Some downwards transfer properties for N2
Some downwards transfer properties for N2
复制标题
N2 的一些向下传递特性
DOI：
10.1016/0001-8708(88)90041-2
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发表时间：
1988
影响因子：
1.7
通讯作者：
R. Laver
中科院分区：
文献类型：
--
作者：
M. Foreman;R. Laver
假设ZFC+的一致性“存在一个巨大的基数”，我们构造了ZFC + GCH的模型，其中基数N具有一些向下传递性质。Chang的猜想（见银[13]和Kunen [9]）是这样一种说法，即每个结构（w，，ol，R），其中R是O2上的可数关系序列，具有形式为（A，B，R r A）的初等子结构，其中A == K，E = N，.该模型将满足Chang的猜想，如[9]。以及其他类似的资产例如，每一个大小和色数为N2的图都将有一个大小和色数为N的顶点导出子图（参见Erdos和Hajnal [3，问题41 B]）。GCH的一致性（相对于单独的ZFC）和关于图的这个陈述失败是Baumgartner [11]的结果。另一个在模型中成立的结果是，如果d =（A，f，），是一个簇中的代数，其中A == K，，使得存在函数h，：（A）<$'，+A（WI <w），使得s $的任何闭于hm's下的Et，-大小的子代数是自由的（特别是当每个K，-大小的子代数是自由的时），那么&是K，-许多自由子代数的并。对于群（这里的假设变成：每个大小为QK的d的子群是自由的）和其他具有某些强性质的变种，Shelah [121]有这个关于Mahlo基数的一致性结果。此外，Shelah已经表明，如果存在静止Sr（c1 <02：cfoc = w）。使得Sny在y中是非平稳的，所有y <w2，则存在大小为Nz的群G，所有6个N1-大小的子群都是自由的，使得G不是X1-许多自由子群的并。有关这些和相关结果，请参阅[2，11，121]。
Assuming the consistency of ZFC+“there is a huge cardinal,” we construct a model of ZFC+ GCH in which the cardinal N, possesses some downwards transfer properties.Chang’s conjecture (see Silver [13] and Kunen [9]) is the statement that every structure (w,, ol, R), with R a countable sequence of relations on 02, has an elementary substructure of the form (A, B, R r A), where A== K, and E= N,. The model will satisfy Chang’s conjecture as in [9]. as well as other properties in this vein. For example, every graph of size and chromatic number N2 will have a vertex induced subgraph of size and chromatic number N,(see Erdos and Hajnal [3, problem 41B]). The consistency (relative to ZFC alone) that GCH holds and this statement about graphs fails is a result of Baumgartner [11. Another result holding in the model is that if d=(A, f,),,, is an algebra in a variety, with A== K,, such that there are functions h,:(A)<‘,+ A (WI< w) so that any Et,-sized subalgebra of s $ closed under the hm’s is free (which happens in particular if every K,-sized subalgebra is free), then & is the union of K,-many free subalgebras. For groups (where the hypothesis just becomes: every subgroup of d of size QK, is free) and other varieties with certain strong properties, Shelah [121 has this consistency result just relative to a Mahlo cardinal. Also, Shelah has shown that if there is a stationary Sr {c1< 02: cfoc= w). such that S ny is nonstationary in y, all y< w2, then there is a group G of size NZ, with all 6 N,-sized subgroups being free, such that G is not the union of X,-many free subgroups. See [2, 11, 121 for these and related results.