Alternatives to proving the uncountability of number between 0 and 1

• robertjford80
In summary, the conversation discusses the concept of countability of real numbers between 0 and 1. It is mentioned that Cantor's diagonal argument is a more sophisticated method to prove uncountability, compared to a simpler method using decimal representations. The conversation also mentions the set of real numbers with finite decimal representations and how it is countable, leading to the conclusion that the set of all real numbers between 0 and 1 is uncountable. The conversation also includes a request for the person to think about the claim before asking for clarification.
robertjford80
I'm aware of Cantor's diagonal argument but can't you prove the uncountability of reals between 0 and 1 using a simpler method? For instance, take the number .1 sooner or later in order to get a list of the reals between 0 and 1, you're going to have to get to .2 but before you can get to .2 you have to write .11 on your list then before you can get to .12 you have to write .111. In other words if you're obligated to open up a new decimal place then you'll never get to .2

Since there are only countably many reals with finite decimal representations, and since this observation only accounts for these representations, it will not suffice. Something more sophisticated like the diagonal argument is needed.

jgens said:
Since there are only countably many reals with finite decimal representations

I need more details as to what you mean.

robertjford80 said:
I need more details as to what you mean.

In the future try thinking about the claim for more than three minutes before asking to have it spelled out for you. In any case, notice that the set An of all real numbers in [0,1] whose decimal expansion has length n is in bijection with the set {0,...,9} x ... x {0,...,9} (this is an n-fold product). Since this latter set is finite, it follows that An is finite. Thus the set A of all real numbers in [0,1] with finite decimal representations is given by ∪An and since a countable union of countable sets is again countable the claim follows.

jgens said:
In the future try thinking about the claim for more than three minutes before asking to have it spelled out for you.
What a nice person you are. Got a love a man that insults people so easily.

In any case, notice that the set An of all real numbers in [0,1] whose decimal expansion has length n is in bijection with the set {0,...,9} x ... x {0,...,9} (this is an n-fold product).
You're going to have to restate this in informal English.

Since this latter set is finite
I'm not convinced that it's finite. Explain.

robertjford80 said:
What a nice person you are. Got a love a man that insults people so easily.

It was not an insult, it was a request. PF is not a place for other people to do the thinking for you. There is an expectation that you think something through before asking.

You're going to have to restate this in informal English.

It would do you good to familiarize yourself with mathspeak, but that point aside I am not sure which part of that sentence is tripping you up, so I need some indication on which parts need clarifying.

I'm not convinced that it's finite. Explain.

Ten minutes is also not a sufficient period of time. Think about the claim some more and if after a couple hours you cannot understand why the set {0,...,9} x ... x {0,...,9} (again an n-fold product here) is finite, then come back and ask.

The OP won't be coming back any time soon, so there's no point in keeping this open.

1. How do we prove that there are an infinite number of numbers between 0 and 1?

One way to prove this is by using the Cantor's diagonal argument. This method involves constructing a decimal number that is not in the given list of numbers between 0 and 1, thus showing that the list is incomplete and there must be more numbers between 0 and 1.

2. Are there any other methods to prove the uncountability of numbers between 0 and 1?

Yes, there are other methods such as using the nested interval theorem or the Bolzano-Weierstrass theorem. These methods also involve constructing a number that is not in the given list, thus proving the infinite uncountability of numbers between 0 and 1.

3. How is the uncountability of numbers between 0 and 1 relevant in mathematics?

The uncountability of numbers between 0 and 1 is relevant in various fields of mathematics, such as analysis, topology, and set theory. It helps in understanding the concept of infinity and plays a significant role in many mathematical proofs and theorems.

4. Is it possible to prove the uncountability of numbers between 0 and 1 without using the diagonal argument?

Yes, it is possible to prove the uncountability of numbers between 0 and 1 without using the diagonal argument. As mentioned before, other methods such as the nested interval theorem or the Bolzano-Weierstrass theorem can be used to prove this concept.

5. Can the uncountability of numbers between 0 and 1 be visualized?

While it may be challenging to visualize an infinite number of numbers, there are visual representations such as the Cantor set or the Sierpinski triangle that demonstrate the uncountability of numbers between 0 and 1. These fractal structures have a finite area but an infinite number of points, showing the infinite nature of numbers between 0 and 1.

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