Am not!
Are too!
Am not infinity!
Are too infinity+1! I call no more infinites. I win.
So it is a well know concept.
The general way to compare infinite sets, by the way, is to pair them off. With finite sets, this is obvious: if you have two sets A and B, just pick one from each and set them next to each other, and do that repeatedly. If you have things from set A left over at the end, set A was bigger; if you have things from set B left over, set B was bigger; if all the elements from sets A and B are paired off with each other, then they were the same size.
For infinite sets, you do a similar thing: if you can come up with a pairing of elements between the sets such that for every element of A, there’s an element of B paired with it (and vice versa), then the sets are the same size. If there’s no way to come up with such a pairing, then one set has to be bigger. Of course, you could never explicitly write down a list of all such pairs if the sets are infinite, but all you have to do is show that this pairing exists; usually this is done by writing down a rule that tells you, given some element of A, which element of B is associated with it.
Just to be clear, the above definition relates to the cardinality of a set–it’s “size” when ordering of objects in the set is not taken into account. If a set has an imposed order that must be preserved, then two sets with the same cardinality may have a different ordinal size (ordinality).
Here’s an easy example. Suppose I have the inifinite set of integers {1, 2, 3, …}. Now suppose I have another set formed as {1, 2, 3, …, 1’, 2’, 3’,…} (i.e. formed by appending another infinite set of ‘primed’ integers). If the order of the two sets is disregarded, I can match 1-1, 2-1’, 3-2, 4-2’, etc. Since every element of the first set can be matched to an element in the second set (and vice versa), the two sets have the same cardinality.
But suppose my matching strategy must preserve the order of the two sets (i.e. if two elements from the first set a1 and a2 are such than a1<a2, then their matches b1 and b2 must also satisfy b1<b2). Then I cannot produce a one-to-one correspondence; the two sets have a different ordinality (in technical terms, if the ordinality of the first set is w, the ordinality of the second is 2w; there is actually a whole, definable arithmetic for these ordinal numbers).
This is a key point for infinite sets: The cardinality and ordinality of finite sets is always the same. For infinite sets, these two values may be different.
So, to the OP: It would be correct and non-problematic to say that the cardinality of the set of real numbers is greater than the cardinality of the set of letters in the English alphabet. If that’s what you mean by “there are more numbers than English letters” (which, it seems to me, is the most natural interpretation), then, yes, that’s correct.
Perhaps a more intuitive way to see a difference in some notion of ‘size’ between the two sets is that even though the set of integers is infinite, no two integers are infinitely far apart; in the expanded set, for instance 1 and 1’ are infinitely far apart (in the sense that there are infinitely many numbers between the two).
Adding to the answers that have been given to the OP, there are even more (real) numbers than there are names you can create through any consistent procedure in English, or any other language with finite terms you may specify. However, it is possible to give every natural number a name (you could, for instance, just pronounce it digit-wise, i.e. 4516 becomes fourfiveonesix).
Infinity is not a number. It’s a concept. ‘Infinity minus 26’ makes as much sense as ‘Santa Clause minus 26’.
Is infinity minus 26 undefined?
It depends on what you mean by “infinity” and “minus”. (I’m assuming the normal definition of “26”, of course). You can have a system where infinity minus 26 is equal to infinity, but a system like that spoils some of the nice properties of the operation “minus”, so mathematicians prefer most of the time not to define operations like addition, subtraction and multiplication on “infinity” – indeed, they would say that “infinity” is not a “real number”, where “real number” is a technical term, not a value judgment.
Not in the sense you mean. Something like ‘26 divided by 0’ is undefined mathematically. Infinity minus 26 is just meaningless.
Speaking roughly, people in a field such as engineering will often just say that infinity minus 26 equals infinity. This bugs mathematicians though, as it isn’t really correct to treat infinity as a number.
To make things more meaningful the actual situation is this: for any number you can think of, there exists a number that is larger than that one. That is what we mean by infinity. In math terms, for any x, x+1 exists. Subtracting 26 has no bearing on that property (if y = x-26 then for any y, y+1 exists).
I think that answers my entire question, thank you.
Saying “infinity minus 26 is just meaningless” and “it isn’t really correct to treat infinity as a number” bugs this mathematician much more than saying “infinity minus 26 equals infinity” does.
That having been said, the kind of explanation offered by your third paragraph of what might be meant by discussion of “infinity” in this context is a very good one. My only qualm is that just because talk of X can be reduced to talk of Y doesn’t mean talk of X is illegitimate (we can always turn talk of rationals into talk of integers, for example, but who would say rationals are therefore not numbers? More to the point, what’s the value in saying such a thing?)
That depends on whether or not you’ve defined it. 
I did a little googling to see if I could find a good site on Cardinal Arithmetic, but I didn’t find anything I liked. I was hoping I could find an authoritative cite, and a clear explanation, that any transfinite cardinal number minus any finite cardinal number equals the transfinite cardinal number—so, for example, c – 26 = c, where c = the continuum (= how many real numbers there are).
In cardinal arithmetic, subtraction (as an inverse to addition) is sometimes not well-defined. For example, letting omega be the cardinality of the natural numbers, 0 + omega = omega = 1 + omega. So does omega - omega = 0 or does omega - omega = 1? Well, it can equal either one, in the same sense that 0/0 can take on multiple values.
But, in the usual context of the axiom of choice (equivalently, the assumption that of any two cardinals, one is at least as big as the other), subtraction is only non-well-defined when you subtract an infinite cardinal from itself. In all other cases, the math is quite trivial, as follows:
In cardinal arithmetic, if X is infinite and Y is anything, then:
X + Y = X * Y = max(X, Y)
X - Y and X/Y are uniquely defined if X is larger than Y (in which case, both X - Y and X/Y equal X), can take on multiple values if X = Y (specifically, X - X can be any value less than or equal to X, and X/X can be any positive value less than or equal to X), and have no value if X is smaller than Y.
Minor correction: X * Y = max(X, Y) for infinite X only if Y is positive; otherwise, X * 0 = 0, of course.
Also, I see now Thudlow Boink wasn’t asking what happens in cardinal arithmetic; just looking for a site that explains to others why what happens happens. Well, hopefully, the above post will still be of use to someone.
So you would say that infinity can be treated as a number? I remember hearing this several times from profs in undergrad calculus courses (for engineers). Primarily in discussions of L’Hopital’s rule, where some limit evaluates to 1/0 and the prof writes “= infinity” on the board but cautions us not to say things like this around math guys.
Well, I thought it was helpful.
Ooh, this is reminding me of some of the discussion in the thread from about a month back: Mathematicians: Is Zero and Infinity Two Sides of the Same Coin?
In a context like the one you mention (limits and L’Hopital’s Rule), it can sometimes be a convenient shorthand to use “infinity” (the word or the symbol) in expressions as if it were a number. For instance, if I were being informal/sloppy, I might write “1/0 = plus or minus infinity.” But it’s dangerous to treat infinity as a number, because it tempts you to apply things that are true of real numbers (like “any real number minus itself equals zero”) to infinity, which is a mistake.
When a mathematician says “infinity is not a number”, what mathematicians really mean is that “infinity is not a number in the same way that you think of a number, so you need to be very careful when using it.” Infinity does not behave like most students would expect a number to behave: the expressions (infinity) - (infinity), (infinity)/(infinity) and 0*(infinity) are not well-defined. Much of algebra relies on the properties x - x = 0, x/x = 1 and 0*x = 0 to hold for any x (with x != 0 for x/x = 1), so if we allow x=infinity, we no longer can count on many algebraic tools.
Since experience shows that many students don’t heed the warnings, it is easier to tell less-experienced students that infinity isn’t a number (and that dy/dx is not a fraction). This is a technically true statement (for our usual understandings) which is useful for pedagogical reasons.