Question about diagonal argument

  • Thread starter cragar
  • Start date
  • Tags
    Argument
In summary, the conversation discusses Cantor's diagonal argument and the inability to line up all decimal numbers with the odd natural numbers. The argument produces an infinite number of numbers that cannot be matched up, and the conversation explores why this is the case. The conversation also touches on the concept of countable infinity and the ability to produce new numbers by making different choices. Ultimately, the conversation concludes that the diagonal argument is a proof by contradiction and cannot be disproven by finding a different way to line up the numbers.
  • #1
cragar
2,552
3
I thought I understood cantors diagonal argument but then I started to rethink it.
Why couldn't i just line up all the decimal numbers with the the odd natural numbers.
Then when we create a new decimal that is on my list I will line it up with an even number because I haven't used any of those yet.
 
Mathematics news on Phys.org
  • #2
Yes, you could do that but you haven't proved anything that way. Cantor's diagonal proof does not produce one number that cannot be matched up, it produces an infinite number of them. You have not yet shown that all of those numbers, that are not matched to the odd numbers, can be matched with the even numbers. In fact, we know, from Cantor's proof, that they can't.
 
  • #3
cragar said:
Why couldn't i just line up all the decimal numbers with the the odd natural numbers.

Why do you think you can even do that?? You can't. That's the whole point of the diagonal argument, you can't line up all the decimal numbers with the odd natural numbers.

The diagonal argument is a proof by contradiction.
 
  • #4
im not saying you can line up all the decimals with the odd numbers , I am
just saying we will start with that, then the new number that you create that is not on
my list I will line that one up with an even number and the next decimal you line
up with the next even number. I could do that with the rational numbers
I could line up the naturals with the naturals and then create a rational that wasn't on my list but we know that the rationals are countable so it is not clear to me why the diagonal argument works now.
 
  • #5
The same does not work with the rationals because the diagonal number is not guaranteed to be rational.
 
  • #6
ok but it still isn't clear to me why the diagonal argument works for the reals because I could just start to line them up with all the powers of 5 and then I would have all the
powers of primes to line up with the new numbers that you create that aren't on my list .
It just isn't clear to me.
 
  • #7
cragar said:
ok but it still isn't clear to me why the diagonal argument works for the reals because I could just start to line them up with all the powers of 5 and then I would have all the
powers of primes to line up with the new numbers that you create that aren't on my list .
It just isn't clear to me.

So that way you won't find a contradiction.
But the usual way, you do find a contradiction.

It's not because you can't find a contradiction if you do something different, that the diagonal argument doesn't work.
 
  • #8
First of is the fluff of requiring each number to be uniquely represented. Next given a countably infinite list it is always possible to produce a number not among them by choosing a feature of each listed number and not matching it. Imagine you have an infinite number of binary options for a sandwich (such as has and does not have). Given an infinite number of sandwiched we can always make a new one by switching one option on each sandwich.

like this
1 has egg
2 does not have mayo
3 does not have onion
4 has pickle
5 has worms
6 has orange peel
7 has yak cheese
8 does not have rhubarb
...To make a new sandwich switch the option given for each existing sandwich. Since the sandwich differs from each existing sandwich in at least one option, it must be new.
 
  • #9
cragar said:
ok but it still isn't clear to me why the diagonal argument works for the reals because I could just start to line them up with all the powers of 5 and then I would have all the
powers of primes to line up with the new numbers that you create that aren't on my list .
It just isn't clear to me.

You can't line up the reals with the powers of 5 or the odd numbers ot any subset of the integers. You are assuming what you want to disprove.
 
  • #10
ok but it still isn't clear to me why the diagonal argument works for the reals because I could just start to line them up with all the powers of 5 and then I would have all the
powers of primes to line up with the new numbers that you create that aren't on my list .
It just isn't clear to me.
Once we've made infinitely many choices to fill up your list completely, I'm going to apply the diagonal argument again to produce yet another number not on your list.

Since your list is full, there's no room for you to add this new number to the list.
 

1. What is the diagonal argument?

The diagonal argument is a mathematical proof technique used to show that a certain set is uncountable, meaning it contains an infinite number of elements. It was first introduced in 1891 by mathematician Georg Cantor.

2. How does the diagonal argument work?

The diagonal argument works by assuming that a set is countable, meaning it can be put in a one-to-one correspondence with the natural numbers. Then, by constructing a new element that is not in this assumed countable set, a contradiction is reached, proving that the set is actually uncountable.

3. What is an example of the diagonal argument?

An example of the diagonal argument can be seen in the proof that the set of real numbers between 0 and 1 is uncountable. Assuming that this set is countable, a new number can be constructed by taking the nth digit after the decimal point of the nth number in the assumed list. This new number is not in the assumed list, leading to a contradiction.

4. What is the significance of the diagonal argument?

The diagonal argument is significant because it provides a way to prove that certain sets are uncountable, which has important implications in mathematics. It also highlights the limitations of counting and the infinite nature of numbers.

5. Are there any criticisms of the diagonal argument?

There are some criticisms of the diagonal argument, particularly in regards to its use in proving the existence of uncountable sets. Some argue that it relies on the assumption that every real number can be represented as an infinite decimal, which may not be true for all numbers. Additionally, the diagonal argument does not work for all types of infinite sets, such as sets of infinite sequences or sets of functions.

Similar threads

I Question about Cantor's Diagonal Proof
    • General Math
Replies
22
Views
2K
B Question about Georg Cantor's Diagonal
    • General Math
Replies
19
Views
1K
B Another consequence of Cantor's diagonal argument
    • Set Theory, Logic, Probability, Statistics
2
Replies
43
Views
4K
B I have an issue with Cantor's diagonal argument
    • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
I Cantor's diagonalization on the rationals
    • Set Theory, Logic, Probability, Statistics
Replies
25
Views
2K
I Maybe all reals can be listed?
    • General Math
Replies
32
Views
2K
What is the problem with the argument for the diagonal length of a square?
    • General Math
Replies
6
Views
2K
B One thing I don't understand about Cantor's diagonal argument
    • Set Theory, Logic, Probability, Statistics
2
Replies
55
Views
4K
B My argument why Hilbert's Hotel is not a veridical Paradox
    • General Math
Replies
8
Views
1K
B Inscribing a circle in an Oblique Square (Drafting)
    • General Math
Replies
9
Views
822

Hot Threads

Recent Insights

Back
Top