in the recent thread about .999… = 1 (which I accept)
someone and i’m not going to read through that monster thread again…
said infinity - 1 is a meaningless statement.
but… isn’t infinity - 1 < infinity? I realize that infinity is not a number… but doesn’t that still hold true?
If infinity is not a number, what does infinity - 1 mean? The sum is, essentially, undefinied.
More directly though: the fact is that if you remove one object from an unbounded number of objects, you still have an unbounded number of objects. That’s what makes infinity infinity. Otherwise it’s just a really big number.
This is maybe not as trivial as it first looks. It is an inability to get over the difference between “big” and “infinite” that leads to so many misconceptions, such as not accepting 0.999… = 1.
Infinity does not mean “bigger than anything you can think of”. It means that no matter how many you remove, you still have as many as you started with.
pan
Or indeed add… 
Grim
Repeat after me: Infinity is not a number. Normal' math doesn't work when dealing with it. Trying to use normal’ math on infinity leads to inconsistent, and therefore meaningless, results.
By some stroke of luck, there’s a very readable treatment of just this subject on kuro5hin.
(Assume `inf’ represents the classic symbol for infinity.)
An example of normal math failing spectacularly when dealing with inifinity? Well, 1/inf = 0. Using `normal’ math, we say inf*(1/inf) = 0*inf, and we conclude that 1 = 0.
WRONG!
The correct way to think about 1/inf = 0 is to read it as lim[sub]x->inf[/sub] 1/x = 0. That is calculus, and it means that as x gets arbitrarily large, 1/x gets arbitrarily close to 0. Infinity itself plays no part in this, except as a notation to say that x can get as large as we want it to be. In reality, x never becomes' infinity, just as 1/x never becomes’ 0.
Let’s think about infinity in terms of sets. (A set is merely a collection of items, in this case numbers.) To begin with, we have the set of natural numbers, or the whole numbers greater than or equal to zero. By convention, we call the set of natural numbers N. Now, N is an infinite set: It contains an infinite number of members, because it is defined to contain the number (x+1) for any number x it already contains.
Another interesting set is the set of all rational numbers, conventionally called Q (for Quotient). A rational number is any number of the form n/m, where n and m are whole numbers (and m cannot, of course, be 0). Q is also an infinite set, as should be obvious.
Now, is Q larger than N? Well, we obviously can’t tally up the members and compare in the usual way. We need the concept of `cardinality’: Imagine a two-year-old trying to number his fingers. He cannot quite grasp the concept of numbering, but he knows he has a certain number of fingers on each hand. If he can touch each finger to its opposite number on his other hand with none left over, he knows he has an equal number of fingers on each hand. That is the ten-cent description of cardinality.
Our method is more subtle than finger-touching. First, we construct a small grid of the positive rational numbers (members of Q[sup]+[/sup]):
1/1 2/1 3/1 4/1 ...
1/2 2/2 3/2 4/2 ...
1/3 2/3 3/3 4/3 ...
...
As you can see, all possible positive numerators are used in the columns, and all possible positive denominators are used in the rows. Next, we draw diagonal lines on the grid to join numbers from the top right to the bottom left, such that the first line only touches 1/1, the next line touches 2/1 and 1/2, and so on. Now, each of those diagonals is finite in length, so we can line them all up (to form something like 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, …) and then number each element of the list using the members of N. Et voila, we’ve proven that N and Q[sup]+[/sup] have equal cardinality! Extending this proof to cover Q in general is trivial: When we labeled the elements of Q[sup]+[/sup], we could have used exclusively the even members of N, reserving the odd members for numbering every negative rational (or every member of Q[sup]-[/sup]).
I’m not going to completely recapitulate the well-written article I’ve linked to. What I’ve just paraphrased will do for your question: Infinity is a concept, not a number, and `normal’ math does not work on it.
There is an extension of the real number system in which the symbol inf is added, and rules of arithmetic are defined for it. However, this is not the standard real number system.
Here are the rules. For any real number r:
r + inf = inf + r = inf
inf - r = inf
r - inf = -inf
inf/r = infsgn®, r != 0 (sgn® is 1 if r > 0, 0 if r = 0, and -1 if r < 0)
infr = inf*sgn®, r != 0
All of these hold if you change the signs on inf.
The expressions 0*inf, inf - inf, and inf/inf are undefined, because there’s no unique value suggested by limits of functions.
This is what I appear to not understand. I don’t see why an unbounded number can’t have one object subtracted from it and therefore be less the original unbounded number.
much the same as
I could see a problem in this case
I don’t understand why my first case isn’t true. (it’s true using limits) but I don’t see why the concept doesn’t carry over to “true” infinity.
Harmonix: Try this on for size.
How many things are in the set of real numbers?
How many things are in the set of positive integers?
Is the set of positive integers a subset of the set of real numbers?
Are there real numbers which are not positive integers?
If so, there must be “more” real numbers than positive integers.
But it’s clear that both sets are infinite.
If you were to subtract the set of positive integers from the set of real numbers, you would still have an infinite set of numbers, because real numbers never end. Infinite does not mean “includes everything” it means “goes on forever.” Clearly, you can have something that is infinite but not all-inclusive.
Well, because there is no real number x = ¥ - 1 such that x + 1 = ¥. If ¥ - 1 is valid, what value would you assign to it?
You might just as well be asking what is the square root of a giraffe.
Messy.
I guess I’m saying the same thing as Derleth, and I considered tossing out this message, but I’ll post it anyway.
One must be careful in talking about infinities, too, with regards to their countability. I’m by no means an expert on math with infinities (if such can be said), but:
Whereas the positive integers are countably infinite, the reals are not.
You can organize the positive integers to count them, 1, 2, 3… in fact, that’s obvious, since they’re the positive integers.
You can even organize the positive even integers this way. Lay out the positive even integers as:
2 4 6 8 10 …
and you can ‘label’ or ‘count’ them as:
1 2 3 4 5 …
That is, you can say the 3rd postive even is 6.
This gives you the true, but totally counterintuitive result that there are the same number of postive even integers as positive integers. You’d think there’d be half as many even integers, but there aren’t…
why?
Because you cannot, the way I’ve ‘counted’ them give me a positive even integer that I cannot give a count number for. They are all, every last one of them, accounted for. There is a direct 1:1 mapping between a specific positive even integer and a positive integer. The 104th (counted) positive even integer is 208. 468 is the 234th (counted) positive even integer. I can go both ways. Exactly one of each category are paired up.
There are, of course, the same number of rational numbers (x/y for some x and y being integers and y not equal to zero (see another message above)) as positive integers, because they, too, are able to be placed in an order for counting.
Reals, on the other hand, cannot be so ordered. There’s some interesting proofs of this, not the least of which is Cantor’s diagonalization argument. So there are ‘more reals’ than positive integers, even though we’ve already proven that the number of positive integers and positive even integers are the same size.
Its messy business. This is why you have to be so brutally careful with infinities.
So when you’re saying infinity - 1, it now has even less meaning. You’re not even classifying your infinity.
The real point, though, is that infinity-1 can only have meaning if you can get some useful non-contraditory information out of defining it to have a meaning, and there really isn’t one.
Consider: In the same way that 0.999… = 1, you could work out the limit:
lim x-1
x->infinity ------ = 1
x
</pre>
This is true and provable in more gory detail, but what it amounts to is that as x approaches infinity, the ratio between x-1 and x approaches 1, so that the limit is 1. This suggests that as x approaches infinity, x-1 approaches x so the limit is that x-1 is equal to x.
I don’t know if this whole math proof holds water, but ultimately, you can take any finite amount from an arbitrarily (infinitely) large value, and have no effect on any math that you could do with them. That is, you could use x-1 anywhere you use x provided x is going to infinity in a limit.
But returning to your regular program…
Wow, I like that kuro5hin site, pretty good explanation.
Friedo
you’re saying that ‡ - ‡ = ‡ no?
but wouldn’t it be more accurate to say
‡1 - ‡2 = ‡3
where
‡1 is the set of real numbers
‡2 is the set of positive numbers?
but that leads to the problem of
‡2 - ‡1 = - ‡4.
my intuition tells me - ‡4 is what it should equal… but I still feel like i’m missing something.
qed
i’d assign it the value ‡5 - 1 = ‡6
and if so… is not ‡5 > ‡6?
gah wtf. my infinity symbol didn’t come out right…
just replace all those “‡” with infinity symbols.
Friedo
you’re saying that infinity - infinity = infinity no?
but wouldn’t it be more accurate to say
infinity1 - infinity2 = infinity3
where
infinity1 is the set of real numbers
infinity2 is the set of positive numbers?
but that leads to the problem of
infinity2 -infinity1 = -infinity4.
my intuition tells me -infinity4 is what it should equal… but I still feel like i’m missing something.
qed
i’d assign it the value infinity5 - 1 = infinity6
and if so… is not infinity5 > infinity6?
Harmonix, maybe this will help.
There are (at least) two types of orders on sets. For one example, we can say:
{1,2,3} < {1,2,3,4},
where “<” means “is a subset of”. With this order, we can say things such as:
R - {1} < R
(where R is the set of real numbers).
All this means is that, if you take the real numbers, delete the number 1 from the set, then the set you get is a subset of the real numbers.
You can think of “is a subset of” to mean “smaller than” if you like, but I don’t think that’s very satisfactory. For example, we have two sets:
{1}, and
{2,3}
Which is the smaller set? Neither set is a subset of the other, so if that’s the criteria we’re using to determine “smaller”, these two sets are incomparable, we can’t say that one is smaller than the other.
Similarly, what about E (the set of even integers) and O (the set of odd integers). Are they the same size? Is one smaller than the other? Again, by the above definition, they’re incomparable, so it doesn’t make sense to talk about the relative sizes of these sets.
However, we can try a new definition. We can say that two sets have the same size (cardinality) if there is a bijection from one set to the other. (See Derleth’s link for more on this, under Comparing Infinite Quantities).
Under this definition, if X is any infinite set, removing any finite number of elements from X will result in a set whose cardinality (“size”) is exactly the same as the that of the original set X.
No, I’m not saying that, you are. 
Your mistake is assuming that the two infinities are somehow different values, and therefore worthy of different labels. They are not. Infinity is infinity. You cannot treat it like a number and apply logical or arithmetical rules to it. It doesn’t work.
To help you understand cardinality, consider this: A lot of members of Q (the set of the rationals from my first post) are equal to members of N (the natural numbers). But a lot of them aren’t, too. You could construct N by removing things from Q. But Q and N have the same cardinality. Using cardinality as a convenient way to represent size, Q and N are the exact same size, even though N could (not that it is or ever would be) be defined as a subset of Q.
(I know I’m repeating others. Maybe by tying this explanation in with my earlier exposition, I’ll make some headway.)
This is hard to impossible to visualize properly. A bijection over two finite sets is easy to see: Just line them up side by side. A bijection over two infinite sets is beyond the capabities of our mind’s eye. I find I have to trust the proofs, which is not a problem for me. Infinity may be beyond my finite experience, but that doesn’t mean I can’t reason about it. I just have to be careful not to let my common sense interfere with my mathematical reason.
I don’t think it’s so difficult to visualize. For the infinity of the integers, lining the integers up side by side with another set to show the bijection (i.e. one-to-one correspondence) is straightforward. For instance, to demonstrate that the set of positive integers has the same cardinality as the set of primes, you need only do this:
{1, 2, 3, 4, 5, 6, … }
{2, 3, 5, 7, 11, 13, … }
Ok, I can see how removing an infinite subset from an infinite set results in an infinite set. I do not however see how you can remove something from set 1 and have the resulting set 2 be exactly the same as set 1.
I can however see how the set of natural numbers is exactly equal to the set of irrational numbers.
don’t infinities exist?
limit x - > 0 1/2x
is different from
limit x -> 0 1/x