, or [13] where the proof is attributed to Baumgartner. It follows that Φ κ + and 2 κ < 2 κ + are equivalent, and ♦ κ + implies Φ κ + . It also shows that it is easy to force Φ κ + to fail.Unlike the case of successor cardinals, it is difficult to force the failure of diamond or weak diamond at an inaccessible cardinal.

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“…(ii) If κ = λ + ≥ ℵ 2 , then κ <κ = κ is equivalent to ♦ κ [She10], and is therefore also equivalent to ♦ i κ . (iii) The failure of ♦ κ at the least inaccessible cardinal is consistent [Gol22] and therefore ♦ κ and ♦ i κ are not equivalent for inaccessible cardinals.…”

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“…To work around this, there is a need for constructing the club and perform the collapses simultaneously; to wit, we shall interleave the Lévy collapses within the Radin forcing using the mechanism of guiding generics. Although the above-described forcing construction has already appeared in previous works [18,22,36], the fact that we are working in the absence of GCH makes the arguments more complicated. This is particularly evident in the construction of the guiding generics (Lemma 2.4).…”

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