Strong downward Loewenheim-Skolem theorems for stationary logics, II: reflection down to the continuum
Strong downward Loewenheim-Skolem theorems for stationary logics, II: reflection down to the continuum
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平稳逻辑的强向下 Loewenheim-Skolem 定理，II：向下反映到连续统
DOI：
10.1007/s00153-020-00751-6
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发表时间：
2021
影响因子：
0.3
通讯作者：
Sakai Hiroshi
中科院分区：
文献类型：
--
作者：
Fuchino Sakae;Ottenbreit Maschio Rodrigues Andre;Sakai Hiroshi
继续（Fuchino等人，Arch Math Log，2020年。网址：//doi. org/10.1007/s 00153 -020-00730-x），我们研究了平稳逻辑的强向下Löwenheim-Skolem定理及其变体.在Fuchino et al.（2020）证明了具有弱二阶参数SDLS（L^Sta_0_ stat，<1，Sta_2）SDLS（L Sta_0，<Sta_2）下至<1，Sta_2 <Sta_2）的一般平稳逻辑的SDLS等价于CH与考克斯的内部俱乐部性对角反射原理的结合。我们证明了无弱二阶参数的平稳逻辑的SDLS^-（L^sta_0_ stat，<1，2^sta_0）SDLS-（L sta_0，< 2 sta_0）下至<1，2^sta_0 < 2 sta_0的SDLS意味着连续统的大小为10_2 sta_2。相比之下，静态逻辑的内部解释可以满足SDLS下降到<1，2^^_0 < 2 ^^_0，在连续统的大小>^_2>^_2的情况下。该SDLS被证明等价于对角反射原理的内部版本，直到大小为<1，2^x_0 < 2 ^x_0的内部固定集合。我们还考虑了一个版本的静态逻辑，并表明，SDLS的这种逻辑在内部解释SDLS^ int _+（L^ PKL _ stat，<1，2^PKL_0）SDLS+ int（L stat PKL，< 2 0），反射低至<1，在ZFC++“存在超紧基数”的相容性假设下，2^<$_0 < 2 <$0是相容的，并且这个SDLS意味着连续统是（至少）弱Mahlo。这三个“公理”的SDLS是后果的三个实例加强一般超紧性，我们称之为拉弗一般超紧性。在这三种情况下，如果存在拉弗-泛属超紧基数，则连续统的基数也分别固定为_1 <$1或_2 <$2或非常大。我们还表明，这些通用的大基数之一的存在意味着“+”版本的相应的强制公理。
Continuing (Fuchino et al. in Arch Math Log, 2020. https://doi. org/10.1007/s00153-020-00730-x), we study the Strong Downward Löwenheim–Skolem Theorems (SDLSs) of the stationary logic and their variations. In Fuchino et al.(2020) it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters SDLS (L^ ℵ _0 _ stat,<\, ℵ _2) SDLS (L stat ℵ 0,< ℵ 2) down to<\, ℵ _2< 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 second-order parameters SDLS^-(L^ ℵ _0 _ stat,<\, 2^ ℵ _0) SDLS-(L stat ℵ 0,< 2 ℵ 0) down to<\, 2^ ℵ _0< 2 ℵ 0 implies that the size of the continuum is ℵ _2 ℵ 2. In contrast, an internal interpretation of the stationary logic can satisfy the SDLS down to<\, 2^ ℵ _0< 2 ℵ 0 under the continuum being of size> ℵ _2> 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< 2 ℵ 0. We also consider a version of the stationary logic and show that the SDLS for this logic in internal interpretation SDLS^ int _+(L^ PKL _ stat,<\, 2^ ℵ _0) SDLS+ int (L stat PKL,< 2 ℵ 0) for reflection down to<\, 2^ ℵ _0< 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 Laver-generic 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 ℵ 1 or ℵ _2 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.