I guarantee that Terry Pratchett did not invent this.
This sounds reminiscent of something James Joyce wrote about his The Portrait of the Artist as a Young Man. In fact, Joyce was recounting how a priest was describing to a class of school boys the concept of eternal (i.e. infinite) damnation (awaiting those who “touched themselves”).
IIRC, the priest asked his young charges to imagine a huge mountain comprised of the tiniest grains of sand. He went to tell them to imagine that every millions of years, a wee bird comes and takes away a single grain, and how many eons you’d wait for it to return and carry away one more grain. And, to imagine how amazingly long it would take for the bird to remove a teaspoonful of the mountain. And, then, to try to begin to imagine how agonizingly long it would be, how many eons upon eons would have to pass before the mountain had disappeared - but then to realize that, in terms of eternal damnation, your stay in Hades would not have been shortened by even a single second.
If you’re curious, there are of course many websites covering the reasons why this is the case. Wikipedia’s article isn’t a bad place to start.
Actually I think that the two problems could be handled similarly. It really depends on how we define the number resulting from an infinite string. When you talk about a decimal followed by an infinite string of 9’s you really mean that you want to consider the series
0.9, 0.99, 0.999, 0.9999, …
and then note that it is converging to a limit as you keep adding 9’s, and the limit that it is approaching is 1.0. So from this definition a decimal followed by an infinite collection of 9’s equals 1.
If we continue with this definition and apply it to the OP, we have the following sequence
9, 99, 999, 9999 …
this is also a well defined sequence and this sequence converges to infinity (for the math purists I use infinity in terms of the 2 point conpactification of the reals). So in answer to the OP’s question, yes an infiniite sequence of 9’s is equal to infinity, but then again so will an infinite sequence of 1’s or an infinite sequence of any digits for which not all of the leading terms are zero.
Someone above said that infinity is not a number. But it’s my understanding that the first infinity is the first number that is greater than every counting number.
I think one point people are trying to make with that statement (“infinity is not a number”) is that infinity is not any kind of integer or real number, and can’t participate in arithmetic with them.
So for example when we speak of “n approaching infinity” in a limit, we don’t mean that n reaches some destination called “infinity” or even gets close to it — or that you can substitute “infinity” for n in any expression you like, as you could with 47 or 2.8, and expect that to be meaningful. Rather, “as n approaches infinity” is shorthand for “as n increases without bound”.
Some of the people you see arguing against 0.(9) equaling 1, for example, will talk about the “inifinitieth” decimal place, or will try to divide or subtract two infinities, or add one to infinity, and of course none of that is valid. Hence, “infinity is not a number.”
One of my nuns used to tell that analogy, concluding it with saying that the time to wear away the mountain is “but a moment of time in God’s Eternity”, which was pretty unsatisfying.
Later I read that the description is apparently derived from a discussion of the Hindu Kalpas, although I’ve never read the original of this.Kalkpas are immense periods in the Brahmanic cycles.
here’s someone referring to this:
Here’s Wikipedia on it, which claims the reference is to a Mahakalpa ("Great Kalpa):
If you accept any of the definitions, they’re still much much smaller than the time Chronos gives. I suspect no one intended the bird and mountain analogy to be some dort of Brahmanic Bureau of Standards accurate definition – it just is supposed to be A Really, Really long Time.
In college, I took a class called History of Mathematics.* We must have spent three weeks on Zeno’s Paradoxes. It’s worth a read if you need a mind-enema.
*Its worth mentioning that it was the best class I’ve ever taken. It was a history class that required a substantial knowledge of base systems, linear algebra, and set theory. It was also writing intensive. Loved it.
This is correct. I would also differentiate between this infinity (i.e. a convenient way to define convergence for a series which increases without bound), and the set theory version of infinity. In the later case, the infinity is used to describe the size of infinite sets. This is the area where you get the multiple levels of infinity (countable, and uncountable etc.). We use the same word for both but really they are defined and used very differently in mathematics.
Not to be obtuse, but while I agree that there’s no such thing as an “infinitieth decimal place,” you can divide and subtract infinities, and add to them.
I did not know there was that distinction! (Between the inifinity of convergence and the infinity of set theory.)
The “infinity of convergence” is the kind of infinity that Bytegeist was talking about in his post about “approaching infinity.” It’s the one that the “sideways 8” infinity symbol denotes. This kind of infinity is not considered a number (at least in standard treatments—see “hyperreal numbers” or “surreal numbers” for attempts at a number system that includes them) and cannot be added, subtracted, divided, etc. (although you do occasionally see things like “∞ + ∞ = ∞” as a sort of shorthand).
The “infinity of set theory” refers to transfinite cardinal numbers. Cardinal numbers specify how many elements are in a given set. The cardinal number for an infinite set is one of the transfinite cardinal numbers, symbolized with Alephs (see link).* You can indeed perform arithmetic (add, divide, etc.) on transfinite cardinal numbers.
There are also transfinite ordinal numbers, which refer to position in an infinite sequence.
(*Speaking of which, wouldn’t it be kickass if Albert Pujols had an Aleph Null on his uniform? Now there’s a Cardinal number!)
Are there reasons for thinking the “infinity of convergence” isn’t the same as one of the transfinite cardinals? Or alternatively, is it just that there are no reasons for thinking they are the same? (Is the “infinity of convergence” considered to be “greater than” the real numbers in some sense? If so, I’d think that’s a reason to think that it’s the same as one of the transfinites.)
I think this article is quite relevant to this subject. It begins with the subject of writing very big finite numbers, then goes off from there to some very interesting things.
Wait, “infinity of convergence”? That gets exactly two other hits on Google. And you can’t talk about “a convenient way to define convergence for a series which increases without bound.” Increases without bound is divergence, not convergence.
Infinite is not a number in any sense, or even a metaphor for a number in any sense. Any talk about its being the largest number or a number greater than or any variation on that just leads to the kind of confusion that plagued the OP.
I’m not sure that orders of infinity should be termed greater than one another either, for that reason, although that language is often used. They can’t be placed in one-to-one correspondence with one another. Almost everything else leads to problems for interpretation. You’re certainly confusing me with this terminology.
If “infinity of convergence” means “The conceptual limiting value of a series of (let’s say) rational numbers which grows arbitrarily large”, then this isn’t the cardinality of any particular infinite collection, in the same way that -1 or 3/5 or what have you aren’t the cardinalities of any particular collection. These aren’t cardinalities. They’re different kinds of measurements. (There are, of course, relationships between different kinds of measurements, but I see no compelling one here crying to be turned into an identification…)
On an unrelated note, I actually agree with Exapno on dispreference for the terminology of cardinalities being greater than one another, though quite possibly for some different reasons in addition to his own. For example, there are various different notions of how to compare cardinalities: A can be mapped injectively into B, B can be mapped surjectively onto A, there is a subset of B which maps surjectively onto A, etc., which do not necessarily all line up in all contexts, or play as nicely with each other as the terminology suggests. There are even contexts (so-called “nonclassical” ones, it is true, but important ones nonetheless) in which one can craft a surjection from a subset of A to B and a surjection from a subset of B to A without there being any bijective correspondence between A and B; should we say A is larger than B? B is larger than A? They’re both at least as large as each other but they don’t have the same size? The terminology is more obfuscatory than helpful, I think, except when one has a really good reason to work in a context shot through with the pervasive assumption that cardinalities are linearly ordered (which of course follows from strong enough choice and classicality principles, but the point is that it is helpful to realize that those principles are nontrivial and that it is both possible and indeed often very useful to look at things in a way which doesn’t require them).
Auugghhh! I still have nightmares about injective and surjective mapping from college. They may have been the beginning of the end of my being a math major. Curses upon you, jectives!
Re: any discussion of “infinity is not a number”, I hate discussion about what things are and are not numbers. “number” is not a precisely defined word, nor should it be; it can be used in describing elements of all kinds of mathematical systems, supposing the right family resemblances to various archetypal uses are present, with none given such a concrete finality as to monopolize the term. I’ve ranted about this before.
Wow, that makes that bit in Good Omens just that much funnier to me.