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Assume the following situation. Suppose I met an alien called Protoss who can speak English. He said he is from Aiur. Now I want to make the following statement.

There exists (a or the) tallest Protoss in Aiur who can speak English.

In school, I learned that I should use "the" before the superlative. But somehow I feel awkward if I say "there exists the ...".

By the way, I know the existence because I already met one, although I might not be able to find him. Aiur is too far. I wonder if this situation affects the answer.

[edit] Thanks to the suggestion by Damkerng T, I write below what I wanted to know. I was actually writing a math sentence. I thought it would be boring to most people, so I tried to write down something which seems more popular. The above was my attempt.

Assume Condition A. Then there exists (a or the) largest integer k satisfying Condition B.

[edit after looking at two more comments] I am trying to modify the original situation more properly.

Let's say Protoss refers to an alien race, the only alien race known, and we call each member a Protoss as well. Let's call the Protoss I met "Fenix". Fenix was quite tall. I like tall creatures, so I wanted to meet the tallest Protoss and then talk to him. But suddenly Fenix began to say nonsense. He said no such Protoss exists. I want to claim that Fenix is wrong. So I told Fenix the sentence in the first box.

This is also awkward. There should be better interpretation and I need to think about it.

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    It might help our answerers more if you added some more information about your context (e.g., In what genre is your writing? Is it creative writing or academic writing? Philosophical or fantasy? Is your sentence the very first sentence? You mention that Protoss is a name of an alien, but it's used like the name of an alien race, why? etc.) -- I believe it's likely that seeing the sentence being written that way, our answerers will suggest rephrasing, which might not be what you're pursuing. Also keep in mind that typical information delivery strategies can be different in different languages. – Damkerng T. Mar 22 '16 at 7:11
  • Either a or the could be grammatical depending on the context and what you want to say. Sentences rarely stand in isolation. What else do you want to say? Is there more than one alien named Protoss in Aiur? in the universe? Also, the proposed sentence is awkward, and it would also be awkward if you used is instead of exists. The grammaticality of the sentence is not affected by using one of these verbs rather than the other. – Alan Carmack Mar 22 '16 at 7:29
  • Am I right to understand that you met an alien from Aiur, and you say the sentence to him? Or is it that you met an alien from Aiur, and he said to you that he is the tallest Protoss of all, and he speaks English? In short, have you met the tallest Protoss, according to your story? – CowperKettle Mar 22 '16 at 7:31
  • Thanks for the comments. In the first box, I was making a general statement, which is true. The audience is not specified. I assumed the situation that I met a Protoss who speaks English, which guarantees that my sentence is correct. – Hwang Mar 22 '16 at 7:44
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    Hwang, I don't think that your prospects as a science fiction writer are very bright: stick to the maths. I have answered your question based on the second example, which is, I believe what you really wanted to know. – JavaLatte Mar 22 '16 at 9:22
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Short answer: you have to use a rather than the because, although you know it exists, you don't know (at this stage) which one it is.

First, a little example that explains why sometimes we can use either a or the, and on other occasions we can only use one of them.

Imagine that your car tyre goes flat. Your passenger very helpfully says "there must be a puncture". He does not, and cannot, say "there must be the puncture". He knows it exists, but has not identified it- he don't know where it is. You go to the tyre repair guy who takes the wheel off and checks it out. He can say "there is a puncture in the side" or "the puncture is in the side". The tyre repair man can use a because he knows that there is only one, and he can use the because he has identified it- he knows where it is.

Normally, when we use superlatives, we are talking about a particular member of a set. It's clear which one we are talking about, for example "Mary is the tallest in her class". When we want to talk about the existence of something defined by a superlative, we may not know which member of the set it is: in these rare cases, we must use a.

First, an example for the laymen, then we will move on to the maths. Imagine a group of improbably well-spoken New York gangsters having a meeting:

A: If we want to grab a bigger share of New York, we need to do something that really gets us noticed

B: We could steal the biggest diamond in New York. That would get us noticed.

A: But we don't know which is the biggest diamond in New York!

B: Maybe, but there must be a "biggest diamond in New York". All we have to do is find it, then steal it.

Now let's move on to the math situation in your question.

It's only possible to talk about a superlative when talking about a closed set, for example "Mary is the tallest in her class". It is possible to leave the set unstated, for example "Mary is the tallest", but the class, or some other set, is implied.

If it's not a closed set, no tallest, or largest can be identified.

If it's an infinite set, then no tallest or largest can exist.

Integers are an infinite set

So, if condition A does something that defines a subset of all integers which is a closed set, and the set is not empty, and the members are unique, then a "largest integer" [within the subset defined by condition A] can, and indeed must exist.

You have to use a rather than the because you know it exists, but you don't know (at this stage) what it is.

If a largest integer must exist, then you can say

Assume Condition A. Then there exists a largest integer k

You could also define the subset using Condition B, then this sentence would also be true and correct:

There exists a largest integer k satisfying Condition B

And finally, if both condition A and condition B together define the subset, the following statement is both true and grammatically correct.

Assume Condition A. Then there exists a largest integer k satisfying Condition B.

QED.

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