The Axiom of Choice, the Well Ordering Principle and Zorn’s Lemma
نویسنده
چکیده
In this note we prove the equivalence between the axiom of choice, the well ordering principle and Zorn’s lemma, and discuss to some extent how large fragment of ZF we need in order to prove the individual implications.
منابع مشابه
A Brief Introduction to Zfc
We present a basic axiomatic development of Zermelo-Fraenkel and Choice set theory, commonly abbreviated ZFC. This paper is aimed in particular at students of mathematics who are familiar with set theory from a “naive” perspective, and are interested in the underlying axiomatic development. We will quickly review some basic concepts of set theory, before focusing on the set theoretic definition...
متن کامل9. the Axiom of Choice and Zorn’s Lemma
§9.1 The Axiom of Choice We come now to the most important part of set theory for other branches of mathematics. Although infinite set theory is technically the foundation for all mathematics, in practice it is perfectly valid for a mathematician to ignore it – with two exceptions. Of course most of the basic set constructions as outlined in chapter 2 (unions, intersections, cartesian products,...
متن کاملA rigorous procedure for generating a well-ordered Set of Reals without use of Axiom of Choice / Well-Ordering Theorem
Well-ordering of the Reals presents a major challenge in Set theory. Under the standard Zermelo Fraenkel Set theory (ZF) with the Axiom of Choice (ZFC), a well-ordering of the Reals is indeed possible. However the Axiom of Choice (AC) had to be introduced to the original ZF theory which is then shown equivalent to the well-ordering theorem. Despite the result however, no way has still been foun...
متن کاملThe Axiom of Choice and Zorn’s Lemma
function on A is thus a choice of an element of the variable set A at each stage; in other words, a choice function on A is just a variable (or global) element of A . The axiom of choice (AC) asserts that if each member of a family A is nonempty, then there is a choice function on A . Metaphorically, then, the axiom of choice asserts that any family of sets with an element at each stage has a v...
متن کاملA Characterization of the Boolean Prime Ideal Theorem in Terms of Forcing Notions
For certain weak versions of the Axiom of Choice (most notably, the Boolean Prime Ideal theorem), we obtain equivalent formulations in terms of partial orders, and filter-like objects within them intersecting certain dense sets or antichains. This allows us to prove some consequences of the Boolean Prime Ideal theorem using arguments in the style of those that use Zorn’s Lemma, or Martin’s Axiom.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012