Proving Uncountability of (0,1): A Puzzling Challenge

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In summary, the conversation is discussing a problem attached as a picture, which involves proving a statement about a set V=(0,1). The individual attempting to solve the problem has tried various approaches, including constructing a sequence of rationals, but has not been successful. They mention that the statement is false and that they are open to other approaches. The other individual asks if a sequence can always be constructed to converge to any arbitrary x in (0,1) and suggests considering an element in (0,1) that is not in U. The first individual notes that the problem is poorly posed and asks for the original statement to be provided.
  • #1
aaaa202
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Homework Statement


The problem is attached as a picture.


Homework Equations


...


The Attempt at a Solution


I have been trying a lot to prove this without any really fruitful approach. At first I thought that the statement was false, or that you could at least construct a sequence of rationals such that V=(0,1) the following way:
Let a1 be a real in (0,1). Since the rationals are dense there exists a rational number b1 such that d(a1,b1)<ε. Let this be the first rational number in the sequence. Now let a2 be another real. Because the rationals are dense there exists a rational b2 such that d(a2,b2)<ε/2 etc. etc. and by successive use of this method I could generate the whole (0,1) with my approach. But this fails because (0,1) is not countable. So I'm open for any other approach to this problem.
 

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  • #2
aaaa202 said:

Homework Statement


The problem is attached as a picture.


Homework Equations


...


The Attempt at a Solution


I have been trying a lot to prove this without any really fruitful approach. At first I thought that the statement was false, or that you could at least construct a sequence of rationals such that V=(0,1) the following way:
Let a1 be a real in (0,1). Since the rationals are dense there exists a rational number b1 such that d(a1,b1)<ε. Let this be the first rational number in the sequence. Now let a2 be another real. Because the rationals are dense there exists a rational b2 such that d(a2,b2)<ε/2 etc. etc. and by successive use of this method I could generate the whole (0,1) with my approach. But this fails because (0,1) is not countable. So I'm open for any other approach to this problem.

Can you always construct a sequence from the enumeration that converges to an arbitrary ##x \in (0,1)##?

Also, showing it's a proper subset isn't too bad. Can you think of an element in ##(0,1)## that isn't in ##U##?
 
  • #3
The statement that you are trying to prove is false (or, rather, not necessarily true) even after fixing it so that epsilon is fixed and the last line refers to V rather than U.

Edit: The problem, as stated, is very poorly posed. Could you maybe copy it verbatim from the source?
 

What is the concept of "proving uncountability"?

Proving uncountability is a mathematical concept that refers to the ability to demonstrate that a set is infinite and cannot be counted or enumerated. In other words, it is impossible to assign a unique number to each element in the set.

What is the significance of proving uncountability in mathematics?

Proving uncountability is important in mathematics because it helps us understand the nature of infinity and the limitations of counting. It also has practical applications in fields such as computer science, where the ability to recognize and work with uncountable sets is crucial.

Why is proving the uncountability of (0,1) considered a "puzzling challenge"?

Proving the uncountability of (0,1) is considered a challenging puzzle because it requires a deep understanding of mathematical concepts such as cardinality, infinity, and set theory. It also involves complex techniques and rigorous logic to arrive at a conclusive proof.

What are some common approaches to proving the uncountability of (0,1)?

Some common approaches to proving the uncountability of (0,1) include using Cantor's diagonal argument, the nested intervals theorem, and the Baire category theorem. These methods involve constructing a bijection (a one-to-one correspondence) between (0,1) and other uncountable sets, such as the real numbers or the power set of natural numbers, to demonstrate their uncountability.

Why is it important to prove the uncountability of (0,1)?

Proving the uncountability of (0,1) is important because it has implications for other mathematical concepts and theories, such as the continuum hypothesis and the theory of infinite sets. It also helps expand our understanding of the nature of infinity and the complexity of mathematical structures.

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