Absurdity of Cantor's Diagonal Slash

[master_coda]

Its not only about listening, its about rational discussion and proof.

The cardinality of \mathbb{N} is infinite (aleph0)

Let card(n) be the cardinality of the set of natural numbers from 1 to n inclusive

Then it is easy to show that card(n) = n

Let the largest possible natural number be Z. According to you, Z is finite. However, in that case card(Z) = Z (a finite number) and not aleph0 (infinity)

Where is the flaw in my logic?

There is no largest possible natural number. By assuming that it exists, you derive a false conclusion.

 

[chronon]

correction : The largest finteger (as defined by chronon) does not exist

Now we're getting somewhere. So do you accept the following

1) There is no largest finteger

2) The set of fintegers is infinite

3) Cantor's diagonal argument shows that set of reals cannot be put into 1-1 correspondence with the set of fintegers.

If you (A) accept these 3 statements and (B) start speaking the same language as everyone else and give the name integers to what I have called fintegers, then you have Cantor's proof

 

[CrankFan]

correction : The largest finteger (as defined by chronon) does not exist

You don't get to decide what the set of integers is, that was decided a long time ago. Everyone here, except you, seems to have a good idea of what the integers are.

If you want to talk about a set that is similar to the set of integers but not actually the same then you have to call that set something other than the integers, otherwise you're just babbling.

If you actually do want to talk about integers then start with a well known definition, something like: the smallest inductive subset of R, closed under negation. Once you settle on a standard definition the rest of the questions can be dealt with. If you can't settle on a standard definition then you simply don't know what the integers are. Sorry.

 

[matt grime]

I never said it was.

you did, here:

The cardinality of N is infinite (aleph0)

Let card(n) be the cardinality of the set of natural numbers from 1 to n inclusive

Then it is easy to show that card(n) = n

Let the largest possible natural number be Z. According to you, Z is finite. However, in that case card(Z) = Z (a finite number) and not aleph0 (infinity)

there is a difference between a set being finite, and its elements being finite.

Do you claim that the relationship card(n) = n (where card(n) is defined as the cardinality of the set of all integers between 1 and n inclusive) is false?

n isn't a priori a set, so it hasn't got a cardinality, but even allowing for this abuse of notation, what on Earth does this have to do with card(N), unless you believe it to be an integer?

 

[HallsofIvy]

It is well established in set theory that each of the sets of natural numbers, integers and rational numbers is countably infinite.

This means that each of these sets has infinite cardinality.

This means that there are infinitely many natural numbers, integers and rational numbers.

Any natural number or integer which is infinite must have an infinite number of digits.

Or perhaps you can explain how an infinite integer can have a finite number of digits?

Yes, there are an infinitely many natural numbers (and so integers and rational numbers).,

Your next line does not follow has a logical "leap": saying that there are infinitely many natural numbers does NOT imply that there must exist "an infinite integer". There exist infinitely many integers, each of which is a finite number.

Since you mention rational numbers, we might point out that there exist infinitely many rational numbers of the form 1/2ⁿ: that is, there exist infinitely many rational numbers, between o and 1, each of which has a finite number of digits (is a terminating decimal).

 

[Hurkyl]

In 1874 Georg Cantor postulated that there is more than one level of infinity. The lowest level he called "countable infinity" and higher levels he called "uncountable infinities." The natural numbers are an example of a countably infinite set and the real numbers are supposed to be an example of an uncountably infinite set.

The postulate of a countably infinite set of integers is well accepted in set theory and number theory.

Agreed.

However, that does not imply the existence of an infinite integer. The set of integers has infinite cardinality, but each individual integer has finite magnitude.

And in which maths bible does it specify an integer can be obtained in this way only a *finite* number of times?

Virtually all of them. Here is a quick heuristic inductive proof of this fact for the natural numbers:

2 can be obtained by adding one to itself a finite number of times.

Suppose n can be obtained by adding one to itself a finite number of times.
Then, (n+1) can be obtained by adding one to itself a finite number of times, and then once more. Thus, (n+1) can be obtained by adding one to itself a finite number of times.

Therefore, by the Law of Mathematical Induction, if n is an integer such that n >= 2 then n can be obtained by adding 1 to itself a finite number of times.

(The Law of Mathematical Induction applies because it is part of the definition of the natural numbers)

I have done this already in Post #7 of this thread, using your own argument to show the absurdity of Cantor's method (if you like, a kind of reductio ad absurdum argument)

An argument is invalid if and only if one of its steps is invalid. So, if your analysis is correct, you should be able to find an invalid step.

Note that if your analysis is correct, and there does not exist an invalid step, you've proven that the axiomatic system you were using is inconsistent.

It is well established in set theory that each of the sets of natural numbers, integers and rational numbers is countably infinite.

This means that each of these sets has infinite cardinality.

This means that there are infinitely many natural numbers, integers and rational numbers.

Any natural number or integer which is infinite must have an infinite number of digits.

As I mentioned above, the fact the set of natural numbers is infinite does not imply that there is a natural number that is infinite.

Since this is an infinite set of integers then the largest member of this set must be infinite

Why do you think there is a largest member? Here's a proof otherwise:

Suppose M is the largest integer.

M + 1 is an integer.
0 < 1
M + 0 < M + 1
M < M + 1

Thus there exists an integer larger than M. This contradicts the assumption that M is the largest integer.

Therefore, there does not exist a largest integer.

Let the largest possible natural number be Z. According to you, Z is finite. However, in that case card(Z) = Z (a finite number) and not aleph0 (infinity)

According to us, card(Z) is not a natural number. It is a cardinal number. In fact, card(S) is a cardinal number for any set S, whether finite or infinite.

It is true, though, that the finite cardinals form a model of the natural numbers.

 

[moving finger]

I am learning lots, and I do appreciate the time that members have taken to explain things. I apologise for wasting your time if it seems many of my questions and ideas seem naïve.

Here is a quick heuristic inductive proof of this fact for the natural numbers:

2 can be obtained by adding one to itself a finite number of times.

Suppose n can be obtained by adding one to itself a finite number of times.
Then, (n+1) can be obtained by adding one to itself a finite number of times, and then once more. Thus, (n+1) can be obtained by adding one to itself a finite number of times.

Therefore, by the Law of Mathematical Induction, if n is an integer such that n >= 2 then n can be obtained by adding 1 to itself a finite number of times.

(The Law of Mathematical Induction applies because it is part of the definition of the natural numbers)

I cannot find any published reference in number theory or set theory which says that either natural numbers or integers are defined in such a way that this derivation may be carried out only a finite number of times, as you suggest above.

If we were to construct the entire of set of natural numbers, starting with zero, from the above rules, then we would end up (because of the stipulation above that the addition may be carried out literally only a finite number of times) with a finite number of natural numbers. Not with an infinite number of natural numbers.

 

[matt grime]

You reallly need to go away and learn some very basic maths, and simple logic.

1 is a natural number. 1+1 is a natural number. The natural numbers are those things that may be obtained by repeatedly adding 1 a finite number of times. That is what the naturals are. Each natural is "finite" but the set of all natural numbers is infinite. You can't find a definition yet everyone else here knows what they are. What does that tell you?

Here's the simple logic: if the set of all naturals were finite then it would have a largest element, M. But M+1 is also a natural and bigger than M, contradicting the assumption that there is a largest natural, ie contradicting the assertion the set of natural numbers is finite, hence it must be infinite.

So far you've not been able to find a flaw in that argument, but only to make unrelated and false assertions and incorrect deductions.

So why don't you go through that argument and find out where it goes wrong?

 

[CrankFan]

I cannot find any published reference in number theory or set theory which says that either natural numbers or integers are defined in such a way that this derivation may be carried out only a finite number of times, as you suggest above.

It's not surprising that you wouldn't find any reference to definitions of naturals or integers in a book on number theory since they usually presuppose that the reader is already familiar with them.

On the other hand, as Hurkyl indicated, set theory books usually have them. What set theory books are you using? Can you name the title and author's? Are you claiming that you can't find a definition of natural numbers in any of the published works that you have access to?

 

[Hurkyl]

Let's look at the definition of a "finite set" or a "countable set".

Definition S is a finite set iff it can be put into a 1-1 correspondence with an initial segment of the natural numbers. That is, the set \{ m \in \mathbb{N} | m &lt; n \} for some natural number n.

Definition S is a countable set iff it is either a finite set, or it can be put into a 1-1 correspondence with \mathbb{N}.


Since | \{ m \in \mathbb{N} | m &lt; n \} | = n, we see that if the size of a set is a natural number, the set is a finite set.


This is typical: finiteness is generally defined in terms of the integers. For example, here's an example from algebra.

Any ordered ring must contain the integers. An element x of an ordered ring is said to be finite if and only if there exists an integer n such that |x| < n. Similarly, x is infinite iff n < |x| for all integers n.

 

[Curious3141]

Incidentally, I believe card(n) = n only applies to *finite* subsets of the natural numbers starting from one.

Aleph-null is the cardinality of the entire set of naturals. It is incorrect to say that the "final" member of this set is itself Aleph-null.

 

[Hurkyl]

Oh, I guess I should specify that in my last post, I was using the convention that 0 is a natural number.

 

[moving finger]

You reallly need to go away and learn some very basic maths, and simple logic.

Matt, please look to your own illogical statements before you start cristicising mine.

The natural numbers are those things that may be obtained by repeatedly adding 1 a finite number of times. That is what the naturals are. Each natural is "finite" but the set of all natural numbers is infinite.

You stipulate that the addition operation may be carried out only a finite number of times, and yet this finite process apparently produces an infinite number of natural numbers? What does that tell you about your logic?

Here's the simple logic: The way you have defined it, the set of natural numbers is not infinitely large, it is arbitrarily large (but still finite in size). An arbitrarily large finite set does not have the same cardinality as an infinite set. The only reason the largest member M cannot be identified is because the actual "number of finite times" that you are allowed to do your above addition is concealed in the smoke & mirrors.

 

[matt grime]

No, any element of the naturals is finite, but can be arbitrarily large. Stop confusing the properties of the set of naturals with the properties of its elements.

Just take any infinire set of "finite objects" to see where you're wrong.

What smoke and mirrors? I've explcitly told you 1 is a natural number, so is 1+1=2, 1+1+1=3, and the set of all naturals is the set of numbers that can be obtained by adding 1 to itself a finite number of times, eg, 4,5,6,7,8,... a list that has no end, is infinite, but everything on that infinitely long list is a "finite number", isn't it?

 

[Hurkyl]

The way you have defined it, the set of natural numbers is not infinitely large, it is arbitrarily large (but still finite in size).

Wrong. The set of natural numbers is one set. It cannot be "arbitrarily large" -- it is as large as it is, no more, no less.

What is true is that there are arbitrarily large (but still finite) natural numbers. The set of all of them is infinite.


Now, I'm going to close this. I love when people want to learn, but I don't love when people just want to play know-it-all.