Mathematical Challenge for all you geniuses.

First of all, let's go to definitions. According to "The American Heritage School Dictionary" published by Houghton Mifflin 1977 the definition is as follows; in.fi.nite adj. 1. Having no limit in space, extent, number or time; endless: "The universe is usually assumed to be infinite". 2.a. Greater in value than any specified number, however large: "an infinite number". b. Having a measure that is without limit: "an infinite plane". 3. Seemingly without limit: "an infinite variety of colors". --n. 1. An infinite thing or quantity. 2. the infinite. God.

So apparently we have two choices; 1. truly infinite or 2. seemingly infinite. Which one are we using?

If we use 1. then there is no challenge other than in someones understanding of the word "infinite".

If we choose 2. then beauty, as it were, is in the eye of the beholder.

I choose 1.

After reading The Fabric of the Cosmos from Brian R. Greene, my understanding is that the space fabric, our 3 dimension reference scheme is moving in the time dimension and evolving while doing so. What we are experiencing is the universe being dragged by the evolving space fabric. So the universe as a whole may be quite static in space, as space itself is expanding.

Although I read some years ago, I brought it out this weekend to explain my 16 years old son why inertia exists in absence of gravity. I referred him to the Higgs Ocean concept. Can't wait for the Swiss Dark Hole Machine.

And by no means do I think this to be an attempt to be more than a little pertinent to this thread and I will gladly stand corrected by experts on these 4 and other dementias.

Hi gigaconcept.com,

"...the space fabric, our 3 dimension reference scheme is moving in the time dimension and evolving while doing so."

Interesting concept.

"So the universe as a whole may be quite static in space, as space itself is expanding."

You threw me here. Did you mean that the universe is static in a larger space outside the universe? If not, then how can it be static if it is expanding?

regards,

S

The analogies used in this book to explain the space-time and other concepts leading to the string theory are quite visual, and most concepts are as simple as possible to understand. For example he uses images such as a sliced loaf of bread to depict our 4 dimension reality, up to the possibility of space travel, the string theories and on.

A very good reading that you and your avatar should enjoy.

As for space expansion vs the (our) universe (as they may be quite a multitude of universes), imagine a portion of space-time as a piece of Lycra fabric on witch you insert sewing pins.

As you stretch ans distort the fabric, the pins will move in respect to each other, while being static relative to the fabric.

So it may be that space itself is expanding, dragging the universe(s) apart (for now).

Hi Shadetree,

We are using truly infinite. That, I'm sure, is what some cosmologists think the universe is. Einstein and others proposed a finite universe. His General theory of relativity calls for curving space. He proposed a 4-dimensional 'sphere' with, I think, an 18 million light year diameter.

'If we use 1. then there is no challenge other than in someones understanding of the word "infinite".'

I take it then that you think an infinite thing can't expand (by logic), the same as me. But where is your mathematical proof?

regards,

S

Let me refer you back to 2.a. Greater in value than any specified number, however large: "an infinite number".

Since I never did 'get' algebra my mind can only really deal with real numbers. Since there are no real numbers that can be used here I felt safe in venturing into these waters.

To my mind you cant put a mathematical equasion to an unknowable volume. Chances are good that you could never prove it one way or the other, at least not to me.

"To my mind you cant put a mathematical equasion to an unknowable volume. Chances are good that you could never prove it one way or the other, at least not to me."

I'm afraid you're right, but I'm not that mathematical. I read where some people got the Nobel Prize in physics for dividing both sides of an equation by infinity. This problem may be as difficult as Fermat's last theorem or even harder.

S

Infinite is not a human experience, therefore ones comprehension is a comparison of something that may be similar, something that expands, a balloon or explosion, etc. These are a visual concepts, they are not infinite as such, a tendency to humanise things, birth, life, death, an event? So a mathematical representation, so how about space = 1/0, the Egyptians had no symbol for zero, perhaps we have something in common with them? I of course don't know the answer, does space expands, or is it the forces in space that vary, and is the result actual or relative? Sorry I've come up with more questions than answers.

Regards JD.

(or that it can, (infinite universe can expand.)if that is what you believe).

Here goes...

Given: an infinitely large existing object. (x)

Given: something additional (1)

To prove: That absolute value of X can increase.

Let x=x;

Let 1=1;

Then x+1=x+1;Additive principle;

X+1>x;

Therefore, x can be expanded by addition.

Check: Given a series of numbers without limit is defined by n, n+1, n+2...N+x;

For any x Addend to the last n+x there can be an x+1. Axiomatic ; definition of number line)

Final evidence: in the beginning was the Word, and it created woman, and Lord, how many words have been uttered (and muttered) since?

milo

Hi Milo,

What you have proved is that you didn't start out with an infinite number!

regards,

S

Not so fast.

please see the first given.

If he was alive, I'd sic Bertrand Russel on you, and he'd do it with sets.

milo

In what sense is the Universe "infinite?" The Universe is a four-dimensional beast, so when we speak of the Universe being "infinitely large," are we speaking of its spacial extent only, or are we including its time dimension as well? Is the Universe infinite in all four dimensions or in less than four? If four, then the time dimension must also be infinite, which implies, then, that the Universe is also infinitely old. The CMB is a strong argument against an infinite time dimension.

'In what sense is the Universe "infinite?"'

I didn't say that it was.

"If four, then the time dimension must also be infinite, which implies, then, that the Universe is also infinitely old"

I made this point a while back in another thread. Space and time are linked. The Big bang sets a starting point for time, so for people who believe in the BB, thinking the space part is infinite is illogical in my view.

"The CMB is a strong argument against an infinite time dimension."

I'm not so sure. I have been reading in a book with another explanation, but I don't want to discuss it now before my next thread.

S

Go round the back and see if there's room....simple

As I understand it the universe is not infinite. That it is expanding in the infinity of space.

Ever since supercomputers were developed we've been trying to find the last decimal of PI. Until we do, the universe will continue to expand.

Then it all comes back together for another big bang and we do it all over again.

Shades of the nine billion names of G-d!

You have an Artur C. Clarke gene showing there, charsley!

nice insight.

milo

We common people believe that universe is infinite. But scientific theory related to expanding universe say that universe is finite. Thus expansion is perfectly possible.

We common people do not understand, if universe is finite, where the hand will go, if we stretch it, standing on the edge of finite universe. Expanding universe theory says, hand will go in finite universe only.

Difficult to understand!

So no chance of winning Nobel !

If I understood a mathematician friend there are an infinite number of infinities. e.g there is an infinity of prime numbers, an infinity of decimal palces of pi and e, Fibonacci numbers, ordinal numbers, cardinal numbers, on and on.

Since space is curved an object within its boundaries (what boundaries?) can expand to those every increasing limits.

Hi all,

I will not enter on cosmological issues, I've read some books about (even some written by people who wonned a Nobel Prize) but I don't consider myself an expert on that.

Just mathematics: Think about limits. If there were only one "infinite" thinking on it as a "number", unthinkable and difficult to understand, but a number, the limit of (3x² + 2x -1)/(x²- 327x+56,7) when X→∞ need necessarily be 1, because as we postulated there are only "one" infinity. But we (well, at least me) know that the answer is 3.

Then this seems refuse our premise that only one infinity is possible. In resume, why in mathematics when we get a result that is ∞/∞ we say that the solution is undetermined?

Some other examples can be put, but if we can assume there are different infinity "sizes", what is the problem?

Cheers

Hi,

It could be useful to explain why such equation [(3x² + 2x -1)/(x²- 327x+56,7)] is going to a limit close to 3 when x->infinity

(1) (3x² + 2x -1)= x²(3+2/x -1/x²)

and

(2) (x²- 327x+56,7)=x²(1-327/x + 56,7/x²)

and x²/x² =1 if x different from 0

Given that 1/x² ---> 0

implies (1) -> 3 as x--->∞

and (2) ->1 as x--->∞

To come back to Cosmology it should be useful to read the Cantor theory on infinite numbers and then go to physics and black holes (infinite mass and 0 space dimension) to better understand that our universe is still very difficult to put into our equations...

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

Cheers

Hi again. I'm loosing memory... I would bet this morning answered and now my answer is missing. Who's deleting my posts?

Well, I know the George Cantor theory about transfinite numbers. I've put this example just trying to simplify.

According to Cantor theories, there are different "infinity sizes", in fact he said there are infinite infinities. For example: the real numbers set is infinite but greater than integer numbers set.

The limit of the example should be easily solved by any engineer which before going to university took math classes and hear about L'Hôpital rule (proposed by this wise man some centuries ago). I put the example to illustrate that quotient between two infinite numbers can be any number from -∞ to +∞ and that this can be interpreted as there is many "infinity sizes".

Coming back to Cantor, remember the "Hotel paradox" in which an hotel with infinite rooms and no vacancies can admit more clients.

So, mathematically speaking, that was the title of original thread post: Taking the Cantor hotel paradox, what can be said against the (math) possibility of universe to be infinite (like hotel) and expand (like the hotel) and being still infinite?

To prove that it did expand, is to prove that it is not infinite.

Here's an idea:

You are here, living your life. You think.

Now imagine that there is another "universe" or "cosmos" which exists in the same space that this universe exists. Except for one concept, which has not been discovered, or "realized."

The other "universe" operates at a different "frequency" than 'our universe.'

Now this frequency might be the frequency of gravity waves. Or it might be the frequency of some other not-yet explained force.

But back to the "other universe." IF this were possible, and given the fact that every THING (matter) makes up very little of any universe, it would be possible for these two "universes" to co-exist IN THE SAME SPACE, because they would not interact with one another. Each universe would retain it's own solidity, yet co-exist within the same space.

NOW, once you imagine this, imagine that there are AN INFINITE NUMBER OF UNIVERSES co-existing inside of each other right now!

Which means that the infiniteness of the universe may not necessarily be described in length x width x height x time.

Just thinking . . . . .

Thats right. Where is it written that the inside of something has to be smaller than the outside?

As evidence, I offer a woman's handbag. There's a heck of alot more stuff in it than it can hold, it seems...

And as all of us wh have done the go halfway on the yard stick trick knows, there is an infinite number of points between two points... its not just about expansion outwardly, its inwardly as well.

Nice insight

milo

Well we're not getting much in mathematical proofs, but the discussion is interesting nevertheless. The Big Bang theory has the possibility of a finite universe in space, but time is an unknown. However, the start of the universe in the BB theory is from a singularity, infinitely small in size and infinitely dense. That is two infinities, so we don't have a theory that doesn't involve infinity!

Infinity is just an unknown or (currently) unmeasurable value.

When people talk of "thinking out of the box", I prefer to think of it as expanding or redefining the box - none of us have the same size or shape of box in which we reside.

The universe is similar - the boundaries we can see are ever increasing, always being redefined as more (infinite?) data is revealed and processed.

Has anyone else noticed the similarities between the solar system and an atom? Was the Big Bang an explosion, and we are standing on a single electron? For all we know, there could be another universe heading in the opposite direction, but travelling at such a speed that we will never notice until we are gone.

What would you see if you were standing on an electron of a Fluorine atom?

"What would you see if you were standing on an electron of a Fluorine atom?"

Depends on which electron, I suppose. You might see the nucleus, another electron (or more), or maybe just the far beyond. The universe in a grain of sand is a topic explored by poets, songwriters, and science fiction authors. Not being a math genius myself, either, I will confine my commentary to prose. Or perhaps poetry...I have been known to rhyme a line or two occasionally.

And would the nucleus look anything like a star??

I have seen much reported about the relative sizes of the nucleus and electrons, but I am unaware of whether there is any proof that all electrons are the same size.

Am I in danger of being infinitely off topic?

"...whether there is any proof that all electrons are the same size"

Or having a surface and being a solid particle.

"What would you see if you were standing on an electron of a Fluorine atom?"

If you look away it will disappear according to Niels Bohr, so you will fall all the way to the turtle's back.

Well, the topmost turtle, anyway... Remember, it's turtles all the way down!

If the univers is infinitly long then it can expand width wise.