A Book of Set Theory by Charles C. Pinter

Author:Charles C. Pinter
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2014-08-13T16:00:00+00:00

5.13 Theorem Let A be a partially ordered set such that (1) A has a least element p and (2) every chain of A has a sup in A. Then there is an element x ∈ A which has no immediate successor.

4 MAXIMAL PRINCIPLES

The propositions we are about to develop are widely used in mathematics to prove theorems by “nonconstructive” methods. They assert the existence of mathematical objects which cannot be constructed. For example, in linear algebra a maximal principle can be used to prove that every vector space has a basis, although, in general, it is impossible to exhibit such a basis.

The maximal principles are consequences of the Axiom of Choice. Furthermore, as we shall verify in Section 6, they are equivalent to the Axiom of Choice.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.