work that out!!!

]]>10!!!...!!! ((10^^^^^^^^^^1000000)!!!...!!! (10! factorials) factorials)

Actually, this might be smaller than G, but it's still enormous.

]]>G1=3^^^^3

G2=3^^...^^3 (G1 up-arrows)

...

G64=3^^...^^3 (G63 up-arrows)

Knowing this, you can say for certain that the number I posted just above is bigger than G.

]]>x1=10^^^^^^^^^^10

x2=10^^^...^^^^^^x1 (x1 up-arrows)

...

x(x1)=10^^^...^^^^^^x(x1-1) (x1-1 up-arrows)

Here's a number definitely larger than G: 100->100->100->100->100->100->100.

]]>Remember, Graham's number grows very rapidly as we move from the first step onwards.

There is reason for me stating this. I had once come up with a number I thought was probably greater than Graham's Number, but the number was more comparable to Moser's. In fact, the number might be lesser than Moser's.

The number I had in my mind was something like

(10^10^10)^^^^^(10^10^10).]]>

Here is a big number:

x1=10!!!!!!!!!!! (10 factorials)

x2=10!!!!!!!!!!! (x1! factorials)

x3=10!!!!!!!!!!! (x2! factorials)

...

x(10^x1)=10!!!!!!!!!! (x((10^x1)-1)! factorials)

The last step is probably larger than graham's number, being the number of iterations.

]]>no element of A is infinite.

Infinite has a whole bunch of different meanings. For example, all elements of A are infinite sets (every real number is). But their magnitude is of course finite.

But I got to say I'm not quite sure where you're going with this.

But by the definition of R (well, it's not really a definition, it's a continuity axiom), the cardinality of [a, b] is infinite and uncountably infinite to boot.

Completeness gives R is uncountable-ness, and it can be proven using the construction of R from the rationals. I looked up continuity axiom and it seems to do with Euclidean space and circles, or Archimedes and the rationals.

]]>You know you were projecting when you said, "Surely you don't need me to show you, do you?", and you know it.

I might if I knew what "projecting" means. Obviously, by your tone, it's a negative thing, but in fact, my comment was intended as flattery - as in, of course you know Cantor's thm.

Anyway, you got me thinking. Ricky, I know you don't need this little tutorial, but here goes, for general consumption.

There is a load of laxity in language regarding "infinity". Consider the subset A = [a, b] of R. By the definition of [ , ], no element of A is infinite. But by the definition of R (well, it's not really a definition, it's a continuity axiom), the *cardinality* of [a, b] is infinite and uncountably infinite to boot.

See what I mean about laxity?

]]>picoseconds??

No, something more like an attosecond which is 10^-18 seconds. To give you an idea, an attometer is roughly the size of a quark...

Here are the prefixes for (relatively small) numbers from 10^18 to 10^-18:

Exa, Peta, Tera, Giga, Mega, Kilo.

Milli, Micro, Nano, Pico, Femto, Atto.

Here are some other ways to compare these:

Comparing a meter to a nanometer is like comparing the circumference of the Earth to that of a marble.

An atom is roughly 10 to 80 nanometers in diameter.

A picosecond is roughly the time between you hitting a key on your keyboard (contact being made) and the computer receiving the signal.

An exameter is over 300000000 times the distance between the Earth and the moon.

Oh yes, and leave of the "your perch" bit if you don't mind - I am tolerant, but not infinitely so

You know you were projecting when you said, "Surely you don't need me to show you, do you?", and you know it.

No, this is wrong. Why do you think this? All finite sets are countable, almost by definition. It's true that a set is said to be countable if there is a bijection to a subset of N, and as N is "countably infinite" again by definition, and as N is always a subset of N, the bijection you refer to may or may not imply a set is countably infinite, it could easily be finite (we don't need the countably bit for finite sets). But it is most certainly not the case that countable implies infinite.

There are two different ways to define countable. I choose the way I've used because it explicitly identifies that finite sets are of a different cardinality than the natural numbers, which I like. After doing some searching, it does seem though that your definition is vastly more common. The book I first got my version from came out of India. Not sure if that has something to do with it.

]]>