How large is infinite?

kgun · #1 (**permalink**) 03-05-2008, 06:20 AM

Don't ask me. But I know it is of utmost importance in mathematics.

Related links:

Graham's number - Wikipedia, the free encyclopedia

If you follow the links in the following:

Theory of Immensity - The Stripy Strudel's Journal - by Alexey Feldgendler

article, you soon end up in Russia.

P.S.
Here

Prototype JavaScript framework: Prototype 1.6.0.2: Bug fixes, performance improvements, and security

is how I found the last article. It is about Prototype 1.6.0.2: Bug fixes, performance improvements, and security .

Prototype is a JavaScript Framework that aims to ease development of dynamic web applications.

Dan Webb is a London-based web developer is one guru user. Among other things, he has written sumo a gemeric microformats parser for JavaScript.

There is always a side note.

This thread is aimed at being a philosophical and number theoretic thread.

jabo · #2 (**permalink**) 03-06-2008, 03:00 AM

can it be measured???
how can you say the size of infinite if it cannot be measured???

kgun · #3 (**permalink**) 03-06-2008, 08:13 AM

Interesting that you answered. You get one positive rep point for that. I did not expect an answer.

The concept of cardinality

"In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {1, 2, 3} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers".

and cardinal number is important.

"In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. For finite sets the cardinality is given by a natural number, being simply the number of elements in the set. There are also transfinite cardinal numbers to describe the sizes of infinite sets. On one hand, a proper subset A of an infinite set S may have the same cardinality as S. On the other hand, perhaps also counterintuitively, not all infinite sets have the same cardinality. There is a formal characterization that explains how some infinite sets have cardinalities that are strictly smaller than other infinite sets".

The cardinal numbers are:

For example the simple Cantor Set is huge.

"The key observation here is that we can specify a mapping, or correspondence, from onto [0,1]. Since is a subset of [0,1], this means that [0,1] and must be the same "size" (or have the same cardinality) -- but let's save the actual details for another day.
Is this cool or what? A set so small that we couldn't possibly draw it on paper (or, in this case, on a computer screen) is still, somehow, as big as an entire interval.
Here's that "big versus small" deal again: The Cantor set is apparently "big", since it has the same cardinality as the interval [0,1] (an uncountable set). If you stick around to read yet more on the Cantor set, you would see that the Cantor set is pretty darn small, too. Of course, if you're tiring of magic and wonder, you could always wimp out".

This is pure mathematics: Very interesting, but my life is too short to study this topic in any detail.

Tubby · #4 (**permalink**) 03-06-2008, 08:43 AM

I do not think infinite actually exists. Infinite is merely a tool to say, 'We do not know'. In math it commonly takes the place for "I do not Know" and there are very many mathematical variations on infinity. 'I do not know' is not as precise as 'infinite' (not that 'infinite' is precise)

One Scientist will tell you today that the universe is Infinite. then the following breath tell you it gets bigger every day.
Infinite - Not a finite thing. If the universe is infinite today' then yesterday it must have at least been somewhat less than infinite. From some point searching for infinity becomes as a religion, some do not believe in it, some do. If you are inclined to a religious belief the acceptance of infinity seems totally acceptable.

I think, size than we could not possibly imagine is a probably a good way to describe it. Or simply look at its varied use in maths. But does it exist. . I personally would rather be dead than live in an infinite place.

The Question is confusing. 'How large is infinite?' It is not big, or small. it is not a finite thing.

Infinity has no boundary or limitations - it is merely a concept that we use as a tool we vary the concept to suit the job at hand. . . infinity has infinite variety, not even bordered by our own imagination..

If you were to define a size for infinity. I could merely imagine it to be bigger and you would simply be wrong... such is the nature of a concept...

**mjtaylor** · #5 (**permalink**) 03-06-2008, 09:33 AM

I'm with Tubby. I don't believe infinity can be measured, as it is boundless and that means without boundaries that can be measured ...

So, I don't perceive 'infinity' as a mathematical concept, since it can't be quantified ... nor does it seem to me to be simply what we don't know. Rather I see it as a metaphysical concept that allows us to label the sheer, unutterable limitlessness of all that is (and isn't?). There is no end in either direction. You can neither reach the end nor hit the wall.

It is a difficult concept to comprehend, since our minds *seem* to have limits, but it is an inherently freeing concept, because once one gets even the smallest notion of what infinity means, one is led to understand that there are always an infinite number of possibilities, and that makes it possible for one to create whatever one desires.

AISI, MJ

kgun · #6 (**permalink**) 03-06-2008, 09:59 AM

Quote:

Originally Posted by Tubby

If the universe is infinite today' then yesterday it must have at least been somewhat less than infinite.

In mathematical set theory that is wrong. You may go from one set with an infinite number of elements to another with an infinite numbers of elements, even if the first set is a true subset of the first.

There are infinite number of natural (also called counting) numbers N

0 ),1,2,3,....

"Mathematicians use N or (an N in blackboard bold, displayed as ℕ in Unicode) to refer to the set of all natural numbers. This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-nullhttp://en.wikipedia.org/wiki/Aleph_number#Aleph-null ()".

My red colouring.

There are infinite number of rational numbers Q: n/m where n and m are positive or negative whole numbers.

There are infinite number of irrational numbers I like SQR(2).

There are infinite number of real numbers R = I U(nion) Q

Example:
N subset of Q that is a subset of R.

There are an over countalble number of elements in the Cantor set mentioned in the above post. That is the Cantor Set is non countable.

You have to switch you brain to the mathematical part to understand this. Accept the building blocks, axioms and definitions. Doubt the definitions, and you may get a new (most probably inconsistent) theory.

The obeservable universe is 30 billion lightyears and that is definitely finite, though. So it may depend on the elements you are counting in your set or space.

Quote:

Originally Posted by Tubby

If the universe is infinite today' then yesterday it must have at least been somewhat less than infinite.

If you were to define a size for infinity. I could merely imagine it to be bigger and you would simply be wrong... such is the nature of a concept...

So you did not read about cardinality above?

kgun · #7 (**permalink**) 03-06-2008, 10:11 AM

Quote:

Originally Posted by mjtaylor

I'm with Tubby. I don't believe infinity can be measured, as it is boundless and that means without boundaries that can be measured ...

So, I don't perceive 'infinity' as a mathematical concept, since it can't be quantified ... nor does it seem to me to be simply what we don't know. Rather I see it as a metaphysical concept that allows us to label the sheer, unutterable limitlessness of all that is (and isn't?). There is no end in either direction. You can neither reach the end nor hit the wall.

It is a difficult concept to comprehend, since our minds *seem* to have limits, but it is an inherently freeing concept, because once one gets even the smallest notion of what infinity means, one is led to understand that there are always an infinite number of possibilities, and that makes it possible for one to create whatever one desires.

AISI, MJ

Great that you quote that thread.

If you stand 1/2 m from the brick wall and you half the distance a finite number of times ( (1/2)^n where n is finite), will you hit the wall?
If you kick your foot against the wall hard enough, will it hurt? If it hurts, you have just proved that infintite is a very real concept. You have halved the distance an infinite number of times since: 1/2 x 1/2 x ...... x 1/2 infinite times = 0. (Your foot hit the wall).
More precisely: lim (n --> infinity) (1/2)^n = 0

Jean-Luc · #8 (**permalink**) 03-06-2008, 11:00 AM

Quote:

Originally Posted by mjtaylor

So, I don't perceive 'infinity' as a mathematical concept, since it can't be quantified ... nor does it seem to me to be simply what we don't know. Rather I see it as a metaphysical concept that allows us to label the sheer, unutterable limitlessness of all that is (and isn't?).

Physics, biology and chemistry are sciences. Mathematics is not a science as its theory is not the result of scientific experimentation. Mathematics is closer to philosophy than to sciences. You could even say that mathematics is a branch of philosophy,.. like metaphysics.

Jean-Luc

Tubby · #9 (**permalink**) 03-06-2008, 06:38 PM

Kgun. said "In mathematical set theory that is wrong. "

Proving one concept. by conformation with another concept ? ? !

Cardinality? . Show me a cardinality, a "real" One!

cw1865 · #10 (**permalink**) 03-06-2008, 07:00 PM

Quote:

Originally Posted by kgun

This is pure mathematics: Very interesting, but my life is too short to study this topic in any detail.

Hmmm, the preceding notwithstanding

Tubby · #11 (**permalink**) 03-06-2008, 07:34 PM

in maths you can create a half, you can divide anything in two. I suspect there is an infinite number of halves.

The first thing a mathematician does (to avoid confusion) is to De-infinitize the half. he creates a boundary -
he starts the half calculation. farmer got 5 cows he sells 1/2 he got 2.5 cows left end of calculation - end of use for that half. the Half 'use' has a start an and end. . It is not infinite.

If there is a failure to create a boundary - 1234567 - to infinity - etc. this Failure creates an infinite number. Maths buffs usually create a boundary. Failure to do so. is the cause of confusion. (foot and the wall) deliberately confusing. There is no infinity. . simply an improper use (belief) of the concept. . everything that exists has a boundary. in math you create the boundary of usage, to make 'an infinity' work.

Tubby · #12 (**permalink**) 03-07-2008, 09:53 AM

I suspect the above is heading towards the right answer Kgun.

The size of an infinity is equal to the boundaries applied to it for the purpose its use is intended.

Well that is my answer today. . (might be different next week)

Tubby · #13 (**permalink**) 03-07-2008, 10:11 AM

P.S.
So If someone said to me "In my mind Gods wisdom is infinite" I have a statement I can accept. This infinity has a clear boundary.

Thank you for asking the question Kgun. It helped me clarify a small muddle..

cw1865 · #14 (**permalink**) 03-07-2008, 11:42 AM

Quote:

Originally Posted by Tubby

everything that exists has a boundary

Ah, and every boundary implies something beyond!

**mjtaylor** · #15 (**permalink**) 03-07-2008, 01:51 PM

Quote:

Originally Posted by kgun

Great that you quote that thread.

If you stand 1/2 m from the brick wall and you half the distance a finite number of times ( (1/2)^n where n is finite), will you hit the wall?
If you kick your foot against the wall hard enough, will it hurt? If it hurts, you have just proved that infintite is a very real concept. You have halved the distance an infinite number of times since: 1/2 x 1/2 x ...... x 1/2 infinite times = 0. (Your foot hit the wall).
More precisely: lim (n --> infinity) (1/2)^n = 0

Yeah, well, I am not a physicist, so I am out of my league here, but I do know that if you have a powerful enough tool you would be able to measure space between my toe and the wall even as I say "ouch."

Quote:

everything that exists has a boundary

Quote:

Originally Posted by cw1865

Ah, and every boundary implies something beyond!

Hear, hear!