Is an open interval in R really uncountable?

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In summary, the conversation discusses the cardinality of a set and the use of a bijection to count the elements in the set. However, it is pointed out that this method only works for rational numbers with finite decimal expansions and does not account for irrational numbers or a large portion of rational numbers. The person speaking is impressed that the error was realized quickly.
  • #1
ak416
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Ok i know it should be because it has the same cardinality as R and R is uncountable. But take for example (0,1). Heres a method I would use to count all its elements:

0.1,...,0.9
0.01,...,0.99
0.001,...,0.999
.
.
.
0.(n-1 zeros)1,...,0.(n 9's)
.
.
.

so count starting from the top left and keep going right till you reach the end of the line then count the in the same direction for the next line and so on.
Doesnt this define a bijection from (0,1) to Z+ ? Take any number in (0,1). If it has n digits after its decimal, go down n lines to find its corresponding integer (it will be 9+99+...+(n-1 9's) + the number * 10^n )

Whats wrong with my reasoning?
 
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  • #2
ok i think i got it. A real number can have an infinite number of decimals!
 
  • #3
Correct!
At each step, you have only counted up rationals with finite decimal expansions.

Not a single irrational number occurs anywhere for any n.
 
  • #4
And only a small fraction of the rational numbers!

ak416, a lot of people just can't grasp that. I'm impressed that you were able to realize the error so quickly.
 

1. What is an open interval in R?

An open interval in R is a set of real numbers that includes all numbers between two given values, but does not include those values themselves. For example, the open interval (0,1) contains all real numbers between 0 and 1, but does not include 0 or 1.

2. How is the uncountability of open intervals in R proven?

The uncountability of open intervals in R is proven using the Cantor diagonalization argument. This method involves constructing a list of all possible real numbers in the interval and then showing that there is always a number missing from the list, indicating that the set is uncountable.

3. Why is the uncountability of open intervals in R significant?

The uncountability of open intervals in R is significant because it demonstrates the infinite nature of the real numbers. It also has important implications in various mathematical fields, such as topology and analysis.

4. Can an open interval in R contain an infinite number of elements?

No, an open interval in R can never contain an infinite number of elements. This is because an infinite number of elements would make the set countable, which contradicts the proof of uncountability using the Cantor diagonalization argument.

5. How does the uncountability of open intervals in R relate to the continuum hypothesis?

The continuum hypothesis states that there is no set with a cardinality between that of the natural numbers and the real numbers. The uncountability of open intervals in R is one of the ways in which this hypothesis is proven, as it shows that there is no countable set of real numbers between any two given real numbers.

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