Is Cantor's Diagonal Argument Flawed in Its Application to Positive Integers?

[Hurkyl]

I don't quite follow this. By -1/9 I take it you are denoting the number that could also be represented as the recurring decimal -0.1111 ...

No, I am not. As I said, - refers to additive inverse, and / refers to multiplication by the multiplicative inverse.

The additive inverse of 1 is ...999. (Because ...999 + 1 = 0)
The multiplicative inverse of 9 is ...8889. (Because ...8889 * 9 = 1)
Finally, ...999 * ...889 = ...111.

As I had said, the numbers in this number system are the left-infinite strings of decimal digits. -0.111... doesn't denote a number.

I'm also wondering what number the symbol oo (infinity) represents (if any). What does it mean when we say 'as x approaches oo' ?

The simplest is to define the phrase "The limit of ____ as __ approaches $\infty$" as a whole; the individual symbols and words aren't given any meaning. This is something you'd probably see in an elementary calculus book.

A slightly more sophisticated treatment would define a new number system (such as the extended real numbers or the extended natural numbers) that contains an element called $+\infty$, and then you can just use the ordinary definition of the phrase "The limit of _____ as __ approaches __".

 

[manker]

No, I am not. As I said, - refers to additive inverse, and / refers to multiplication by the multiplicative inverse...

Yes, I see this now.

The simplest is to define the phrase "The limit of ____ as __ approaches $\infty$" as a whole; ...

I guess I was trying to find the distinction in meaning between the symbols $$\infty$$ and $$\aleph_0$$. Would I be right in thinking that while $$\aleph_0$$ is the cardinality of the set of integers (amongst others), $$\infty$$ is (potentially at least) an arbitrarily large member of that set? Is there any sensible way of comparing $$\aleph_0$$ and $$\infty$$? After all the cardinality of {1, 2, 3} is also a member of the set.

 

[Hurkyl]

Yes, I see this now.

I guess I was trying to find the distinction in meaning between the symbols $$\infty$$ and $$\aleph_0$$. Would I be right in thinking that while $$\aleph_0$$ is the cardinality of the set of integers (amongst others),

Essentially yes -- that is the most common intended meaning for that symbol.

$$\infty$$ is (potentially at least) an arbitrarily large member of that set?

No. Scratch that -- emphatically no. Can you even define what you mean by an "arbitrarily large member of a set"?

Again, it's just another symbol. But unlike $\aleph$, there are actually several different mathematical structures that make use of the symbol $\infty$ to denote an element. The most common structures you'd encounter are the projective real numbers and the extended real numbers (Look them up!) (In the latter, $\infty$ is really just shorthand for $+\infty$).

Is there any sensible way of comparing $$\aleph_0$$ and $$\infty$$? After all the cardinality of {1, 2, 3} is also a member of the set.

No and yes. The most common uses of those symbols lie in different structures, without any "default" conventional way to convert between them. But that said, it is occasionally useful to observe the fact that the closed interval $[0, \aleph_0]$ of cardinal numbers and the interval $[0, +\infty]$ of extended natural numbers are isomorphic as ordered sets. (Equivalently, as topological spaces, when given the order topology)

 

[proton]

Hoping I may be permitted to make a belated contribution here.

Summarising the key points from some of the other posts:
The key to the problem is that the new 'number' generated by the diagonal argument will have an infinite number of non-zero digits resulting from the infinite number of leading zeros in the original numbers. Such a number is not an integer since, although the set of integers has an infinite number of members, each individual member is finite and must have a finite number of non-zero digits.

i was actually having a ton of trouble figuring out what everyone was saying until i read this post. thanks alot!