Linked Questions

1
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2answers
20k views

Can somebody explain to me Cantor's diagonalization argument? [duplicate]

Like..can somebody explain this to me as if I was a 5 year old or something? Every explanation I read repeats the same exact thing that I simply do not understand. This is what my book says: "The ...
1
vote
3answers
313 views

I don't understand the concept of different sizes of infinity. [duplicate]

The argument to prove that there are different sizes of infinity is by saying .. no matter how many decimal numbers you write you can always come up with new one different from all others listed ...
1
vote
3answers
739 views

Cantor's Diagonalization: Impossible to formulate the decimal expansion in (0, 1) that serves as the contradiction? [duplicate]

Possible Duplicate: How does Cantor's diagonal argument work? It seems to me that if you have an infinite list of unending decimal numbers between $(0,1)$: ...
0
votes
2answers
122 views

Can somebody explain why the interval $\left ( 0,1 \right )$ is not countable? [duplicate]

I cannot seem to understand the proof of why the interval $\left ( 0,1 \right )$ is not countable. The proof that is written in my book using the method of Reductio ad absurdum. It starts with the ...
0
votes
1answer
199 views

Can someone explain the reasoning behind Cantor's diagonal argument? [duplicate]

I'm taking a class that's covering cardinalities, and I was introduced to Cantor's diagonal argument today, and I'm having trouble following the logic. The theorem states, "If s[1], s[2], … , s[n], … ...
0
votes
1answer
118 views

How does Cantor's diagonal argument actually prove that the set of real numbers is larger than that of natural numbers? [duplicate]

I have looked into Cantor's diagonal argument, but I am not entirely convinced. Instead of starting with 1 for the natural numbers and working our way up, we could instead try and pair random, ...
-2
votes
2answers
91 views

A difficulty in the proof that $2^{\mathbb{N}}$ is uncountable. [duplicate]

This is the explanation of the book, but for me I did not catch the idea of the set $v$, and I need a numerical example for it and the set $u_{k}$, could anyone help me please? Is there a systematic ...
0
votes
0answers
75 views

Why are positive rational numbers countable but real numbers are not? [duplicate]

If we can say that any positive rational number is countable or listable by showing that every positive rational number is the quotient of p/q of two positive ...
1
vote
1answer
37 views

By using a diagonal argument, show that the powerset $P(N) = (S|S ⊆ N)$ is uncountable. [duplicate]

Any tips or solutions for this one? By using a diagonal argument, show that the powerset $P(N) = (S|S ⊆ N)$ is uncountable.
35
votes
2answers
8k views

Why doesn't Cantor's diagonal argument also apply to natural numbers?

In my understanding of Cantor's diagonal argument, we start by representing each of a set of real numbers as an infinite bit string. My question is: why can't we begin by representing each natural ...
4
votes
5answers
4k views

Prove that the open interval $(0, 1)$ contains uncountably infinite numbers.

Prove that the open interval $(0, 1)$ contains uncountably infinite numbers. Apparently, there is a way to prove this proposition using Cantor's diagonalization argument. How does that work? How ...
8
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7answers
2k views

Improving my understanding of Cantor's Diagonal Argument

I studied Cantor's Diagonal Argument in school years ago and it's always bothered me (as I'm sure it does many others). In my head I have two counter-arguments to Cantor's Diagonal Argument. I'm not ...
6
votes
3answers
804 views

Understanding Cantor's diagonal argument

I'm trying to grasp Cantor's diagonal argument to understand the proof that the power set of the natural numbers is uncountable. On Wikipedia, there is the following illustration: The explanation of ...
0
votes
4answers
6k views

Why is the cardinality of irrational numbers greater than rational numbers?

This was asked by blogegog on a YouTube comment (gasp!): [Regarding Cantor's diagonal argument:] Couldn't I just make the same statement about rational numbers and say, 'take the largest ...
0
votes
6answers
380 views

Issue with Cantor's Diagonal Argument and square matrices

I have an issue with Cantor's Diagonal Argument. We suppose we have an infinite list with every real in $(0, 1)$ listed. Then we take the diagonal and change every digit in a predefined way (e.g. $+1 ...

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