: Real and Abstract Analysis (Graduate Texts in Mathematics) (v. 25) : Edwin Hewitt, Karl Stromberg. Real and Abstract Analysis. Edwin Hewitt and Karl Stromberg His mathematical interests are number theory and classical analysis. Real and Abstract Analysis: A modern treatment of the theory of functions of E. Hewitt,K. Stromberg Limited preview –
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You made infinitely many steps, therefore you had to use the axiom of choice. And indeed, it is consistent with the failure hewigt the axiom of choice that there are infinite sets which do not have a countably infinite subset. One classical example is an infinite set which cannot be written as a disjoint union of two infinite sets meaning, every subset is finite or its complement is finite.
Real and abstract analysis : a modern treatment of the theory of functions of a real variable
Home Questions Tags Users Unanswered. Following is Theorem 4. Every infinite set has a countably infinite subset.
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I do not understand the use of Axiom of Choice in the proof. Steve 3 This requires the axiom of choice.
Alternatively, you can prove without using AC that every Dedekind-infinite set has a subset that satisfies Peano’s axioms, i. The proof here, however, chooses and glues up such subsets into a countably infinite subset.
First, let us remind ourselves of what the Axiom of Choice is.
It can be written as: It might be worth analysiw out that the axiom of countable of choice is not sufficient for an inductive proof. One has to modify the proof a little bit to get it to work. AsafKaragila is right; countable choice is not sufficient for the proof in this answer.
absteact It is, however, sufficient for a different proof of the result, as follows. Right you are guys, thanks! Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.