# “less” or “fewer” for countable and uncountable infinities [duplicate]

I feel like this is too grammatical for the math stack exchange, but I am sorry if it is too mathematical for this stack exchange.

In math there are several different types of infinity, some of which are known as "countable" and others which are known as "uncountable." I have always heard that one should use fewer to denote instances where one can count the objects to compare and less when one cannot count the objects. A beginning math student will often make the error of saying that the number of integers which exist is smaller than the number of rational numbers which exist.

Would it be proper for this student to say "There are less integers than rational numbers," or should he say "There are fewer integers than rational numbers"? Both of these sets are countably infinite, so one could theoretically count them. However this student may make a similar error, saying that there are fewer numbers between one and two than between two and three.

Would this student be correct to say "There are fewer numbers between one and two than between one and three," or should he say "There are less numbers between one and two than between one and three"? Both of these sets are uncountably infinite, so one could not count them.

Do the definitions of countable and uncountable make a difference in this case? Which would be the correct versions of these sentences? This is not a question about the general definition of the words "less" and "fewer," it is a question about whether the mathematical definitions of "countable" and "uncountable" will hold for the standard English definitions of "countable" and "uncountable."

## marked as duplicate by FumbleFingers, Edwin Ashworth, ScotM, Marv Mills, Tushar RajJun 23 '15 at 17:54

• I would disagree, because this question deals more with the grammatical weight of mathematical definitions than the general use of the words "less" and "fewer." – nosyarg Jun 15 '15 at 13:41
• IMHO any "distinction" in use is nothing but pedantry, and I've no idea what you might mean by the "grammatical weight of mathematical definitions", or how that could somehow elevate this to a different question. Your issue is simply a matter of confusing mathematically "uncountable" infinities with grammatically uncountable nouns. – FumbleFingers Jun 15 '15 at 13:44
• (Note from the above that grammatically speaking, even multiple infinities are in fact countable! :) – FumbleFingers Jun 15 '15 at 13:50
• The fact that "fewer-etc" issues are pedantic: let's accept that. So, what's the correct (pedantic) answer? – Fattie Jun 15 '15 at 14:16
• Regarding duplication, this question is unrelated. The OP here understands the answer to that general question already, and is asking what is the nature of the Things in question here. – Fattie Jun 15 '15 at 14:16

(Edit: To be clear, this answer assumes that you want to use "less" and "fewer" in the way that many authorities, such as the Oxford Dictionaries page quoted below, recommend. It seems to me that the OP does want to do that, otherwise he or she wouldn't have asked what's clearly a well-informed question. For what it's worth, I do think the "rule" is prescriptivist - but I also think that's a completely different question.)

My degree's in mathematics, so I completely understand the question. It hinges on the mathematical terms countable and uncountable. I won't go into the meaning of those terms here, because it's off-topic for this group - I'll just point to this Wikipedia article to prove that the terms do exist and have a clear definition.

This page from Oxford Dictionaries says that you should use less "when you’re referring to something that can’t be counted", and fewer otherwise. This "rule" is not universally accepted, as comments on the OP make clear, but if you choose to follow it then you might reason as follows.

On face value, it seems you should use fewer for members of a countable set such as integers, and less for members of an uncountable set such as real numbers. However, there is a clash of terminology here. When talking about grammar, countable and uncountable have a specific meaning, which is different from the mathematical meaning.

In grammar, this Wikipedia article neatly summarises what countable means: "common nouns that can take a plural, can combine with numerals or counting quantifiers (e.g., one, two, several, every, most), and can take an indefinite article such as a or an (in languages which have such articles)".

Now, does a real number (whether rational or not) meet all those criteria? Yes, definitely. In particular, you can talk about (say) "three real numbers between two and three", in a way that you can't talk about (say) "three water".

So, grammatically, "real number" is a countable noun-phrase. Therefore, if you do wish to follow the controversial "rule", the correct choice in this case is fewer.

• Good answer. The difficulties I see are that (1) all noun usages are not neatly analysable as 'count' or 'noncount' (eg 'That was one eerie silence' / 'There was an eerie silence'); (2) the ODO 'rule' is not binding. 'That's one less problem to worry about' is far more frequently used than the alternative using 'fewer'. The 'rule' was apparently foisted upon the populace by a chap called Baker. Wasn't he the one who couldn't count to twelve? – Edwin Ashworth Jun 15 '15 at 22:41
• Just to reiterate, I'm not trying to defend the less/fewer distinction, which is a completely different question. However, if you do want to follow the purported "rule", then the word to use in this particular case is unambiguously fewer. I'll the answer (again) to try to make that (even) clearer. – Morton Jun 15 '15 at 22:50
• Whoops, inexplicable typo in my comment above - in case anyone's puzzling over it, it should have read "I'll edit the answer". – Morton Jun 15 '15 at 23:10
• The difficulties are not with your answer! Certainly, OP is confusing polysemes. But I'm adding that the Wikipedia article contains at least one flaw. The different tests for (grammatical) countness do not always give the same results. However, I've not come across an article I'm happy with covering the difficult cases (There was a deathly silence / *?There were two deathly silences). See A blinding light Blinding sunlight A blinding sunlight. – Edwin Ashworth Jun 15 '15 at 23:13

I think that once you've conceptualized something as a set of things, the members of the set are grammatically countable, regardless of the sort of infinity. To say that there are fewer numbers between 1 and 2 than there are between 2 and 3 is an appropriate way to formulate this false proposition. But of course many people would use "less" here, rather than "fewer", regardless of the count/mass distinction, so it is largely academic.

And we should be careful about that term "uncountable", which in math doesn't actually mean you can't count things -- it means you can't count all of them. This is not at all what "mass" means in grammar.

• A superb question and not one but two superb answers. By God! – Fattie Jun 15 '15 at 14:17
• Some mass usages involve referents that in some contexts are mathematically countable (but for some of these, counting the individual elements would be senseless in almost all circumstances: rice (formally singular and taking singular agreement) and confetti (formally plural – confetto does exist – but taking singular agreement)). Furniture and cutlery. – Edwin Ashworth Jun 15 '15 at 23:25