2

I know

There is an infinite number of prime numbers

is correct. But is

There are infinite prime numbers

correct as well?

carllacan
  • 341
  • 1
  • 3
  • 7
  • 1
    @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 06 '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 06 '13 at 21:21
  • 4
    @FumbleFingers: No, it's actually a dupe of Which is correct: “…infinite ways…” or “…an infinite number of ways…”? – ruakh Feb 06 '13 at 22:04
  • @ruakh: Well found! I can't change the target of my closevote, unfortunately, but I completely agree yours is the right one. – FumbleFingers Feb 06 '13 at 22:10

6 Answers6

5

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.

Peter Shor
  • 88,407
3

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.

J.R.
  • 58,828
  • 5
  • 95
  • 196
Jay
  • 36,223
  • 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 06 '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
2

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

Community
  • 1
John Lawler
  • 107,887
  • What is the source of your first statement and your first quote? – Canis Lupus Feb 06 '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 06 '13 at 19:54
  • 1
    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 06 '13 at 20:23
  • I disagree about "only matters to mathematicians". Countable vs. uncountable is an important distinction with a very clear and tangible meaning: You can start counting "zero, one, two, three..." and continue forever, but with the real numbers you can't even describe most of them and there's no way to list or enumerate them comprehensively at all. (That said, it's not really essential to the question as asked.) – camccann Feb 06 '13 at 20:24
  • Yes, yes, some people are fond of saying "infinity is not a number". That depends on your definition of "number". I'm not sure how such a discusion enhances our understanding of mathematics. I recall a math book I read years ago that insisted that fractions are not numbers, but rather just "the solution to a division problem". Apparently the writer's position was that only integers are "true" numbers, as far as I could figure out. – Jay Feb 06 '13 at 20:28
  • @camccann: O.K., but the rational numbers (the numbers that can be expressed in the form p/q, where p and q are both integers; so, for example, 3/4 and 4/3, but not π or √2) also form a countably infinite set, because it's possible to re-arrange them into a list. Is this relevant to usage? Would you distinguish, in ordinary English, between the number of primes/naturals/integers/rationals and the number of reals? – ruakh Feb 06 '13 at 22:11
  • 1
    @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 06 '13 at 22:19
  • @ruakh: In natural language. The rational numbers (and in fact the algebraic numbers -- the set of all solutions to algebraic equations with rational coefficients -- a superset of the rationals which is also countably infinite) are the stars in the sky on a dark night, going on forever. The real numbers are the darkness separating the stars. All of it. – John Lawler Feb 07 '13 at 00:24
1

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.

Kristina Lopez
  • 26,536
  • 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 06 '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 06 '13 at 19:19
  • 2
    @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 06 '13 at 20:09
  • 3
    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 06 '13 at 20:20
  • @John, however Cantor demonstred that there is no contradiction where you see a contradiction. Nevertheless I'm not saying you are wrong under a linguistical perspective. –  Feb 06 '13 at 21:34
  • @camccann: Are you saying that only finite primes exist, or that the term infinite can't be applied to numbers / sets of numbers? In which case, what does this AHDEL definition mean is existing? 3. Mathematics a. Existing beyond or being greater than any arbitrarily large value. ....pseudo-numbers? – Edwin Ashworth Feb 06 '13 at 22:16
  • 2
    @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 06 '13 at 22:27
  • Yes, though it's set by metaphor for most people. Folks who learn addition and subtraction by the Arithmetic Is Object Collection metaphor theme (There are 4 5's in 20) have a tough time with algebra because zero isn't a number in that metaphor -- it's the absence of anything. And negative numbers are proverbially difficult, looked at that way. But if you learn by the Arithmetic is Motion theme (Count backwards from 20), zero is a distinguished number, and negative numbers are just numbers moving in the other direction. – John Lawler Feb 07 '13 at 00:19
0

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.

camccann
  • 479
  • 2
    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 06 '13 at 22:23
-1

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

Patrick T. Randolph
  • 1,795