Exploring Multiple Infinities: Beyond Cantor's Proof

[meemoe_uk]

Is there another convincing way, other than the one orginally used by cantor, of prooving that there exists infinitys greater than alpha zero?

I ask because cantor's proof seems a bit shaky to me, at least the way I've read it. I hear that there is some discontent amongst top maths dudes circles with it as well.

I wonder that the reason why that big conjecture about infinitys existing between alpha zero and one is undecidable is because there is only one infinity, therefore invalidating all the concepts which different size infinitys rely upon.

There maybe some infinite sets which are uncountable, but maybe that doesn`t imply more than one infinity.

 

[moshek]

Organic use to write here quite a lot about this interesting question.
And you can look also at : www.as.huji.ac.il/midrasha04.htm[/URL]

Best
Moshek :smile:

 

[mathman]

There is a simple way to resolve this, using measure theory (a generalization of length).
1) Measure of any countable set is 0.
2) Measure of unit interval is 1.

Therefore no. points in unit interval is not countable (> aleph-0).

 

[master_coda]

Is there another convincing way, other than the one orginally used by cantor, of prooving that there exists infinitys greater than alpha zero?

I ask because cantor's proof seems a bit shaky to me, at least the way I've read it. I hear that there is some discontent amongst top maths dudes circles with it as well.

I wonder that the reason why that big conjecture about infinitys existing between alpha zero and one is undecidable is because there is only one infinity, therefore invalidating all the concepts which different size infinitys rely upon.

There maybe some infinite sets which are uncountable, but maybe that doesn`t imply more than one infinity.

Given the definition of the natural numbers the definition of the real numbers and definition of cardinality, you can prove that the cardinality of the real numbers is greater than the cardinality of the natural numbers. This is not a matter of debate among mathematicians.

There is some mathematicians who study the idea of changing the fundamentals of set theory and/or cardinality but the validity of the current definitions and results produced from them are not controvertial.


I can't comment on the idea of how many infinities there are without knowing what definition of infinity you happen to be using.

 

[Hurkyl]

Assume that N has the same cardinality as 2^N; the set of all functions from the natural numbers into {0, 1}.

That means there is a bijection from N to 2^N. Let's call it f.

Let's define a function, g, by g(n) = 1 - f(n)(n). (For those unfamiliar with this type of thing, allow me to try and clarify; f is a function from N to 2^N, so f(n) is an element of 2^N. Elements of 2^N are functions from N into {0, 1}, so we can evaluate f(n) at some number m. We write this as f(n)(m))

Now, g(n) is a function from N into 2^N, so there must exist an x such that g = f(x). (Because f is a bijection)

However, g(x) != f(x)(x), so g != f(x). This is a contradiction, so our assumption is incorrect.

I wonder that the reason why that big conjecture about infinitys existing between alpha zero and one

Actually, there is no question about this; aleph one is by definition the smallest cardinal greater than aleph 0. The conjecture to which you are referring is the continuum hypothesis: aleph one = c. (c is the cardinality of the real numbers)

There maybe some infinite sets which are uncountable, but maybe that doesn`t imply more than one infinity.

Assuming you mean "There maybe some infinite sets which are uncountable, but maybe that doesn`t imply more than one infinite cardinal number," you are incorrect by the very definition of the terms involved. Two cardinal numbers, by definition, are equal if and only if any two sets they represent have a bijection between them. If an infinite set is uncountable, that means there is no bijection between that set and the natural numbers, thus the cardinality of this set must be different than the cardinality of the natural numbers.



Now there are some esoteric things you can do with logic... for instance, it is possible to arrange things so that you have a "small" set theory and a "big" set theory. While the "small" versions of the natural numbers and real numbers, of course, have no "small" bijection between them, there is a bijection in the "big" theory. So, the "small" real numbers form a countable set in the "big" theory.

 

[matt grime]

and just to add more weight to it, do not follow moshek's link as it is just to another crakpot rant form someone who doesn't understand mathematics. no mathematicians have any problem with this issue. to read more about the thing hurkyl mentions, it's called skolem's paradox.

 

[moshek]

It could be better for your name now, if you were looking on the names that appear in the link i suggest in this thread for meemoe_uk before you wrote what you wrote in this forum!

Here it again just for you Matt:

www.as.huji.ac.il/midrasha04.htm[/URL]

 

[matt grime]

Thank you Moshek, I naively and stupidly assumed you were posting a link to Organic again, a crank in anyone's language, having mixed it up with a reply in another thread. I retract what I said about this link unequivocally.

 

[moshek]

Matt , I am glad you can see your mistakes and also admit with that. Now tell me really and i meen to that! please give me deep answer to why you use not nice word to someone you don’t agree with his attitude to mathematics (Organic).

Moshek

www.gurdjieff-internet.com/books_template.php?authID=121

 

[matt grime]

Look at baez's crackpot index and score any of organic's threads in a serious and impartial manner, read the www.crank.net stuff on maths cranks, appreciate that organic has at no point managed to admit that maths, as has been practised by many people far cleverer than he, has any useful points, nor has he ever admitted he is wrong in any way despite the copious evidence to the contrary. see that at no point has he managed to offer any proofs or evidence supporting his position, that nothing he has written has any practical purpose. that is sufficient proof of crank status to anyone.

 

[moshek]

Matt: Way you choose your name as matt grime ?

"Organic mathematics" will be the name to the Non-Euclidian mathematics that will be declared and accepted during the next 10 years. Very fundamental point will be a new definition to the concept of number as Organic Share also with this forum.

Moshek

www.geocities.com/complementarytheory/CATpage.html