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On another site it sometimes happens that I give answers like:

The first formula is correct but not the second one.

This is probably grammatical, but I find the formulation a bit clumsy, perhaps because of the influence of other languages I am familiar with, where the pronoun one is not used.

So my question is whether a formulation like

The first formula is correct but not the second .

or some other variant exists, which would be perceived as more elegant but still natural by a native speaker of English.

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Note while the latter is slightly "prettier", the former is less ambiguous due to English frequently using the same words for adjectives and nouns, so adding "one" makes sure the word before it is understood as adjective. – SF. Jun 12 '12 at 8:17
@SF.: In this specific case, the only alternative reading I see is if you understand the second version to be saying "The first formula is correct, but [the first formula] is not the second [formula]". Which isn't really a very plausible interpretation, so you could hardly say it introduces ambiguity. Come to that, you could just as easily impose that perverse interpretation on OP's first version. – FumbleFingers Jun 12 '12 at 20:40
@FumbleFingers: Yes, in this case there is no doubt, because there is no sense in determining the correctness of the unit of time of 1/60th minute. But if you state: "sometimes first attempts last less than second" you'd better qualify it with "ones"... – SF. Jun 13 '12 at 0:28
@SF.: No ambiguity is possible there. It would have to be "...last less than a second" to be read as duration. – FumbleFingers Jun 13 '12 at 1:09

Omitting a word that the reader can retrieve from the text in this way is a common feature of English. There is no need to be hesitant about using it if you think your meaning remains clear.

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Both sentences are correct and sound natural. In the first, second functions as a determiner. In the second, second functions as a noun.

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It has not escaped my attention that in your second and third sentences you omitted the pronoun one. So you are doing as you preach: +1 ! – Georges Elencwajg Jun 12 '12 at 8:14
Or, for example, you might analyise it as a transitive vs intransitive determiner/quantifier. (It seems slightly unsatisfactory to say that "second" changes category.) – Neil Coffey Jun 12 '12 at 9:41
I remember that I learned that in a sentence like "I like the red dress, but not the blue one" needs the "one" and that adjective "blue" must not stand alone. That's somehow ambivalent. What is different in the example above or is it wrong what I learned? – Em1 Jun 12 '12 at 10:26
@Em1 I think I'd judge that both are grammatical-- it's essentially a stylistic preference. But I think it is probably true that the ellipsis is more common with ordinals. – Neil Coffey Jun 12 '12 at 11:29

Omissions like this are very common and natural. The meaning is clear with or without one. Your sentence already involves a handful of reductions from the fullest form:

The first formula is correct but the second formula is not correct.

English even requires omitting certain comparative constructions. Reduction is the norm, as long as clarity isn't compromised. Things get repeated only for rhetorical effect.

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