Two kinds of infinity- how can we understand them?

[MCdread]

An interesting idea you might want to think about is that of a fractal object: having a perimeter of infinite length, but a finite area, such as the Koch Star. This is different from your line or Zeno's distance, whose length is nevertheless finite even though it is a sum of infinitesimal components.
Another twist to confuse the troubled soul of your story is that the curve is continuous but not differentiable (you can't draw a tangent to any point). In other words, the line will never look straight no matter how much you zoom into it.
I think the discovery of such a line would be a seminal moment in your character's life, the equivalent of seeing God in his/her world.

 

[Kyriakos]

Thank you all again for the comments

McDread: can you elaborate more on the god part? I was thinking of including a similar sentiment, since people with autistic worries do tend to have complicated ideas about a deity, if for no other reason then because they cannot stand not being able to calculate something, and a god is by definition incalculable and therefore they have a systemic allowance to fail to calculate whatever they project onto the god-idea.

Finite and infinite appear to me to be at the very core of one's ability to form thoughts. I view them as linked to an equation of synthesis and analysis. I have not yet decided how old the narrator of the story (the friend of the kid who has the problem) would be. This will decide a lot of things on the structure of the story, since if it is an adult i can have as complicated thoughts as i want to, but if it is another child then i have to limit the scope of the thoughts somewhat.

 

[LulThyme]

For correctness: The question, whether a set exists with more elements then Aleph_0 but less then Aleph_1 is undecidable within Zermelo-Fraenkel-Choice.

No, Aleph_1 is the next cardinal after Aleph_0. (I think you need the axiom of choice for this to be true strictly, but some version of it is true in ZF.)
What IS undecidable is whether the continuum (which is also 2^(Aleph_0)) is equal to Aleph_1, in other words, if there is a cardinal between Aleph_0 and 2^(Aleph_0)=c. This is called the continuum hypothesis.

 

[Zelig]

Take some math classes.

 

[warpus]

I'm not sure if anyone's pointed this out yet or not, but in mathematics you can divide infinite numbers into 2 categories - countable or uncountable. That is a far more mathematically intuitive way of categorizing infinite numbers than the one you propose in the OP kyriakos.

The set of real numbers is uncountably inifinite, while the set of integers is countably infinite. I'm too tired to explain the difference any better than that

 

[red_elk]

The set of real numbers is uncountably inifinite, while the set of integers is countably infinite. I'm too tired to explain the difference any better than that

I was going to write something similar, but several people have already written about aleph-0, aleph-1 and continuum. Nothing to add.

 

[Save_Ferris]

The number of atoms, their isotopes and their compounds is unknown, but it is probably not infinite. Let's call this number x.

Given infinite dimensions, the total of all space is x*Infinity.
Given that each dimension has infinite values, the total of all space is x*Infinity^Infinity.

 

[uppi]

The number of atoms, their isotopes and their compounds is unknown, but it is probably not infinite. Let's call this number x.

Given infinite dimensions, the total of all space is x*Infinity.
Given that each dimension has infinite values, the total of all space is x*Infinity^Infinity.

You cannot just assert that there are infinite dimensions, nor that these somehow have infinite "values".

 

[Zelig]

You cannot just assert that there are infinite dimensions, nor that these somehow have infinite "values".

He just did.

 

[Thorgalaeg]

Well, about the real existence of inifinite, as long as the universe is finite (if it indeed is as we are not sure about it) everything countable and contained in it would be finite. So aleph-null would be not real. However we dont know if everything is countable. For instance take space itself (and therefore time). Is there a minimal portion of space or is it a continuous and divisible in an infinite number of infinitely small portions and there are infinite possible locations? According to quantum mechanics it is discrete too, Plank length being the minimal distance beyond which it does not make sense to speak about space itself. So apparently any infinity has not existence in the real world, only in mathematical world. Of course we could discuss now how real is indeed mathematical world...

 

[red_elk]

Well, about the real existence of inifinite, as long as the universe is finite (if it indeed is as we are not sure about it) everything countable and contained in it would be finite. So aleph-null would be not real. However we dont know if everything is countable. For instance take space itself (and therefore time). Is there a minimal portion of space or is it a continuous and divisible in an infinite number of infinitely small portions and there are infinite possible locations? According to quantum mechanics it is discrete too, Plank length being the minimal distance beyond which it does not make sense to speak about space itself. So apparently any infinity has not existence in the real world, only in mathematical world. Of course we could discuss now how real is indeed mathematical world...

Mathematical abstractions are not physical objects and they don't exist in physical world. Not only aleph-null, but also line, triangle, sinusoid, etc. don't exist in the same sense as "moon" or "water" do.

 

[Thorgalaeg]

Mathematical abstractions are not physical objects and they don't exist in physical world. Not only aleph-null, but also line, triangle, sinusoid, etc. don't exist in the same sense as "moon" or "water" do.

Yep, it seems something obvious, however some guys as Platon and some modern scientists and mathematics think that mathematical objects have existence by itself and therefore are as real as our "real" universe or more. There is even this guy Tagmark that developed a multiple universe theory who thinks that mathematical structures are the only reality, so some sort of modern platonism.

 

[Save_Ferris]

So what if the universe is finite if there are infinite universes? I'd assume that across infinite universes there are infinite dimensions.

 

[Kyriakos]

Another question:

If something is infinite, does it have to include all possible variations of everything?

I ask this because, for example, there are infinite natural numbers, but there are also infinite numbers between any two natural numbers. (?)

So, if the universe is infinite (making an assumption) does it have to include at any one time all variations of all phenomena in existence? Or does it merely have to include an infinite number of phenomena, but not the absolute infinity that would have all of their variations?

An example: Lets say there exists a book that has 10.000 words. 9,999 of them are the three letters "iii". The last is the word "human". Would an infinite universe contain all variations of the book, from a complete 10.000 "iii" to every position for the word "human"? Or would it merely need to have a finite number of chance variations, along with an infinite other number of different books?

 

[Valka D'Ur]

So, if the universe is infinite (making an assumption) does it have to include at any one time all variations of all phenomena in existence?

I should think the answer would have to be "yes"; otherwise, it's not infinite. The moment you put any sort of limit on a quantity of something, it becomes finite.

 

[Thorgalaeg]

So what if the universe is finite if there are infinite universes? I'd assume that across infinite universes there are infinite dimensions.

It depends of which sort of multiple universes we are speaking about, as there are several possible kinds of "multiverses". Our own common universe being infinite impplies an infinite number of sub-universal spheres isolated one from others by the speed of light limit called Hubble spheres which would be universes per se, but all of the same dimenssions as our own Hubble sphere of course.

There are other possible multiverse models, though.

Another question:

If something is infinite, does it have to include all possible variations of everything?

I ask this because, for example, there are infinite natural numbers, but there are also infinite numbers between any two natural numbers. (?)

So, if the universe is infinite (making an assumption) does it have to include at any one time all variations of all phenomena in existence? Or does it merely have to include an infinite number of phenomena, but not the absolute infinity that would have all of their variations?

An example: Lets say there exists a book that has 10.000 words. 9,999 of them are the three letters "iii". The last is the word "human". Would an infinite universe contain all variations of the book, from a complete 10.000 "iii" to every position for the word "human"? Or would it merely need to have a finite number of chance variations, along with an infinite other number of different books?

It is not only possible but unavoidable. And not only any variation but inifinite copies of every variation. In fact it can be calculated that at 10^10^28 meters away from here there is other Kyriakos writing that same post.

BTW have you read Borges's "The Library of Babel"?

 

[Thorgalaeg]

Duoble post

 

[Kyriakos]

Yes i have, it was a clear influence on my question

But would there be an indentical other me, writing the identical post, on an identical internet, or would there just be a sole existence for anything in all its specific parameters, and then infinite variations of them, including (or not) a periodical perfect copy?

 

[Thorgalaeg]

In an infinite universe there would be infinite copies of our own Hubble Sphere and all its possible variations. It does not make much sense to ask which would be the "original" copy, if i understand your post right.

 

[red_elk]

If something is infinite, does it have to include all possible variations of everything?
...
I ask this because, for example, there are infinite natural numbers, but there are also infinite numbers between any two natural numbers. (?)

Set of natural numbers is infinite (but countable). There are no natural numbers between two members of it which differs by 1. Set is infinite, but it doesn't include all possible variations of everything - for example it doesn't include 1/2.