Strong axioms of infinity and elementary embeddings

On tall cardinals and some related generalizations

“…We mention that we are assuming some familiarity with the large cardinal notions of measurability, measurable cardinals of high Mitchell order, tallness, hypermeasurability, strongness, strong compactness, and supercompactness. Interested readers may consult [13], [17], [18], or [21]. In addition, we are assuming some familiarity with basic inner model and core model theory, as presented in [22] and [19].…”

Indestructibility under adding Cohen subsets and level by level equivalence

“…By [22,Theorem 4.8] and the succeeding remarks, if γ is η strongly compact and η is regular, then η must reflect stationary sets of ordinals of cofinality ω. Consequently, P "δ is not δ strongly compact". Thus, by the preceding three sentences, P "λ < δ ", and R "δ is λ strongly compact".…”

On level by level equivalence and inequivalence between strong compactness and supercompactness

“…δ -strategically closed and adds a non-reflecting stationary set of ordinals of cofinality ω to θ +6 δ ", the closure properties of P 1,δ together with an application of Theorem 4.8 of [19] and the succeeding remarks yield V…”

“…Interested readers may consult [14] or [19] for further details. We only note that the cardinal κ is <λ supercompact if κ is δ supercompact for every cardinal δ < λ.…”