ペーパーバック – 2001/2/9
3件中1 - 3件目のレビューを表示
In that week I made heavy weather of the text. The material ranges from third year introductory topics to advanced conjectures that are merely relevant to the doctoral student! Even to the topic of large cardinals ... relevant to the existence of a consistent theory of the hyperreal system proposed by H. Jerome Keisler ...
As someone inclined to Mathematical Platonism with my beliefs that the real numbers really exist and that in this universe the laws of physics express these real numbers ... I have a contempt for finitistic theories of mathematics ... and this means that, in considering the situational logic of set theory as described in chapter nine, I have an appreciation of the proven problem of the non existence of the set of all sets, with the implication that in some deep sense the sets must therefore exist in some series of Platonic spheres in another realm ... accessible by contemplation by our brain singularities' intuition ... wherein there is the theory expressed in an existent model of Peano's Laws Of Arithmetic (chapter eight), based on which in turn there exists a model of the real numbers ... the argument from the Dedekind Cuts proving the existence thereof ... in some sort of hierarchical Russellian type theory ... Mathematical Platonism: The True Nature Of Reality!
More recently physicists have been reexamining the old questions ... their phrases bring back memories ... computational nature of reality ... the cellular automata finitistic unverse?? ... OR ... a universe based on the real numbers??
Two other reviews explained the strong points of this book very well. I would just like to agree particularly that the presentation of the constructible universe model for ZF is excellent. The explanation in this book is significantly better than explanations in other books, both earlier and later than the 1967 date of publication. It is interesting to compare Shoenfield's constructible universe (pages 270–281) with two briefer presentations of it in "Set Theory and the Continuum Hypothesis" by Paul Cohen. The Shoenfield "construction" maps the ordinal numbers to the constructible universe. This has the advantage that you only have to accept the existence and properties of ordinal numbers in order to create a set-universe for the model, whereas some other constructions start with the von Neumann universe V and choose elements of that universe to represent constructible sets (by restricting sets in V). Building up the universe from ordinals seems to me much better than pruning back the more metaphysical von Neumann universe. (That's just my personal bias.)
Shoenfield gives some presentation of Paul Cohen's forcing method for ZF models and the consequent independence proofs (pages 282-303), but otherwise he gives little coverage to the bewildering menagerie of ZF models which appear in "Consequences of the Axiom of Choice" by Howard and Rubin, and the interesting menagerie of models in "The Axiom of Choice" by Jech.
The typesetting is excellent, the paper is bright white, and the binding is good. The notations and definitions are modern.
Although the writing is good, that doesn't mean it is easy. He progresses deliberately through the details, rarely giving an overview. I think he is just expecting that you already have a good sense of context from the undergrad logic course you took (didn't you?). Sometimes he seems to belabor a point. There is also a dearth of examples, just five in the whole book, three of them in the appendix. There are no references at all. The age of the book makes it, not wrong, but inadequate in some areas. Still, I have looked at alternatives and haven't found something better for a graduate survey text in English.