- #1
Rasalhague
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"A real number is a Dedekind cut in the set Q of rational numbers: a partition of Q into a pair of nonempty disjoint subsets <L,U> with every element of L less than every element of U and L having no largest member. Thus 21/2 can be identified with the cut: L = {q in Q: q2 < 2}, U = {q in Q: q2 > 2" (Goldblatt: Lectures on the Hyperreals, p. 12).
"A number d in an ordered field is called infinitesimal if it satisfies 1/2 > 1/3 > ... > 1/m > |d| for any natural counting number m = 1,2,3..." (Stroyan: Foundations of Infinitesimal Calculus, p. 9).
How can we tell that no more than one real number is defined by a Dedekind cut; is the answer that real numbers are simply defined as equal if they can be represented by a single Dedekind cut?
I don't understand Stroyan's proof, on p. 10, that there are no positive infinitesimals in the reals. He uses a Dedekind cut (A,B), where A = {a < 1/m: m = 1,2,3...} and B = {1/m < b: m = 1,2,3...}, then claims that "zero is at the gap in the reals and every positive real number is in B." But it seems to me that zero is in A rather than the gap. His cut seems to disprove the statement that there are no positive infinitesimals by defining one. What am I missing here?
"A number d in an ordered field is called infinitesimal if it satisfies 1/2 > 1/3 > ... > 1/m > |d| for any natural counting number m = 1,2,3..." (Stroyan: Foundations of Infinitesimal Calculus, p. 9).
How can we tell that no more than one real number is defined by a Dedekind cut; is the answer that real numbers are simply defined as equal if they can be represented by a single Dedekind cut?
I don't understand Stroyan's proof, on p. 10, that there are no positive infinitesimals in the reals. He uses a Dedekind cut (A,B), where A = {a < 1/m: m = 1,2,3...} and B = {1/m < b: m = 1,2,3...}, then claims that "zero is at the gap in the reals and every positive real number is in B." But it seems to me that zero is in A rather than the gap. His cut seems to disprove the statement that there are no positive infinitesimals by defining one. What am I missing here?