The axiom of choice for linearly ordered families
نویسندگان
چکیده
منابع مشابه
The Axiom of Choice for Well-Ordered Families and for Familes of Well-Orderable Sets
We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well ordered families of sets and the axiom of choice for sets for well orderable sets are both true, but the axiom of choice is false. We are concerned with the following two consequences of the axiom of choice: C(WO,∞): Every well ordered collection of sets has a choice function. C(∞,WO): ...
متن کاملWell-ordered Sets, Ordinalities and the Axiom of Choice
The discussion of cardinalities in Chapter 2 suggests that the most interesting thing about them is their order relation, namely that any set of cardinalities forms a well-ordered set. So in this section we shall embark upon a systematic study of well-ordered sets. Remarkably, we will see that the problem of classifying sets up to bijection is literally contained in the problem of classifying w...
متن کاملThe Axiom of Choice
We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X...
متن کاملThe Axiom of Choice
We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X...
متن کاملA Substitute for the Axiom of Choice
This theorem is well known to be equivalent to the axiom of choice (though there does not seem to be a proof of this fact in the literature) and it has been suggested as an alternative for this axiom. The purpose of this note (which is purely methodological) is to propose a simpler but equivalent formulation of (A) as a substitute for the Zermelo axiom. The simplicity lies in the fact that we m...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Fundamenta Mathematicae
سال: 1978
ISSN: 0016-2736,1730-6329
DOI: 10.4064/fm-99-2-133-139