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I know

There is an infinite number of prime numbers

is correct. But is

There are infinite prime numbers

correct as well?

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possible duplicate of A number of questions "has been" or "have been" asked? –  FumbleFingers Feb 6 '13 at 19:11
@FumbleFingers, sorry to disagree but this question is different than the other question you cited in that the meaning is drastically changed by the variations in the two sentences. (Thanks for tip on searching for dupe's though!) –  Kristina Lopez Feb 6 '13 at 19:24
@Kristina: Well, as John's "?*There are infinite perfect numbers" indicates, OP's second sentence wouldn't be considered a valid construction by many/most native speakers. I almost get the feeling we're seeing more of these ultra-basic questions lately, even though ELL is now up & running in public beta mode. I'd rather see ELU more concerned with looking into things that might interest native speakers, rather than disabusing learners of basic errors. –  FumbleFingers Feb 6 '13 at 21:21
@FumbleFingers: No, it's actually a dupe of Which is correct: “…infinite ways…” or “…an infinite number of ways…”? –  ruakh Feb 6 '13 at 22:04
@FumbleFingers, I agree and let's hope that ELL starts to attract these questions and we'll all have to try to steer suitable ones that way. –  Kristina Lopez Feb 6 '13 at 22:04

6 Answers 6

Your first sentence is correct, but has the stylistic disadvantage of repeating the word "number". Your second sentence is wrong; "infinite" is usually only applied as an adjective to uncountable nouns (e.g., "infinite space"). The standard way to rephrase the first sentence without repeating the word "number" is

There are infinitely many prime numbers.

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Note that this is the exact phrasing used on Wikipedia. –  camccann Feb 6 '13 at 20:27
+1 for the "uncountable nouns" bit. One way to look at it might be that infinite can substitute for the not-very-grammatical infinitely much - but where the sense is infinitely many, that's how we say it. –  FumbleFingers Feb 6 '13 at 21:31

The 2nd sentence is a bit ambiguous in that it can be interpreted to mean there are prime numbers that are infinite. Now, logically that may not make any sense, but the 1st sentence clearly states that the number of prime numbers are infinite.

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You have enough rep now to closevote duplicates, so I think you should do a quick search before answering really basic questions which are likely to have come up before on ELU. As I didn't realise for a long time, in a case like this it's better to do a site-specific Google search if you want to look for, say, number is are, because the built-in ELU search ignores those all-important little verb forms. –  FumbleFingers Feb 6 '13 at 19:16
Not having infinitely large primes would mean that there must be a largest prime. But there are infinitely many primes. –  Edwin Ashworth Feb 6 '13 at 19:19
@EdwinAshworth: Just because there is no finite upper bound for prime numbers doesn't mean there's a prime number which is itself infinitely large. Precision is important when talking about infinities because it's very easy to make a misstep. –  camccann Feb 6 '13 at 20:09
A prime number must be an integer, which is a number. Infinity is not a number; it's a limit. The concept of a "prime number which is itself infinitely large" is a contradiction in terms. –  John Lawler Feb 6 '13 at 20:20
@EdwinAshworth: The former. The concept of an infinitely large prime number doesn't really make sense. Infinite numbers are perfectly reasonable, with many caveats about their mathematical properties. "Number" is an incredibly vague term, mathematically. –  camccann Feb 6 '13 at 22:27

First, there is only a countably infinite number of prime numbers.

There are infinitely more real numbers, for instance, than there are prime numbers, because the real numbers are not countably infinite. So, saying an infinite number is ambiguous. Though that probably only matters to mathematicians; countably infinite is still infinite.

Second, infinity is not the name of a number; it's the name of a mathematical concept. And infinite isn't the name of a number either; it's an adjective meaning 'without end' and has special descriptive uses in set theory, and in any branch of mathematics derived from set theory (which means pretty much everything).

Consequently neither word can be used as a quantifier, the way a number name can, so

  • There are (more than) 3,406,295,004 perfect numbers.
    is OK, while
  • ?*There are infinite perfect numbers.
    is at least odd, and certainly not the way mathematicians talk.

Third, infinite, as you point out, can modify number, and takes an article (usually indefinite) when it does so:

"We've all heard that an infinite number of monkeys at an infinite number of keyboards will eventually produce the complete works of Shakespeare; now, thanks to the Internet, we know that this is not true."
-- Robert Wilensky

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What is the source of your first statement and your first quote? –  Canis Lupus Feb 6 '13 at 19:50
The set of prime numbers is a proper subset of the set of integers. The set of integers is countably infinite. Therefore the set of prime numbers cannot be uncountably infinite. Though it is certainly an infinite set, as Euclid proved. As to the quotation, it's off the net. –  John Lawler Feb 6 '13 at 19:54
Oh. Okay. Thanks. –  Canis Lupus Feb 6 '13 at 20:11
Hmm, but exactly what is the point of your first statement in context? It is perfectly correct to say that there are an infinite number of primes. Yes, it is true that some infinite sets are larger than others (like their are more real than integers, but there are no more integers than there are primes) but that doesn't make the first statement false. It's like someone saying, "Bob is from Europe" and you reply, "No he's not, he's from France." –  Jay Feb 6 '13 at 20:23
@ruakh: That's as dependent on specific context as it is off-topic for this site. Ignoring the distinction is fine as far as language use goes; I was only quibbling over the importance of the concept. :] –  camccann Feb 6 '13 at 22:19

I think people are split on whether you should say, "There is an infinite number of ..." or "There are an infinite number of ..."

I think this problem generally exists when using "number" and similar words to express a quantity. Here's the Google Ngram on "is a large number" versus "are a large number"; note "are" has about twice as many.

Similarly, in this Ngram "are an infinite number of" beats "is an infinite number of".

As others note, you can't say "There are infinite prime numbers" in the sense you mean. that statement as worded would mean that there exist prime numbers that are infinite, which doesn't really make sense. There is no single word that I know of that can be fit into that sentence in place of "infinite" to express the idea that you want. Dictionary definitions might lead you to write, "There are infinity prime numbers", but no one actually says that.

If you dislike repeating the word "number", you could always say, "There are an infinite number of primes." Of course that solution relies on the fact that "prime" can be used as a noun synonymous with "prime number", a fact that wouldn't work in other cases. Like, "There are an infinite number of perfect numbers". You can't say, "There are an infinite number of perfects", that's just not an accepted term.

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There is the language spoken in bars, cars, and playgrounds, and there is language spoken by mathematicians. The "correct" way to say this is "There are infinitely many," not, "There are an infinite number of," although only those with fairly extensive mathematical backgrounds are likely to notice the difference, or wince at the error when they hear it. –  J.R. Feb 6 '13 at 23:22
Well, in general there are often differences between the technical language used by professionals and the language used by people in ordinary conversations. Indeed the professionals often speak differently in conversation outside the job than they do in conversation with other professionals. One of my pet peeves is when people say that the ordinary use of a word is "wrong" because it doesn't match some technical definition -- like when I was in school a teacher said that it was wrong to say that study was a lot of work because "work" is a measure of force expressed over distance, not ... –  Jay Feb 11 '13 at 18:05
... mental effort. Even as a kid I realized that was silly: I'm sure even professional physicists say they "did a lot of work on this research", meaning mental effort, not Joules expended. As to your specific point: if you say so -- when it comes to mathematics I'm an interested amateur. –  Jay Feb 11 '13 at 18:06
I tried to be measured in my comment. I agree with you: in a discussion about language, I think it's worth pointing out the stringently "correct" answer (i.e., the one I'd use during my thesis defense), but there's nothing wrong with discussing how non-experts might also try to express the same thing. I hope I didn't trip your peeve – if so, I didn't mean to sound pedantic. As for your teacher and the word "work" – my goodness! That teacher really ought to get a dictionary and see how many ways that versatile word can be used. :^) –  J.R. Feb 11 '13 at 18:10
@J.R. I wasn't disagreeing with you. I think we're pretty much in agreement on this point, actually. –  Jay Feb 12 '13 at 15:05

As pointed out in other answers, "There are infinite prime numbers" doesn't work because it reads as both "infinite" and "prime" modifying "numbers". "There are prime infinite numbers" is equivalent (but the adjective order is fishy).

In the phrase "There are an infinite number of prime numbers", "infinite" is modifying "number", and the phrase "an infinite number" expresses a non-specific quantity, much like saying "There are lots of prime numbers." I can't recall any single word that would work here.

The previously suggested phrase "There are infinitely many prime numbers" is similar, and if your motivation is avoiding the repetition of "number" is probably the best choice.

However, the structure of the phrase "There are infinite prime numbers" would work if "infinite" were replaced by an explicit quantity, such as "eleven". The word "infinity" is plausible in this case, but is idiomatically dubious. Generally, "infinity" in this sense is used only as an overall category ("some infinities are larger than others") or when speaking of limits ("as X goes to infinity"). As such, the phrasing "There are infinity prime numbers" is, while clear and comprehensible, technically wrong and is likely to irritate anyone who knows that.

If your motivation is a word that represents an infinite quantity the same way "eleven" represents a quantity, the term for that is a cardinal number. There are a variety of cardinal numbers representing infinite quantities, but the cardinality of the prime numbers specifically is aleph-null, which is the "smallest infinity". It's not a particularly common use, but the phrase "There are aleph-null prime numbers" is grammatically and mathematically correct, if somewhat informal.

If your audience isn't likely to know what aleph-null means, you should probably stick with the "infinitely many" phrasing.

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I can't agree that "infinite prime numbers" is wrong simply because infinite could apply to both the numbers themselves, and the quantity of numbers. "There may be infinite stars in the universe" is also just as wrong to my ear. But not everyone agrees with my thinking - in Google Books I find 36 instances of are infinitely many stars, but that's somewhat undermined by 11 instances of *are infinite stars. Even so, I know what sounds "right" to me, so (nothing personal), I'm downvoting this and upvoting John's answer saying infinite here is generally not considered "acceptable". –  FumbleFingers Feb 6 '13 at 22:23

Yes, the first one is more specific. But I would say, "Prime numbers are infinite."

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You mean: "Prime numbers are infinite in number"- Five is not infinite. –  Jim Feb 7 '13 at 6:27
Check out George Cantor. All numbers can theoretically be infinite. –  Patrick T. Randolph Feb 7 '13 at 18:10
No, that's not it. Please read his work. Cheers. –  Patrick T. Randolph Feb 8 '13 at 4:32
@PatrickT.Randolph: This discussion is about English, not mathematics. If you want to assert that "Prime numbers are infinite" is a valid way to express something, then I'd suggest you elaborate and explain. "Please read Cantor" is hardly a good way to substantiate something that doesn't seem intuitive, particularly when the O.P.'s question concerns the correct way to describe the number of primes, not the prime numbers themselves. –  J.R. Feb 8 '13 at 16:00

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