Hurkyl said:Intuitively, how does a Cauchy sequence specify a real number? And how does a Dedekind cut specify a real number?
The bijection should be nearly obvious -- once you know what it is, you just have to go through the motions of writing it down, showing it really is well-defined, a bijection, et cetera. (it's a lot of motions, though)
Hurkyl said:Of course, you could just prove them both isomorphic to the decimal numbers. Or prove that all complete ordered fields are isomorphic, and both happen to be complete ordered fields. You don't have to compare them directly to each other.
qspeechc said:Chapter 1 of Pugh's Real Mathematical Analysis constructs the real numbers from Dedekind cuts, and then proves that complete ordered fields are unique up to an isomorphism. Later on, in chapter 2, he constructs the real numbers from Cauchy sequences, and then because complete ordered fields are unique we get that the two constructions give essentially the same thing. So you might be interested in geting Pugh's book from the library.
sevenlite said:Ive found this book, thank you!
Cauchy real and Dedekind real are two different ways of defining real numbers in mathematics. Cauchy real numbers are defined based on the concept of sequences, while Dedekind real numbers are defined based on the concept of cuts. Both definitions aim to capture the same set of real numbers, but they differ in their approach.
Yes, Cauchy real and Dedekind real are considered equivalent or isomorphic, as they both define the same set of real numbers and can be translated into each other. This means that any mathematical statement about one system of real numbers can be translated into the other system without changing its truth value.
The equivalence or isomorphism between Cauchy real and Dedekind real is significant because it allows for a more rigorous and comprehensive understanding of real numbers. It also allows for the application of results and theorems from one system to the other, making mathematical analysis more efficient and powerful.
One example of a Cauchy real number is the sequence 0.9, 0.99, 0.999, 0.9999, ... which represents the real number 1. On the other hand, the corresponding Dedekind real number would be the cut {(x in Q | x < 1) | (x in Q | x > 1)}. This cut divides the rational numbers into two sets, one containing all numbers less than 1 and the other containing all numbers greater than 1.
Yes, there are other systems of defining real numbers, such as surreal numbers, hyperreal numbers, and constructive real numbers. Each system has its own unique approach and properties, but they all aim to define the same set of real numbers.