Dedekind Cuts & the Real Line: A Countable Set?

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SUMMARY

The discussion centers on the nature of Dedekind cuts and their relationship to the real line. It is established that while Dedekind cuts are defined using rational numbers, they can also represent irrational numbers, thus producing the entire real line. A specific example provided is the set T = { x ∈ Q: x² < 2 or x < 0 }, which illustrates that not all Dedekind cuts correspond to rational numbers. This distinction is crucial for understanding the completeness of the real number system.

PREREQUISITES
  • Understanding of Dedekind cuts in real analysis
  • Familiarity with rational and irrational numbers
  • Knowledge of decimal expansions of real numbers
  • Basic concepts of set theory and properties of real numbers
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  • Study the properties of Dedekind cuts in detail
  • Explore the concept of completeness in the real number system
  • Investigate the implications of irrational numbers in set theory
  • Learn about the construction of real numbers using Cauchy sequences
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Mathematicians, students of real analysis, and anyone interested in the foundational aspects of the real number system and its properties.

cragar
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If every Dedekind cut is at a rational it seems that these cuts would only produce a countable set and would not produce the whole real line. So how should I think about it.
 
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cragar said:
If every Dedekind cut is at a rational it seems that these cuts would only produce a countable set and would not produce the whole real line. So how should I think about it.

The point is that there are Dedekind cuts that are NOT at rationals.
For example, take any real number A and consider its decimal expansion. A sequence of rational numbers An can be defined by taking n terms of the expansion. Let the cut be defined by all rationals less than any term in the sequence. This cut gives the real number A.
 
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Even though all Dedekind cuts consist of only rational numbers, all are not rational cuts.
T = { x \in Q: x^2 < 2 or x < 0 } is a dedekind cut, you can check that all the properties hold, but it can not be a rational cut.
 

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