0# and some forcing principles
0# and some forcing principles
复制标题
0
DOI：
10.2307/2273940
复制
发表时间：
1986
影响因子：
0.6
通讯作者：
S. Shelah
中科院分区：
文献类型：
--
作者：
M. Foreman;M. Magidor;S. Shelah
它一直被认为是可取的许多集理论家找到极大性的性质，国家的宇宙在某种意义上有“许多集”。到目前为止，孤立的性质往往是相互一致的（据我们所知）。例如，人们普遍认为超紧基数的存在性与L（R）中的决定性公理是一致的。这种一致性被认为是这些性质真实性的证据。考虑到这一点，第一作者提出了以下建议：极大性原理如果P是偏序且G <$P是V-泛超滤子，则要么a）存在真实的r ∈ V [G]且r <$V，要么B）存在序数α使得α是V中的基数，但不是V[G]中的基数。这个极大性原理应用于花园变种偏序对V的结构有着惊人的结果。例如，如果对某些，则P =<${p：p <$κ，<$p <$< κ}，则<$k既不增加真实的也不坍缩基数。因此，从极大性原理我们可以推出，G。C. H.到处都失败了，没有不可接近的红衣主教。（因此，这一原则与大枢机主教相矛盾。类似地，我们可以证明在任何基数κ上都没有Suslin树。这些结果有助于证明标题“最大化原则”。由于极大性原则意味着G. C. H.它在强奇异极限基数上失败，它至少具有“许多大基数”的一致性强度。(See[M].）另一方面，相对于任何假设，它不知道是一致的。
It has been considered desirable by many set theorists to find maximality properties which state that the universe has in some sense “many sets”. The properties isolated thus far have tended to be consistent with each other (as far as we know). For example it is a widely held view that the existence of a supercompact cardinal is consistent with the axiom of determinacy holding in L(R). This consistency has been held to be evidence for the truth of these properties. It is with this in mind that the first author suggested the following: Maximality Principle If P is a partial ordering and G ⊆ P is a V-generic ultrafilter then either a) there is a real number r ∈ V [G] with r ∉ V, or b) there is an ordinal α such that α is a cardinal in V but not in V[G]. This maximality principle applied to garden variety partial orderings has startling results for the structure of V. For example, if for some , then P = 〈{p: p ⊆ κ, ∣p∣ < κ}, ⊆〉 neither adds a real nor collapses a cardinal. Thus from the maximality principle we can deduce that the G. C. H. fails everywhere and there are no inaccessible cardinals. (Hence this principle contradicts large cardinals.) Similarly one can show that there are no Suslin trees on any cardinal κ. These consequences help justify the title “maximality principle”. Since the maximality principle implies that the G. C. H. fails at strong singular limit cardinals it has consistency strength at least that of “many large cardinals”. (See [M].) On the other hand it is not known to be consistent, relative to any assumptions.