Any Mathematicians? Want your opinion.

[giant_robo]

Hi paul, I'm not a mathematician but would be interested. From what I understand there are two differenct classes of infinity. The first is countable (you can enumerate), and the most basic example would be the natural numbers: 1,2,3,4,...
There are an infinite number of them, but, by definition, you can count them.
Then there are those infinities that cannot be enumerated and counted up with an infinite series. I guess the real numbers in a finite interval (sat 0 and 1) would fall in this case.

I would guess that you have probably shown that the set of rational numbers can be counted. Rational numbers almost seem like irrationals to the laymen sometimes, because for practical purposes they do the same job up to any desired accuracy you want.

An irrational number consists of two integers, numerator and denominator. So it's equivalent to a 2 dimensional vector of integers (n,m). These are countable.

(1,1) (1,2) (1,3) (1,4) ...
(2,1) (2,2) (2,3) .........
(3,1) (3,2) (3,3) .........
(4,1) (4,2) ...............
...........................

You can enumerate this 2-d matrix structure which extends infinitely in two dimensions. Just count across diagonally, from the corner.

(1,1),
(2,1), (1,2),
(3,1), (2,2), (1,3),
(4,1), (3,2), ...
.... and so on

Since it can be enumerated in a series it's countable. So there are a countable number of rationals.

 

[TallPaul]

I was reading a lot of more-or-less popular mathematics books a couple years ago and got to thinking about infinity. Cantor, as I recall, showed that you could not do a one-on-one correspondence of the reals from 0 to 1 with the integers. I believe he therefore concluded they were different levels of infinity. I think I can show that there are the same number of reals between 0 and 1 as there are integers. I wrote something up on it about a page long and wondered if anyone is interested in reviewing it. If so, maybe I could email it to you.

 

[TallPaul]

giant_robo: I think I have shown that the reals between 0 and 1, while not countable, are exactly equivalent in number to the integers. It involves a tricky transformation. Hopefully I am using the terms correctly. I used integers (natural numbers and zero) and reals between 0 and 1 (rational, irrational, and whatever else--can't remember). It's been a year since I last looked at this stuff. I'll email it to you, but it won't be until Tuesday as it is on a different computer that I can't get to right now. Now, chances are that I am wrong because what I am claiming contradicts the great mathematician, Cantor, and I an in no way a mathematician, or even all that mathematically adept. What I need is someone to show me the hole in my thinking, because if I really were to show Cantor to be wrong, it would be a remarkable event. I doubt it.

 

[giant_robo]

Paul, I would be quite interested. It's unlikely that Cantor is wrong, but there could be some interesting ideas. My email is laubwsky~freeshell.org . Replace the ~ with an "at" sign.

 

[UncleS2]

?? A RATIONAL # can be written exactly as the quotient of two integers. An IRRATIONAL number cannot. Just a typo?

Before we spend any more time on this, let's make sure we're all on the same page.

Complex numbers(C): The largest set of numbers, includes all other sets. Allows us to find square roots of negative numbers. Of the form a+bi, where a & b are real #'s, a is the real component, b is the imaginary coefficent, & i^2 = -1. Ex: 3 - 2i, where a=3 & b=-2. "i" is the imaginary unit of the square root of -1.

Real Numbers(R): All Complex numbers where the imaginary coefficent, b, =0. Restricted to Real #'s, we have no square roots of negative numbers, we cannot factor the sum of two squares, etc. Reals include both rational & irrational numbers.

Irrational numbers: Real numbers that cannot be written in exact value form as the quotient of two integers. Ex: Pi, the square root of two.

Rational numbers(Q): Consists of all real numbers that can be written exactly as the quotient of two integers. Ex: 2/3, -5. (-5 = -5/1, so even though it didn't look like it at first, it fulfills our requirements for this set.)

Integers(Z): rational numbers where all the denominators = 1, & so are not needed. Extend from negative infinity to positive infinity. Made up of the set Z={...-3,-2,-1,0,1,2,3...}

Whole numbers(W): Take the Integers, throw away the negatives, & you're left with the Whole Number set: W={0,1,2,3...}

Natural Numbers(N)(also called the counting numbers): Exactly how you began to count things as a small child. Consists of the positive integers only. N={1,2,3,4...}

OK, that's Sets of Numbers 101 for today.

And: Iff: means If and only if; commonly used in definitions in math.

Now- infinity. You're referring to degrees of infinity. Definitionthis is *IMPORTANT*): A set is COUNTABLY INFINITE Iff there is a one to one correspondence between the Natural Numbers(N) and the set in question.

It's pretty easy to show, using a matrix type arrangement, that W, Z, & Q are all countably infinite. The real #'s are uncountably infinite, as are(this is a little mind boggling) the irrational numbers, which are a subset of the real. It escapes me now, but a common proof is to show that there are more real numbers between 0 and 1 than there are natural numbers.

If you seem to be establishing a 1-1 correspondence between the Naturals & the Reals, something is wrong somewhere. Are you leaving out the Irrationals, as suggested above?

In proofs, remember: A thousand examples do not constitute proof of anything. But *ONE* Counterexample is all it takes to Disprove something.

 

[giant_robo]

yes, there is a typo Stuart. Sorry for the confusion. I guess rationals and irrationals are so similar that I subconsiously mistook them myself.

quote:

---

I would guess that you have probably shown that the set of rational numbers can be counted. Rational numbers almost seem like irrationals to the laymen sometimes, because for practical purposes they do the same job up to any desired accuracy you want.

An irrational (--should be rational--) number consists of two integers, numerator and denominator. So it's equivalent to a 2 dimensional vector of integers (n,m). These are countable.

---

 

[TallPaul]

Wow! Thanks Stewart. I will email you my so-called proof. A co-worker listed out all the number types for me, but I misplaced the list. I really meant whole numbers, not integers. My thing seems to show that the reals between 0 and 1 are countably infinite. Looks like I will be able to email this to you Monday. Looking forward to your response.