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Which of the following is more correct to use in a scientific paper?

Consider the set A..., Recall that the set A..., Put the set A...

or

Let us consider the set A..., Let us recall that the set A..., Let us put the set A...

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Please give us more context. The choice of phrase is based on the situation where it will be used. –  Matt Эллен Mar 8 '12 at 13:41
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It's common practice in academic/scientific writing to avoid personal pronouns as much as possible (though one would also wish to avoid excessive use of the passive voice which this might lead to). So unquestionably go for the first version. –  FumbleFingers Mar 8 '12 at 15:47
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2 Answers

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In a scientific paper, it is needless verbosity to repeat "let us" so many times, as in your second example. (You might use "let us" as a transitional phrase, as in "Now let us consider..."; but usually not otherwise.) For that matter, if the interval between mentions of A is not long and you have not overloaded the symbol A, don't repeat the words "the set". Thus:

Consider set A..., Recall that A..., Let A...

The reason for defining a symbol A as standing for a certain set is to allow brevity and clarity in further exposition that depends on that set.

In a comment you say that you use put as in put A = {...}. My preference is to write "let" rather than "put " when defining a set; for example, "Let A be (1,4)" if A is the open set of reals between 1 and 4. I would write (eg) "Put x=7" if something special happens in the case where x is 7, and otherwise would write "Let x=7" or "Take x=7", but this may be just a matter of personal preference.

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Thank you very much –  user18889 Mar 8 '12 at 18:51
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"Consider the set" is fine
"Let us consider the set" is also fine, but a bit wordy
"Recall that the set" or "Let us recall that the set" are used only if that particular set was referenced before and you want to mention it again
"Put the set " or "Let us put the set" are not good

The simplest and most frequently used way of saying it is Let A be the set (of) ...

edit: in the view of the new edit, those expressions that include "let us" are used less in scientific literature than the ones without (more than 10 times less*), but they are used often enough to consider them correct. The reason "let us" is used is for deference, shifting the attention from "you" to a more polite "we".
*search on Google or Google Scholar

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Thanks, the last is also used for example to say: put A = {...}. - "Reccall" (or others mentioned above) means that you recall or that we recall ? –  user18889 Mar 8 '12 at 14:57
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"Put the set" is fine if it's not used to define the set. However, I wouldn't say "put A = {...}" unless you're describing an algorithm. –  Peter Shor Mar 8 '12 at 16:45
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