As a child i was always told infinity was the largest number there was cause it went on forever.
Well thats all fair enough but how do you get there.
and when you reach it how have you truly reached it cause there may be more objects to count after that like grains of sand. do you just keep going 1234 infinity infinity infinity.
so really i guess how do you get there?
Zaphod
Infinity is not the largest number, because it goes on forever.
Infinity is not a number.
Infinity is the name for the concept of “there is no largest number because numbers go on forever.”
You can never get there because there is no “there” there.
well ok then is it possible to count the grains of sand on a beach, i know of the number a googol or a googolplex whatever it is but then how the hell do you get there?
Sure, it’s possible, though I doubt one person could do it on his own in one lifetime.
A googol is 1 x 10[sup]100[/sup] - within an order of magnitude or so of the number of parotons in the universe.
A googolplex is Google[sup]Google[/sup] - really, really freaking huge. But nowhere close to infinite. Just because you can’[t count to it, doesn’t make it an infinity.
Errr…protons.
Ever watch Seasame Street? The Count is your friend.
One! One grain of sand.
Two! Two grains of sand.
Three! Three grains of sand.
…
Nine hundred and eighty four billion! Nine hundred and eighty four billion grains of sand!
(Assuming, of course, that you are counting in increments of one. You could, of course, use pretty much any increment you want.)
Q.E.D. writes:
> A googolplex is Google [to the] Google [power.]
Actually, googleplex (note the spelling) is 10 to the google power.
he he, knew google was a decendant of the name googol, but i also thought google may also be a number.
Interesting that you would choose sand, as did Archimedes: The Sand Reckoner
.
A truly brilliant concept, but not infinity.
[quote]
Zaphod7: …but then how the hell do you get there?
When counting to infinity, I have always found it helpful to count by two’s. It takes half the time…sort of.
From a mathematical perspective, I guess the best way to answer the question is to talk about the ordinal numbers.
(One trick that may help in understanding the following is to think of the process of counting as being rather than doing. If for example, there are a googol elementary particles in our universe, one need not count them individually for it to be so–they just are. If you think of counting as a “doing” process, how far can you count? Well, if you count as fast as you can throughout your entire life, I doubt you’d ever get through the billions. Hell, even if you are immortal, with your birth coinciding with the big bang, and you have counted as fast as you can your entire life up until this point, it’s doubtful you would have managed to get anywhere near even a googol. Large numbers, such as a googol or googolplex, are not something that must be reached, they simply are.)
Anyway, if that parenthetical paragraph made any sense, I guess it’s safe to proceed. First, we have the counting numbers we’re all familiar with: 0, 1, 2, 3, 4,… And that’s all we’ve got of the ordinal numbers, unless we introduce a “new” axiom (actually, it’s not a “new” axiom–it’s been around for years in standard set theory, but for perspective I’m calling it a new axiom)–the Axiom of Infinity. The Axiom of Infinity says, basically, that there exists an infinite set (or, equivalently, that the collection of (finite) ordinals thus far, {0,1,2,3,…} is itself a set).
And now we have the first infinite ordinal, w (to be read as “omega”, but I don’t know how to type Greek letters, so “w” will have to do).
And so now we have a whole bunch of new ordinals. They start as they originally did:
0, 1, 2, 3, 4,…
and immediately following all of the finite ones, we have w (recall that it may be helpful to think of w not as a point to be reached, it is just there.)
We can still keep adding one, so we’re not done yet. So now our ordinals look like:
0, 1, 2, 3, 4,…, w, w + 1, w + 2, w + 3,…, w + w, w + w + 1, w +w + 2,…, w + w + w,…
And now we’re just getting started. These ordinals are all actually still pretty small (even though the latter ones are infinite). They are called the countable ordinals, since each of them can be put into a one-to-one correspondence with the original counting (finite) numbers 0, 1, 2, 3,…
With the Power Set Axiom (which says that, given a set X, the collection P(X) of all subsets of X is itself a set), we get an ordinal that is uncountable (meaning that it’s actually too big to be put into one-to-one correspondence with the original counting numbers). The first uncountable ordinal comes immediately after all of the countable ordinals, and is called w[sub]1[/sub] (omega-one). So now our ordinals look like:
0, 1, 2, 3, 4,…, w, w + 1, w + 2, w + 3,…, w + w, w + w + 1, w +w + 2,…, w + w + w, w + w + w + 1,…, w[sup]2[/sup], w[sup]2[/sup] + 1, w[sup]2[/sup] + 2,…w[sup]2[/sup] + w,…, w[sup]2[/sup] + w + w,…,w[sup]3[/sup],…,w[sup]4[/sup],…,w[sup]5[/sup],…,w[sup]6[/sup],… w[sup]w[/sup],…,w[sup]w[sup]w[/sup][/sup],…, w[sup]w[sup]w[sup]w[/sup][/sup][/sup],…, w[sub]1[/sub]
Of course, I have omitted several ordinals along the way, the vast majority of which are inexpressible in the notation I’m currently using (or in any notation, for that matter).
And now we’re getting started (just).
Certain ordinals are known as cardinals. Cardinals cannot be put into a 1-1 correspondence with any ordinal preceding them. All of the finite ordinals (0, 1, 2, 3,…) are cardinals, w is the next cardinal following those, and we’ve just seen the next cardinal, w[sub]1[/sub]. Cardinals can be thought of as an actual “jump” in magnitude along the ordinal line. And they also keep on going. The cardinals:
0, 1, 2, 3,…, w, w[sub]1[/sub], w[sub]2[/sub], w[sub]3[/sub], w[sub]3[/sub],…, w[sub]w[/sub], w[sub]w+1[/sub], w[sub]w+2[/sub],…w[sub]w+w[/sub],…, w[sub]w[sup]2[/sup][/sub],…, w[sub]w[sup]w[/sup][/sub],…w[sub]w[sub]1[/sub][/sub],…, w[sub]w[sub]2[/sub][/sub],…, w[sub]w[sub]w[/sub][/sub],…, w[sub]w[sub]w[sub]w[/sub][/sub][/sub],…
And on and on and on…
You may notice that the cardinals themselves are indexed by the ordinals. Also, that last list was of cardinals only (and, as before, I omitted many)–the ordinals are also in-between all of these cardinals, as I showed in some greater detail earlier.
And they go on forever. Also, just as at the beginning we had the Axiom of Infinity, you can introduce new axioms (“Large Cardinal Axioms”) that get you even bigger and bigger cardinals than we already have. And that’s when it really gets weird.
Zoe: The cardinality of the evens is the same as the cardinality of the integers. That is a compact way of saying that there are just as many even integers as integers in general.
If you want to understand infinity, it is vitally important you grasp that fact.
Now, how did I reach that bizarre conclusion? Well, now is a good time to get out some paper, or at least enter the right frame of mind to begin visualizing things. Imagine the set of all positive integers — that is, the set which includes zero and all the numbers you can construct by adding one to zero one or more times. Call it N. Now, construct the set of all positive even integers — that is, the set of all members of N which, when divided by two, leave a remainder of zero. Call it E.
Here comes the visuals. Line N and E up side by side: For each element of N, find an element of E to match it up with. Keep going. Keep going further. Go past one billion. (Hope you have access to a couple reams of paper, if you can’t do it in your head. ;)) Notice that the bijection, or the matching, never falters: For each element of N, there is always an element of E to match it up with.
Now, math types aren’t happy unless you can write an equation. Here it comes: y = 2x, where x is any element of N and y is the element of E x matches up with. (In case it’s been too long since high school math, or in case you haven’t even gotten there yet, 2x' means two times x’.)
That equation cinches it: It proves, in its own simple way, that there will always be enough of E to match up with N.* E matches up with N even though we’ve `thrown out’ half of N to create E. We’ve subtracted an infinity from an infinity and we’ve come up with an infinity of the exact same size. A little thought shows that not only does O (the set of all odd positive integers) have the same cardinality as N, but it also has the same cardinality as E.
*(Well, not prove-prove. We haven’t defined multiplication, after all, but a little handwave and we know that it won’t fail us now (in math-geek terms, it’s closed under both N and E).)
Now, are all infinities the same size? Not at all. Construct a new set, the set of all numbers without an imaginary part. Call it R, meaning reals. R includes all of N, all of the negative integers, all of the fractions like 5/1 and 2/3 and 1/1000 (called the rationals, and included in the set Q), and all of the weird numbers like pi and the square root of two, which are the irrationals (and, sometimes, the transcendentals). Does R have the same cardinality as N?
Here I come to one of the most beautiful and most subtle arguments I’ve yet heard in mathematics. It is called Cantor’s Diagonal Argument.
First, assume that R does have the same cardinality as N. That is, assume that for each element of R, there is an element of N to match it up with. That would imply that R is countably infinite, which means that we can traverse the set in a series of discrete jumps, just like we can traverse N simply by adding one to the last number we saw, and we can traverse E by adding two to the last number we saw. That property implies we can list R out in an enumeration, or a list, and in that enumeration we’ll capture all of R.
Since we can’t even try to construct a full enumeration of R, if it is indeed possible to do so, do this: Construct a square grid of numbers by writing multiple decimal numbers, all between one and zero, one on top of the other. It should look something like this:
0 . 1 5 7 8 . . .
0 . 2 0 2 1 . . .
0 . 8 6 0 5 . . .
0 . 6 0 2 3 . . .
. . .
(None of those numbers are special in any way. They were picked out of a convenient orifice. And remember, this is infinite in both directions: Real numbers don’t have a finite representation.)
Now, we construct a new decimal number between one and zero. We go from the top left of the grid, just past the decimal point, and move in a diagonal down to the bottom right. If the number we come across in that simple line is equal to 0, the number we add on to our new number is 1. If the number we come across is not 0, the number we add on to our new number is 0.
Here is our new number:
0 . 0 1 1 0 . . .
This number certainly looks new. In fact, it’s different in important ways from anything in the grid we have above. It’s different from every number in the grid in at least one decimal place, in fact, meaning it cannot be in the enumeration.
Since it cannot be in the enumeration, the enumeration cannot be complete. Even if we enlarge our view of R, we can still construct a Cantor number that will not be in that view. Guaranteed.
That blows our assumption out of the water. R cannot be enumerated, therefore it is not countably infinite. R is not countably infinite, therefore it has a higher cardinality than N. R is `more infinite’ than N.
Math types have names for these cardinalities. N has cardinality aleph-null. R has cardinality aleph-one. I don’t think too much about what might have cardinality aleph-two. The notion of cardinality aleph-aleph-null frightens me and makes me want to think about a nice, cozy padded room.
Wikipedia has a very nice article about Cantor’s wonderful argument, and it has a nice set of other articles available via hyperlinks and a search engine. Check it out.
Oh, and doing Greek in HTML is easy: Simply write the name of the character between an ampersand (&) and a semicolon (;).
ω gets you ω, Ω gets you Ω.
Easy as π.
Thanks for the tip about Greek, Derleth.
Actually, that’s the old Continuum Hypothesis, which is undecidable in standard (ZFC) set theory. It’s consistent to assume that R has cardinality aleph-one (sidenote: the cardinal aleph-one is the same cardinal I referred to earlier as ω[sub]1[/sub]–you may see either notation used for cardinals: the one I used (using ω) or the other, which replaces the ω with aleph (now how do I do Hebrew on this thing?) (ordinal notation almost always sticks with ω, however)). Anyway, like I was saying, it’s consistent to assume that R has cardinality aleph-one, but it’s also consistent to assume that R has any cardinality greater than aleph-naught (=ω), so long as that cardinality has cofinality greater than ω. In fact, most set theorists believe that the conituum hypothesis is false, and that the cardinality of R may be much larger than aleph-one (=ω[sub]1[/sub]).
In general, regardless of the continuum hypothesis, any cardinal can be described in the following manner–it’s the (cardinality of) the set of all ordinals with lesser cardinalities. In other words, ω is the set of all finite ordinals; ω[sub]1[/sub] is the set of all countable ordinals; ω[sub]2[/sub] is the set of all ordinals with cardinality at most ω[sub]1[/sub], and so forth.
ℵ gives you ℵ, which isn’t really aleph (or alef, whichever)* but seems to be a concession to us math geeks. It’s a symbol that bears a striking resemblence to a certain Hebrew letter, is all.
*For one thing, it doesn’t go right to left. 
Oh, and I’m certainly not up on the latest on set theory. I was unaware (well, dimly aware) that what I was saying touched on the continuum hypothesis, or that there would be any connection between the ℵ and the ω.
τΗαℵκ γομ 
I wrote:
> Actually, googleplex (note the spelling) is 10 to the google
> power.
I can’t believe I made a mistake this stupid. “Googol” and “googolplex” are the numbers, while “Google” is the search engine. Thanks for the correction, Brutus.
:smack: Am I the only one who spotted that Zoe was being ironic? I’d heard that Americans don’t understand irony…
All I see is a square box (Win98, IE6).