Continuing [6], we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary logic and their variations. In [6], it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters SDLS(L ℵ 0 stat , < ℵ 2) down to < ℵ 2 is equivalent to the conjunction of CH and Cox's Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak secondorder parameters SDLS − (L ℵ 0 stat , < 2 ℵ 0) down to < 2 ℵ 0 implies that the size of the continuum is ℵ 2. In contrast, an internal interpretation of the stationary logic can satisfy the SDLS down to < 2 ℵ 0 under the continuum being of size > ℵ 2. This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size < 2 ℵ 0. We also consider a P κ (λ) version of the stationary logic and show that the SDLS for this logic in internal interpretation SDLS int + (L P KL stat , < 2 ℵ 0) for reflection down to < 2 ℵ 0 is consistent under the assumption of the consistency of ZFC + "the existence of a supercompact cardinal" and this SDLS implies that the continuum is (at least) weakly Mahlo. These three "axioms" in terms of SDLS are consequences of three instances of a strengthening of generic supercompactness which we call Lavergeneric supercompactness. Existence of a Laver-generic supercompact cardinal in each of these three instances also fixes the cardinality of the continuum to be ℵ 1 or ℵ 2 or very large respectively. We also show that the existence of one of these generic large cardinals implies the "++" version of the corresponding forcing axiom.

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Strong reflection principles with the reflection cardinal ≤ ℵ1 or < 2 ℵ 0 imply that the size of the continuum is either ℵ1 or ℵ2 or very large. Thus, the stipulation, that a strong reflection principle should hold, seems to support the trichotomy on the possible size of the continuum. In this article, we examine the situation with the reflection principles and related notions of generic large cardinals.

Gostaria de agradecer primeiramente aos meus pais por todo o apoio e compreensão, sempre me incentivando e me permitindo avançar em meus estudos. Agradeço a minha orientadora, Prof a Lúcia Renato Junqueira, pela paciência e todo o apoio e atenção. Agradeço a todos os meus amigos e colegas do IME-USP por me acompanharem por tantos momentos desde a graduação até o mestrado. Agradeço aos meus companheiro de república, com quem eu compartilhei um teto e tantas lembranças durante os últimos anos. Agradeço a todos os professores do IME-USP que me ensinaram praticamente tudo o que sei sobre Matemática, em especial à professora Iryna Kashuba pelo apoio durante a iniciação científica. Agradeço a todos os colegas e professores do Grupo de Topologia, tanto os do IME quanto os do ICMC em São Carlos e da UFBA em Salvador. Agradeço à CAPES e à CNPq pelo apoio financeiro durante a produção deste trabalho. Agradeço também a Prof a Ofélia Teresa Alas e Gabriel Zanetti Nunes Fernandes pelas contribuições a este trabalho. Em especial a Santi Spadaro pelo resultado do teorema 1.2.22. i Resumo

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