Most mathematicians are happy to add the axiom of

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Most mathematicians are happy to add the axiom of choice to the stan- dard axioms and this is what we shall do. Note that if we prove something using the standard axioms and the axiom of choice then we will be unable to find a counter-example using only the standard axioms. Note also that, when dealing with specific systems we may be able to prove the result for that system without using the axiom of choice. The axiom of choice is not very easy to use in the form that we have stated it and it is usually more convenient to use an equivalent formulation called Zorn’s lemma. Definition 31. Suppose X is a non-empty set. We say that is partial order on X , that is to say, that is a relation on X with (i) x y , y z implies x z , (ii) x y and y x implies x = y , (iii) x x for all x , y , z . We say that a subset C of X is a chain if, for every x, y C at least one of the statements x y , y x is true. If Y is a non-empty subset of X we say that z X is an upper bound for Y if z y for all y Y . We say that m is a maximal element for ( X, ) if x m implies x = m . You must be able to do the following exercise. Exercise 32. (i) Give an example of a partially ordered set which is not a chain. (ii) Give an example of a partially ordered set and a chain C such that (a) the chain has an upper bound lying in C , (b) the chain has an upper bound but no upper bound within C , (c) the chain has no upper bound. (iii) If a chain C has an upper bound lying in C , show that it is unique. Give an example to show that, even in this case C may have infinitely many upper bounds (not lying in C ). 11
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(iv) Give examples of partially ordered sets which have (a) no maximal elements, (b) exactly one maximal element, (b) infinitely many maximal el- ements. (v) how should a minimal element be defined? Give examples of partially ordered sets which have (a) no maximal or minimal elements, (b) exactly one maximal element and no minimal element, (c) infinitely many maximal elements and infinitely many minimal elements. Axiom 33 (Zorn’s lemma). Let ( X, ) be a partially ordered set. If every chain in X has an upper bound then X contains a maximal element. Zorn’s lemma is associated with a proof routine which we illustrate in Lemmas 34 and 36 Lemma 34. Zorn’s lemma implies the axiom of choice. The converse result is less important to us but we prove it for complete- ness. Lemma 35. The axiom of choice implies Zorn’s lemma. Proof. (Since the proof we use is non-standard, I give it in detail.) Let X be a non-empty set with a partial order having no maximal elements. We show that the assumption that every chain has a upper bound leads to a contradiction. We write x ´ y if x y and x 6 = y . If C is a chain we write C x = { c C : x ´ c } . Observe that, if C is a chain in X , we can find an x X such that x ´ c for all c C . (By assumption, C has an upper bound, x 0 , say. Since X has no maximal elements, we can find an x X such that x ´ x 0 .) We shall take to be a well ordered chain.
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