Freethought & Rationalism Archive

Big Spoon · 03-17-2003, 01:53 AM

I've recently been researching into the mathematics of Georg Cantor. He proved that there are infinite different sizes of infinty, eg the natural numbers, the real numbers etc. He also postulated that the greatest of all the infinities was "absolute infinity", and this was the "size" of God.

He then claimed that absolute infinity could not be investigated mathematically because inconsistencies and contradictions would arise due to the divine nature of this number.

If there are infinite different sizes of infinity, how could there be a greatest? It's like saying that there is a greatest integer.

After the controvesy his theories caused, he then took a rather cowardly way out by saying that he wasn't responsible for the theories, he was just speaking on god's behalf.

Nowhere357 · 03-17-2003, 08:34 AM

Infinity and eternity as concepts make me wary. IMO they represent reality characteristics which just don't "fit in our brains". What they SEEM to be ("forever", "always") is not in fact what they are.

Example: Zeno's Paradox. Obviously, there are an infinite number of mathematical "points" between any two points.

In order to travel from A to B, you must first travel half that distance. To travel that far, you'll have to travel half that distance, and so on. You can never reach your destination because you'll always have half the distance to travel before you can get to any point. Logically sound, but completely wrong, since we easily go from A to B in reality.

This is a real paradox that went unanswered until Newton invented calculus. However, math is a tool used to describe reality; math is NOT a motivating force- and the paradox stands.

Physics gives us the Planck length. This very small measurement (about 10^-35m) can be considered the quantum of length, and therefore there's no such distance in the macro world as half a Planck length.

This at least gives me a handle on the situation. The answer to Xeno is that the premise is wrong: there is NOT an infinity of points between two points. IMO infinity and eternity are always illusory, not reality.

So the concept of "an infinity of infinities" is an illusion of illusions. How can we expect that to EVER describe reality in a meaningful way? Infinity is not a number.

paris · 03-18-2003, 01:05 PM

There is worst in these stories of infinite sets.
There is a natural question which cannot be answered from usual axioms of mathematics.
This question is: is there a set contining N contained in R which is not in bijection with N or R.

So, in the set theory of god's creation: is this property true or not ?
Of couse, Cantor ignored that this problem could
not be proven...

Anyway, these kinds of contradictions and aporias
show the non-sense of these questions on "god's creation" and so on...

AdamWho · 03-20-2003, 10:08 PM

The levels of infinity are well defined in math and are not really a subject of philosophical debate; they are definitions

Countable infinity means that you can put what ever you are counting into a 1 to 1 relationship with the positive integers.

Uncountable infinity means that you can't.

As far as the Cantor claim of "absolute infinity", didn't Russel take care of this claim when he showed that the set of all sets cannot contain itself?

John Page · 03-22-2003, 07:21 AM

Quote:

Originally posted by AdamWho
As far as the Cantor claim of "absolute infinity", didn't Russel take care of this claim when he showed that the set of all sets cannot contain itself?

Well, he showed that a paradox is produced by set theory when making the assumption that the set of all sets does contain itself.

The issue seems to be how to avoid Russell's Antinomy without producing infinite regression as a result.

Cheers, John

Big Spoon · 03-24-2003, 03:27 AM

Another way of defining countable infinity is whether or not you can count the members of a set in a systematic way.
ie. the naturals, integers and rationals are countable; the real numbers are not.

Nowhere357, can you describe a number, say 3, without any kind of self-reference? I doubt that you can because numbers themselves are abstract. The aim of pure mathematics is not to describe reality but to describe the way in which numbers behave. The infinity of infinities is not some whimsical conjecture that you can discard - it has been proven unequivocally to be true.

Nowhere357 · 03-24-2003, 01:06 PM

Quote:

Originally posted by Big Spoon
Another way of defining countable infinity is whether or not you can count the members of a set in a systematic way.
ie. the naturals, integers and rationals are countable; the real numbers are not.

Nowhere357, can you describe a number, say 3, without any kind of self-reference? I doubt that you can because numbers themselves are abstract. The aim of pure mathematics is not to describe reality but to describe the way in which numbers behave. The infinity of infinities is not some whimsical conjecture that you can discard - it has been proven unequivocally to be true.

Well I agree with everything you say here. I think I failed to explain my point well. You say "The aim of pure mathematics is not to describe reality but to describe the way in which numbers behave." Exactly! That's why I say that the concept of "infinity" does not accurately describe anything that exists physically.

Big Spoon · 03-25-2003, 02:05 AM

By the tone of your reply it sounded like you meant that not describing anything physical was a bad thing. I'm sorry if that's not how you meant it!
As far as I'm concerned, the more abstract the better.

03-17-2003, 01:53 AM	#1
Big Spoon Junior Member Join Date: Feb 2003 Location: UK Posts: 74	Infinity and Cantorian set theory I've recently been researching into the mathematics of Georg Cantor. He proved that there are infinite different sizes of infinty, eg the natural numbers, the real numbers etc. He also postulated that the greatest of all the infinities was "absolute infinity", and this was the "size" of God. He then claimed that absolute infinity could not be investigated mathematically because inconsistencies and contradictions would arise due to the divine nature of this number. If there are infinite different sizes of infinity, how could there be a greatest? It's like saying that there is a greatest integer. After the controvesy his theories caused, he then took a rather cowardly way out by saying that he wasn't responsible for the theories, he was just speaking on god's behalf.

Thread Tools	Search this Thread
Show Printable Version	Search this Thread: Advanced Search

03-17-2003, 08:34 AM	#2
Nowhere357 Veteran Member Join Date: Mar 2003 Location: Grand Junction CO Posts: 2,231	Infinity and eternity as concepts make me wary. IMO they represent reality characteristics which just don't "fit in our brains". What they SEEM to be ("forever", "always") is not in fact what they are. Example: Zeno's Paradox. Obviously, there are an infinite number of mathematical "points" between any two points. In order to travel from A to B, you must first travel half that distance. To travel that far, you'll have to travel half that distance, and so on. You can never reach your destination because you'll always have half the distance to travel before you can get to any point. Logically sound, but completely wrong, since we easily go from A to B in reality. This is a real paradox that went unanswered until Newton invented calculus. However, math is a tool used to describe reality; math is NOT a motivating force- and the paradox stands. Physics gives us the Planck length. This very small measurement (about 10^-35m) can be considered the quantum of length, and therefore there's no such distance in the macro world as half a Planck length. This at least gives me a handle on the situation. The answer to Xeno is that the premise is wrong: there is NOT an infinity of points between two points. IMO infinity and eternity are always illusory, not reality. So the concept of "an infinity of infinities" is an illusion of illusions. How can we expect that to EVER describe reality in a meaningful way? Infinity is not a number.

03-18-2003, 01:05 PM	#3
paris Junior Member Join Date: Mar 2003 Location: France Posts: 16	There is worst in these stories of infinite sets. There is a natural question which cannot be answered from usual axioms of mathematics. This question is: is there a set contining N contained in R which is not in bijection with N or R. So, in the set theory of god's creation: is this property true or not ? Of couse, Cantor ignored that this problem could not be proven... Anyway, these kinds of contradictions and aporias show the non-sense of these questions on "god's creation" and so on...

03-20-2003, 10:08 PM	#4
AdamWho Contributor Join Date: May 2001 Location: San Jose, CA Posts: 13,389	The levels of infinity are well defined in math and are not really a subject of philosophical debate; they are definitions Countable infinity means that you can put what ever you are counting into a 1 to 1 relationship with the positive integers. Uncountable infinity means that you can't. As far as the Cantor claim of "absolute infinity", didn't Russel take care of this claim when he showed that the set of all sets cannot contain itself?

03-24-2003, 03:27 AM	#6
Big Spoon Junior Member Join Date: Feb 2003 Location: UK Posts: 74	Another way of defining countable infinity is whether or not you can count the members of a set in a systematic way. ie. the naturals, integers and rationals are countable; the real numbers are not. Nowhere357, can you describe a number, say 3, without any kind of self-reference? I doubt that you can because numbers themselves are abstract. The aim of pure mathematics is not to describe reality but to describe the way in which numbers behave. The infinity of infinities is not some whimsical conjecture that you can discard - it has been proven unequivocally to be true.

03-25-2003, 02:05 AM	#8
Big Spoon Junior Member Join Date: Feb 2003 Location: UK Posts: 74	By the tone of your reply it sounded like you meant that not describing anything physical was a bad thing. I'm sorry if that's not how you meant it! As far as I'm concerned, the more abstract the better.

Freethought & Rationalism Archive

The archives are read only.