Cantor, infinity and cosmology

[dkotschessaa]

I should start a new thread for my questions rather than hijack others...

Posted this on another thread but it didn't get any response, so bear with me if you've seen it. Has anybody looked into Cantor's works on infinity and seen how they relate to the question of an infinite universe?

In short, and crudely stated, Cantor proved that there wasn't just one "infinity" but different degrees of infinity, i.e. the set of all real numbers is larger than the set of natural numbers, though both are infinite. Does this have any bearing on cosmological questions of an infinite universe and on singularities? I know a guy who claims it does, but he's not a mathematician or physicist and mostly just some whack job, but I think it's an interesting question. (his claim is that infinity can't exist because Cantor said it can't. I don't think that was Cantor's conclusion).

-DaveK

 

[Hurkyl]

Cantor's work on set theory showed that there is more than cardinal number of infinite size.

The $+\infty$ and $-\infty$ symbols that appear in real analysis are extended real numbers; they have nothing to do with the notion of cardinality.

 

[dkotschessaa]

Meaning that there is really no relation to anything in physics?

(Sorry if that's a stupid question, but I need to be able to explain it to someone else in terms we both understand).

-DaveK

 

[mathman]

Meaning that there is really no relation to anything in physics?

(Sorry if that's a stupid question, but I need to be able to explain it to someone else in terms we both understand).

-DaveK

Cantor's work on cardinality, etc. is pure mathematics. It has nothing to do with physics.

 

[dkotschessaa]

Cantor's work on cardinality, etc. is pure mathematics. It has nothing to do with physics.

That's how I understand it. The quack who I let pull me into arguments on the internet reasons thusly: Cantor proved that infinity is impossible, therefore there is nothing infinite in the universe. While the second part may be true, I do not believe it follows from the first.

Nonetheless it has made me interested in Cantor's work and I'm interested in pursuing it further. Is there a way to read Cantor's original papers in English? I'm having a hard time locating them and my German isn't what it's going to be. (Will start on that next year...)

-DaveK

 

[ImaLooser]

I should start a new thread for my questions rather than hijack others...

Posted this on another thread but it didn't get any response, so bear with me if you've seen it. Has anybody looked into Cantor's works on infinity and seen how they relate to the question of an infinite universe?

In short, and crudely stated, Cantor proved that there wasn't just one "infinity" but different degrees of infinity, i.e. the set of all real numbers is larger than the set of natural numbers, though both are infinite. Does this have any bearing on cosmological questions of an infinite universe and on singularities? I know a guy who claims it does, but he's not a mathematician or physicist and mostly just some whack job, but I think it's an interesting question. (his claim is that infinity can't exist because Cantor said it can't. I don't think that was Cantor's conclusion).

-DaveK

The infinities Cantor was talking about were integers and real numbers. These are completely imaginary things.

I think that no infinity can physically exist in our visible universe. However our universe is much larger than the physical universe, and astronomers think that it is infinite. So then an infinity physically exists.

 

[twofish-quant]

Posted this on another thread but it didn't get any response, so bear with me if you've seen it. Has anybody looked into Cantor's works on infinity and seen how they relate to the question of an infinite universe?

As far as we can tell. They don't. Cantor was talking about math. Cosmology is about physics.

Does this have any bearing on cosmological questions of an infinite universe and on singularities?

As far as we can tell. No, it doesn't.

 

[twofish-quant]

Meaning that there is really no relation to anything in physics?

(Sorry if that's a stupid question, but I need to be able to explain it to someone else in terms we both understand).

Mathematicians create tools for physicists. So a physicist sees something that you want to describe, and you go into your toolbox and pull out some tool that a mathematician has come up with.

Now, Cantor created some very interesting tools, but so far, no one has needed them. You look in your toolbox, and you see a hammer, a screwdriver, and some green round plastic thing with blinking lights that's cool to look at, but doesn't seem to be useful for anything you can think of.

Cantor proved that infinity is impossible, therefore there is nothing infinite in the universe. While the second part may be true, I do not believe it follows from the first.

The first is false. Cantor proved that there were different levels of infinities.

 

[dkotschessaa]

I agree. Just arming myself against quackjobs.

So, if I can ask again - does anybody know where I can I possibly read Cantor's papers in English?

-Dave K

 

[dkotschessaa]

Mathematicians create tools for physicists. So a physicist sees something that you want to describe, and you go into your toolbox and pull out some tool that a mathematician has come up with.

This is actually one of the reasons I'm asking. I thought perhaps that physicists might look at Cantor's work or work derived from it as a tool for dealing with some of the extremes found in cosmology, like dealing with singularities. Perhaps this is naive as I still have a ways to go before I even get to Cantor in my own studies.

It's just that when I hear physicists talk about the math that we "don't have yet" to deal with things like black holes, I often wonder what that math would have to look like.

-DaveK

 

[mathman]

This is actually one of the reasons I'm asking. I thought perhaps that physicists might look at Cantor's work or work derived from it as a tool for dealing with some of the extremes found in cosmology, like dealing with singularities. Perhaps this is naive as I still have a ways to go before I even get to Cantor in my own studies.

It's just that when I hear physicists talk about the math that we "don't have yet" to deal with things like black holes, I often wonder what that math would have to look like.

-DaveK

I suggest you stop looking for any connection between Cantor's work and physics. You could get a lot of material about Cantor's work from the internet - Google "Cardinality Cantor".

The black hole problem for physicists is that General Relativity and Quantum theory both apply, but the results from trying to do both at the same time lead to nonsense.

 

[dkotschessaa]

I suggest you stop looking for any connection between Cantor's work and physics.

You are probably right about that.

You could get a lot of material about Cantor's work from the internet - Google "Cardinality Cantor".

I've found plenty *about* his work, but unfortunately I've been unable to find his original work anywhere.

-Dave K

 

[mathman]

You are probably right about that.



I've found plenty *about* his work, but unfortunately I've been unable to find his original work anywhere.

-Dave K

I am curious as to why you feel you need to see Cantor's original work (you probably would have to visit a major University Library or the Library of Congress(?) to see his original papers). The content is readily available from Wikipedia and other sources.

 

[dkotschessaa]

I am curious as to why you feel you need to see Cantor's original work (you probably would have to visit a major University Library or the Library of Congress(?) to see his original papers). The content is readily available from Wikipedia and other sources.

I prefer original sources to peoples regurgitations.

-DaveK

 

[dkotschessaa]

Why is that even surprising?

 

[mathman]

Why is that even surprising?

There are many math textbooks which cover Cantor's works well and probably are a lot clearer than the original. Going back to the original seems to me a task for someone interested in the history of the subject, but (to me at least) a lot of unnecessary work to learn the material.

 

[ImaLooser]

There are many math textbooks which cover Cantor's works well and probably are a lot clearer than the original. Going back to the original seems to me a task for someone interested in the history of the subject, but (to me at least) a lot of unnecessary work to learn the material.

I have read Cantor's original paper and found it quite clearly written. I've also read original papers by Godel and Einstein and found them superior to any explanation.

I read Cantor at least 20 years ago in a book that was a collection of famous papers. I'm surprised its not available online.

 

[Chronos]

The only collection of original papers by Cantor I know of is in a book by Ewald.

 

[dkotschessaa]

There are many math textbooks which cover Cantor's works well and probably are a lot clearer than the original. Going back to the original seems to me a task for someone interested in the history of the subject, but (to me at least) a lot of unnecessary work to learn the material.

I'm coming from a background of disciplines other than mathematics where relying on second hand sources of information is considered irresponsible. So it's weird for me to hear someone say "Just read it somewhere else." Perhaps in most math, because of it's rigorous nature, this is not as much of a problem.

Of course, I think for most mathematics I wouldn't be as eager, but Cantor's work seems especially prone to misinterpretation, including a lot of woo-woo philosophical ideas (usually by amateurs and non mathematicians).

-DaveK

 

[twofish-quant]

This is actually one of the reasons I'm asking. I thought perhaps that physicists might look at Cantor's work or work derived from it as a tool for dealing with some of the extremes found in cosmology, like dealing with singularities.

People have looked at it. It turns out not to be useful. One problem is that even when dealing with real numbers you are already using an approximation to reality.

It's just that when I hear physicists talk about the math that we "don't have yet" to deal with things like black holes, I often wonder what that math would have to look like.

The problem isn't the math. The problem is that we don't have observations to let us know what happens.

 

[twofish-quant]

I'm coming from a background of disciplines other than mathematics where relying on second hand sources of information is considered irresponsible. So it's weird for me to hear someone say "Just read it somewhere else." Perhaps in most math, because of it's rigorous nature, this is not as much of a problem.

In history, you have to go to primary sources because a retelling of a story is different from the original. This isn't the situation in math or physics. I can explain Newtonian mechanics or Cantor's infinity in ways that are different than the original, but I'm still talking about the same thing and explaining it in a way that is better.

This makes math something that works very will with wikipedia.

Cantor's work seems especially prone to misinterpretation, including a lot of woo-woo philosophical ideas (usually by amateurs and non mathematicians).

If you go to a summary of Cantor's work then sure. If you go to the original proofs, and the proofs are not difficult and can be understood by most high school students, then there's no possibility for misinterpretation.

http://en.wikipedia.org/wiki/Cardinality

 

[Chronos]

Cantor also had a metaphysical bent, which is allegedly apparent in his papers. This may explain why reception of his work during his lifetime was rather tepid.

 

[dkotschessaa]

Cantor also had a metaphysical bent, which is allegedly apparent in his papers. This may explain why reception of his work during his lifetime was rather tepid.

And just to reiterate, I am involved with one of those "There's someone wrong on the internet" situations where someone is claiming an particular interpretation of Cantor's work that I believe (well, that we know) is false. Usually in such a situation I find it best to quote original work. Despite the fact that the person that I'm arguing with is a bit of a nut, it did get me looking into Cantor, so I have him to thank for that.

-DaveK

 

[mathman]

General comment: Cantor's work is pure mathematics - the discussion makes more sense in one of the math forums.

 

[ImaLooser]

Cantor also had a metaphysical bent, which is allegedly apparent in his papers. This may explain why reception of his work during his lifetime was rather tepid.

I read the Kronecker disliked Cantor's work. Kronecker was quite influential at the time. I get the impression that K did not much care for the real numbers.