Review: H. Jerome Keisler, Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers
Review: H. Jerome Keisler, Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers
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评论：H. Jerome Keisler，无限逻辑模型理论。
DOI：
10.2307/2273064
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发表时间：
1973
影响因子：
0.6
通讯作者：
E. López
中科院分区：
文献类型：
--
作者：
E. López
(BR)以及命题演算的Godel-Mal'cev定理（GM）（它说每个形式上相容的公式集都是可满足的）。证明BR意味着GM只是简单地勾勒出来，因为它密切遵循Rasiowa和Sikorski的XVII 72中使用的论点。为了证明GM蕴含BR，Henkin对给定的布尔代数21显式构造了一个命题演算f>«，其满足函数集是所需的Stone空间。结构非常简单。洛斯评论说，由于布尔表示定理和每个布尔代数都有一个素理想（PI）的定理是等价的，人们可以通过证明GM蕴涵PI来获得亨金定理。这很容易，因为每个布尔代数都同构于适合的Lindenbaum-Tarski代数21的商代数21/3（每个自由布尔代数都是某个命题演算的Lindenbaum-Tarski代数）。21/3中素理想的存在性由GM得出，因为GM暗示21中素理想3 ' => 3的存在性，并且3'/3在21/3中是素的。在后一部分，他的论文亨金给出了一个条件下，人们可以找到一个布尔表示一个给定的布尔代数与基数的单位元素的代表性小于基数的给定布尔代数。他给，事实上，两个条件，其中之一，需要公理的选择在其证明，并要求是否可以证明他们等价没有公理的选择。洛杉矶显示如何可以证明他们等价的唯一使用公理的选择紧凑的拓扑空间，一个较弱的公理，这是相当于PI（洛杉矶和Ryll-Nardzewski XXXVIII 540）。安·S费雷比
(BR) and the Godel-Mal'cev theorem (GM) for propositional calculus (which says that every formally consistent set of formulas is satisfiable). The proof that BR implies GM is only briefly sketched as it follows closely the line of argument used in Rasiowa and Sikorski's XVII 72. For the proof that GM implies BR Henkin constructs explicitly, for a given Boolean algebra 21, a propositional calculus />« whose set of satisfaction functions is the Stone space required. The construction is quite straightforward. Los comments that, as the Boolean representation theorem and the theorem that every Boolean algebra has a prime ideal (PI) are equivalent, one could alternatively obtain Henkin's theorem by proving GM implies PI. This is easy, as every Boolean algebra is isomorphic to a quotient algebra 21/3 for suitable Lindenbaum-Tarski algebra 21 (every free Boolean algebra is the Lindenbaum-Tarski algebra of some propositional calculus). The existence of a prime ideal in 21/3 follows from GM, since GM implies the existence of a prime ideal 3 ' => 3 in 21 and 3'/3 is prime in 21/3. In the latter part of his paper Henkin gives a condition under which one can find a Boolean representation of a given Boolean algebra with the cardinality of the unit element of the representation being less than the cardinality of the given Boolean algebra. He gives, in fact, two conditions, one of which requires the axiom of choice in its proof, and asks whether they can be proved equivalent without the axiom of choice. Los shows how they can be proved equivalent with only the use of the axiom of choice for compact topological spaces, a weaker axiom which is equivalent to PI (Los and Ryll-Nardzewski XXXVIII540). ANN S. FEREBEE