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I sometimes encounter proofs as the following:

$\exists n\in\mathbb N$ such that …. Then $n$…

But the fact in that $n$ is not defined in the second sentence. A correct proof would contain instead:

$\exists n\in\mathbb N$ such that …. Let such a $n$. Then $n$…

Is there a notation to mean "there exists and let"?

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    In my opinion, writing "there exists..." in full letters is sufficiently understandable. – TheSilverDoe Oct 21 '20 at 10:38
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    Interesting observation. However, I don't believe I've seen the first version much, maybe not ever, at least not in an actual publication as opposed to internet blogs, discussion groups, and unpolished manuscripts. In thinking about how I handle this, I think I usually say something like: "Choose $n$ $\ldots$" or Let $n$ be given". However, to me it seems quite unnatural to state that (or to observe why we know that) $n$ exists and then follow by using $n$ without introducing $n,$ so it's not something I have to think about to avoid. Indeed, doing it like you began with seems strange to me. – Dave L. Renfro Oct 21 '20 at 10:51
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    Does this answer your question? Basic question about proofs with the goal $\exists x P(x)$ – Mauro ALLEGRANZA Oct 21 '20 at 10:52

3 Answers3

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You are asking about notation for "there exists and let". Note that the first part of this, "there exists," is a statement telling how something is, while "let" is an imperative telling the reader what to do. They have different roles in the proof and they have somewhat different formal languages.

The formal notation of statements contains things like quantifiers ($\forall,\ \exists$) and implications ($\implies$), while the formal notation for a proof rather is a deduction tree containing statements: $$ \dfrac { \begin{matrix}\\A \lor B\end{matrix} \quad \begin{matrix}\\ [A]\end{matrix} \quad \dfrac{[B] \quad B \rightarrow A}{A} } { A } $$ (The example proves $A$ from $A \lor B$ and $B \rightarrow A.$)

In a proof, "there exists" will be true either by some definition or by some other theorem or lemma, and the text would read "By definition/theorem $X,$ there exists $x$ such that $P(x).$ Take such an $x.$ Then ..." Here's a deduction tree for that: $$ \dfrac { \begin{matrix}\\ \triangledown \\ \hline \exists x\ P(x)\end{matrix} \quad \begin{matrix} [P(x)] \\ \vdots \\ Q \end{matrix} } { Q } $$ where $\triangledown$ denotes "definition/theorem $X$" and $\vdots$ that there is some deduction tree deriving $Q$ from $P(x).$ The left part above the horizontal line corresponds to "By definition/theorem $X,$ there exists $x$ such that $P(x),$ and the right part corresponds to "Take such an $x.$ Then $P(x)$ so ... Thus $Q(x).$"

md2perpe
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In $(\exists x \in X)R(x)$ in $R(x)$ $x$ is actually defined, because it is same as $(\exists x) (x \in X \land R(x))$.

Addition. If we consider $(\exists x \in X)R(x) \land S(x)$, then it same as $(\exists y \in X)R(y) \land S(x)$.

zkutch
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    I believe the concern is more about things like "$(\exists x\in X)R(x)$. Then note that $x+1>\pi$." – Mark S. Oct 21 '20 at 10:49
  • $R(x)$ is exactly something like "$x+1>\pi$". What you wrote I understand as $(\exists x\in X)(x+1>\pi)$ and it is same with $(\exists x)(x\in X \land (x+1>\pi))$. Or you mean something else? – zkutch Oct 21 '20 at 10:53
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    As written by @Mark S, the scope of the quantifier in $(\exists x\in X)R(x)$ ends with $R(x).$ Indeed, this is further reinforced linguistically since the English sentence ends (note the period, followed by a capital letter used in "Then") and a new sentence begins with "Then note that". – Dave L. Renfro Oct 21 '20 at 10:57
  • Of course "scope of the quantifier $(\exists x\in X)R(x)$ ends with $R(x)$" is correct and I mean that all we want to say about $x$ should be in $R(x)$. If/when we start new sentence, then it will be under some other quantifiers. – zkutch Oct 21 '20 at 11:03
  • Constructive critique is welcome. – zkutch Oct 21 '20 at 11:04
  • I had started writing my answer before yours appeared, but was pulled away. I think/hope that it's the sort of thing the OP was looking for, whether or not you agree that "choose" is the best solution (or even that a solution is needed). – Mark S. Oct 21 '20 at 11:06
  • @Mark S. I wrote answer to you comment seeing only one sentence. As I wrote above, I see solution to include everything what we want to say about $x$ in $R(x)$. And if/when we start new sentence, then it should have its quantifiers. If your example rewrite as "$(\exists x\in X)(R(x) \land$ (Then note that $x+1>\pi$))", then we have not ambiguities with $x$. If you start new sentence independently from $(\exists x\in X)(R(x)$, then, of course you should have new condition on $x$, otherwise it have no sense. And let me say, that imho I expect down votes after discussion, not before. – zkutch Oct 21 '20 at 11:16
  • I don't think it matters who downvoted, but for the record it wasn't me (and the downvote also appeared after my initial comment). But someone who agrees with me that your answer probably doesn't address the intended question is within their rights to downvote and not comment. "...otherwise it have no sense." I believe you now express the same objection to improper writing that the OP was trying to bring up. Since "$(\exists x\in X)(R(x))$. Then note that $x$..." would "have no sense", how should the writing be fixed? – Mark S. Oct 21 '20 at 11:34
  • It is already in my previous comment. As to your last question, then I already added it to answer: if is written $(\exists x \in X)R(x) \land S(x)$, then it same as $(\exists y \in X)R(y) \land S(x)$ and I see possible ways, depending on what is wanted to be expressed: include $S$ in $R$, write separate quantifier to $S$, write general quantifier to whole sentence - do you see some other? First way is my answer, how I understand text without additional info. – zkutch Oct 21 '20 at 12:05
  • "do you see some other?" Yes, as given in my answer. "Choose" is a standard way to have a quantifier-like meaning that can span several sentences. – Mark S. Oct 21 '20 at 12:29
  • De facto your "Choose", and especially, that you confirm your desire to extend/span quantifier to several sentences, is same as I have in my answer: you include $S$ in $R$. Here I see suggestion to move discussion to chat - I am not against, if you also agree. – zkutch Oct 21 '20 at 12:39
  • @zkutch I think you did not understand the OP's question. The question is not "is $x$ defined in $R(x)$ if I write ($\exists x,R(x)$) ?". The question is rather the following : if you want to deal with a $x$ satisfying a propoerty $R(x)$, it is not sufficient to tell that $x$ exists (this is what the statement ($\exists x,R(x)$) means) : you also have to pick such a $x$. The OP's question is : is there a short way to say "such a $x$ exists and I pick it". – TheSilverDoe Oct 21 '20 at 15:18
  • I've been away for a few hours, and just returned. FYI, I didn't downvote either (indeed, I very, very rarely downvote), but my understanding about this is the same as explained by @TheSilverDoe just before this comment. – Dave L. Renfro Oct 21 '20 at 15:41
  • @TheSilverDoe. Do not hurry with accusals - let's discuss facts. For words "pick it", as well as for word "choose", we have formal way "exists" quantifier. So, I am telling, that $R(x)$ or include also your "I pick it" or it have no sense separately from first sentence. Continuing your ideas, when we said, that $x$ exists and then we want to deal with $x$, then we include next sentences under first quantifier. Do you see any other way? – zkutch Oct 21 '20 at 15:42
  • @zkutch Well, I disagree that the quantifier $\exists$ includes the choice of a particular element. For me, it is the same as saying that a set is non-empty. Just stating that the set is non-empty does not mean that you picked an element in it. It means that you can pick an element in it, but not that you actually do. – TheSilverDoe Oct 21 '20 at 15:47
  • @Dave L. Renfro. I am not worring about down voting itself, but about down voting without arguments and with wrong arguments. So, lets analyse. As I wrote sentence $(\exists y \in X)R(y) \land S(x)$ can be formalized in 3 ways: include $S$ in $R$, write separate quantifier to $S$, write general quantifier to whole sentence. We have not "quantifier-like" "choose" or "pick it". We have existence quantifier and for me most native is first way, which is my answer. – zkutch Oct 21 '20 at 15:51
  • @TheSilverDoe. Let's take more working example with set having elements. When we say " $[01]$ is not empty and now take $x$ from $[01]$, such that $x>\frac{1}{2}$" it mean $(\exists x \in [01])(x>\frac{1}{2})$. Can you write other formalization? – zkutch Oct 21 '20 at 15:59
  • @zkutch No : the sentence "$(\exists x \in [0,1])(x>\frac{1}{2})$" means only that "$\lbrace x \in [0,1], x > 1/2 \rbrace$ is not empty", it does not mean that you take an element in it. – TheSilverDoe Oct 21 '20 at 16:01
  • I would add something : writing ($\exists x, R(x)$) is absolutely equivalent to writing ($\exists y, R(y)$). The variable $x$ (or $y$) is a bound variable, which means that the sentence does not define a particular element. – TheSilverDoe Oct 21 '20 at 16:05
  • @TheSilverDoe. Of course properties of $x$ defines some sets(when define) and when we deal with $x$, formally we are considering exactly these sets. Can you write your formalization of "take"? And for second comment: above I wrote already, that $(\exists x \in X)R(x) \land S(x)$ is same with $(\exists y \in X)R(y) \land S(x)$ and we are discussing ways to give sense to $S(x)$ i.e. where to put quantifier for $x$. I brought 3 ways. Look up, please. And again, I see suggestion to move discussion to chat - I am not against, if you also agree. – zkutch Oct 21 '20 at 16:12
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I agree with your concern about $n$ not being defined in the second sentence of the first quote. Because the scope of $\exists$ would be to the end of the logical statement/sentence, and not extend/bind $n$ beyond that.

And while you'd be understood if you wrote "There exists" in words as suggested by TheSilverDoe in a comment, doing that produces a rare distinction between symbols and their usual readings, which I would prefer to avoid.

I prefer using the word "choose" (and then reserving "let" for universal quantification), as in "Choose $n\in \mathbb N$ such that $n>N+3$. Then note that $n\ge N+2$, so that...". "Choose a natural $n$ such that..." is fine as well. Or after it has been justified that something with a property exists, "So we can/may choose $n$ so that..."

You can see usage of "choose" recommended/exemplified in Doug West's mathematical writing guide The Grammar According to West in his points on expressions as units and "let x,y be".

Mark S.
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  • If you write "Choose $n \in \mathbb{N}$ such that $n > N+3$", you should theoretically prove before that such a $n$ exists... – TheSilverDoe Oct 21 '20 at 11:15
  • @TheSilverDoe I basically agree, and examples almost like that come up when I teach an introduction to proofs or similar. In less artificial examples, the context hopefully justifies the condition in an obvious way. – Mark S. Oct 21 '20 at 11:24