The infinite universe - in both directions?

[AndiFlatland]

Scientists and philosophers frequently claim that the universe is infinite, and I am pretty much in support of that view. The most plausible theory is that it is 'finite but unbounded'.
However, one extrapolation from that theory which I've not heard mentioned is this: That the universe is infinite not only on the very large scale, but possibly down to the smallest, as well. At first sight, this probably sounds daft, but it seems to me, from all the ongoing developments in physics, that the limit has not been reached, and may possibly never be reached. At first, we only believed in the smallest things being what we could see, down to grains of dust. Microscopy showed us far deeper levels of detail, and eventually electron scanning revealed to us the true structure of the atom. So for a while, it was believed that the smallest components of matter had been found, and that was the end of the road. But since then, we have found evidence of particles like quarks, gluons, photons and a whole menagerie of even smaller things. Now we believe, although not conclusively proven, that the lower scale ends with superstrings. How long before those are predicted to be only the next, and not the last stage?
I really do wonder if the universe could be so profoundly unfathomable to our present or foreseeably future technology and understanding, that the levels go all the way to infinity. After all, a universe which is 'infinite' in only one direction is... not really infinite... is it???

[jake-the-peg]

Not really a commonly held view

Most physicists believe the Planck length (http://en.wikipedia.org/wiki/Planck_length) is the smallest length that is meaningfull

But tell me why you think the Universe is finite but unbounded - that's certainly an opinion that was pretty common some years ago but seems distinctly unfasionable now

[Clanad]

"One direction"... Really? And where is that measured from? What's the norm, so to speak?
Fact is, nothing with a beginning can be "infinite" and present theory provides sufficient evidence for a beginning a la "Big Bang" (probably one of the worst misnomers ever foisted on science).

JTP and me always seem to collide (pun only slightly intended) in the area of Planck Scale... but that scale involves not only dimensionality but time as well. Planck length
(1.6 X 10 ^-35 meters) as well as h mass and h time are dependant on at least 1 if not more constants. These constants are the subject of much speculation not unlike Einsteins artificially introduced cosmological constant in his theory of general relativity, only to admit that it was the "worst mistake of his career"... So, time will tell, no?

[jake-the-peg]

Nothing with a begining can be infinite?

Live positive integers?

[R1Geezer]

There is an infinite number of numbers starting at 1 and ending at 2. I think I'd rethink the whole "Beginning" theory if I was you Clanad!

[Clanad]

Aha! So you want to talk philosophy, eh? I'm all for that... First we establish a context and meaning for "infinite" that's not circular, Ok? Most dictionary descriptions of "infinite" or "infinity" include the same words... boundless, endless, eternal. Beyond those few discriptors, the explanation tends towards additional adjectives meaning the same thing.

So, in the context of my example, once a beginning is established, that which has begun is no longer considered infinite, since it has a boundary, that being its beginning (circular, but reasonably fair, would you agree?). Additionally, in our human experience, nothing with a beginning can be endless (accountable to the Laws of Thermodynamics, dontchaknow?).

Now, as to your examples...
Infinity isn’t a number in any conventional sense; you can’t count to infinity. If you could, it wouldn't be, well... infinity. So, first you both are tasked with proving that the repitition of "symbols" (which are what numbers are) does not have an end.

But even more intriguing is the fact that that a number, be it 1,2 or 1700... means nothing unless it counts something. One apple, three oranges.... you get the point. Mathmatician George Cantor has stated "... the most common formal rule for determining if numerical sets have equal numerosity (are numerically equal) is if the elements of one set can be placed in one-to-one correspondence with another set..." Cantor uses this rule to show that when numbers of one set (intergers) are compared to another set (real numbers) it can never be shown of certain that one set is "more infinite" than the other. If a supposed infinity of anything can be compared to the next higher "number" then it, by definition, ceases to be infinite...(Contd.)

[Clanad]

(Contd.)

However, lest we be forced to live with this dichotomy, Cantor (and others) also prove that the set of all "real" numbers (the rationals plus the irrationals) can be placed in one-to-one correspondence with points on a line. "...Thus numbers can represent all possible line segments..." Why is this important, you ask? Simply put, " it links the concept of number with the concept of linear magnitude...", which is the basis or "conceptual foundation" for the calculus... (Source: my well worn copy of "Formal Arithmatic and the Childs Understanding of the Number")

Cantor does differentiate between "uncountable infinity" and "countable infinity", but that's enough for the day... besides the water pump in pasture 8 is broken and the cattle are probably thirsty...

Next?

[jake-the-peg]

Oh I can't let you get away with that Clanad!

Cantor showed that there was more than one infinity and some were larger than others.

The idea that that invalidates the infinity of the smaller is entirely of your own construction I think!

A set is infinite if when you add to it you have the same number

As I pointed out there is an infinity of positive integers - it is an infinite set. It very clearly has a start.

I also fear that your well thumbed maths book is a little too well thumbed - the idea that calculus is underpinned by the notion of the infinitessimal was really making people unhappy in the nineteenth century. It was reformulated at that timed based on the notion of the "limit" which avoids nasty infinitessimals.

- for those who don't follow this imagine I am trying to find the area of a circle.

I approximate it by drawing a polygon around it and joining the corvers to the center to make a number of triangels

Like this:http://www.descarta2d.com/BookHTML/Chapt
ers/HTMLFiles/area_28.gif


Now I calculate the area of one of the triangles and add them up to get an approximate area.

----Now the naughty bit-----

I say lets make the triangles infinitely thin and add them up - presto I get the area of the circle

The basis of calculus

----------- Woah there Cowboy! ------------

Let's just back up there.

These "infinitessimal" triangles - they either have thickness or they do not

If they do they are not an exact equivilent of the circle
If they do not - you can add as many as you like you'll always get 0

This is why calculus had to be "re-engineered"

[R1Geezer]

I'm staying out of this heavy weight bout!

[beso]

I gave up worrying a about infinity when I discovered that mathmaticians had proved that some infinities are larger than others.

[R1Geezer]

primary school children realise that beso!

[mibn2cweus]

To say that anything is infinite is to plead ignorance of its nature. That, my friends, is a contradiction. Sure, you can argue the case for infinity until the cows come home to roost . . . but unless and until one lays an egg you'll be none the wiser.