Cauchy real and dedekind real are equivalent or isomorphic

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Discussion Overview

The discussion revolves around the relationship between Cauchy real numbers and Dedekind real numbers, specifically whether they are isomorphic or equivalent. Participants explore the concepts of these two constructions of real numbers, seeking proofs or references that demonstrate their connection. The scope includes theoretical aspects of real analysis and mathematical proofs.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses a need for a proof that Cauchy reals and Dedekind reals are isomorphic, acknowledging that they are not equal but seeking a mapping between them.
  • Another participant questions how Cauchy sequences and Dedekind cuts specify real numbers, suggesting that a bijection exists but requires formal verification.
  • Some participants propose proving that both constructions are isomorphic to decimal numbers or that all complete ordered fields are isomorphic, rather than comparing Cauchy and Dedekind directly.
  • A participant mentions Pugh's Real Mathematical Analysis as a resource that constructs real numbers from both Dedekind cuts and Cauchy sequences, indicating that it shows these constructions yield the same result due to the uniqueness of complete ordered fields.
  • Another participant suggests a line of reasoning that recognizes the relationship between Dedekind cuts and equivalence classes of Cauchy sequences, indicating a potential method for proof.
  • One participant expresses skepticism about the simplification of proofs by seeking a "direct" approach.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to prove the isomorphism between Cauchy and Dedekind reals. Multiple viewpoints and methods are presented, indicating ongoing debate and exploration of the topic.

Contextual Notes

Some participants express a lack of knowledge regarding the underlying concepts, which may limit their ability to engage fully with the proofs or references discussed. There is also mention of the need for clarity on the definitions and properties of complete ordered fields.

sevenlite
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Hiya, I am looking for the proof for cauchy real and dedekind real are equal (isomorphic). I know they are not equal (CR \= DR) but I need to prove them point to the same real number or mapping from CR -> DR, DR -> CR. I have looked at the textbooks on number system, real analysis and calculus. and i can't find any. what I know is that its a classic proof. Can anyone piont out any book contains this proof or some hints about how to prove it? Thank you
 
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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)
 
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.
 
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)

I know how to define to CR and DR , I've been told that both surjections Q -> R exist. would u show me just one how its bijection exists? i don't have much knowledge on this. thank you
 
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.

the same problem, lack of knowledage. would u point out some books that contain one is complete ordered filed? I can look at. thank you
 
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.
 
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.

Ive found this book, thank you!
 
sevenlite said:
Ive found this book, thank you!

This method of proof is not direct. Why not try a line of reasoning that recognizes that each Dedekind cut determines an equivalence class of Cauchy sequences.

Conversely, each equivalence class of Cauchy sequences either has a limit or determines a Dedekind cut.
 
I don't think a "direct" proof simplifies things any.
 

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