Is there a standard (or at least common) symbol in computability theory used to indicate that $x$ enters the c.e. set $W\_e$ at stage $s$, i.e., $x \in W\_{e,s} - W\_{e,s-1}$ (at least for $s \neq 0$)? 


I was thinking of using $x \searrow\_s W\_e$ but I was worried that would be too confusing with its use in the automorphisms of $\mathscr{E}$ (i.e. $X \searrow Y$ are the elements that enter $X$ and then $Y$ (and I vaguely remember that $X \searrow\_s Y$ is used for something).


Any suggestions for good notation? I'd consider using just use $x \in W\_{e,s} - W\_{e,s-1}$ (though a binary relation would be nicer) except for the annoying need to adopt the convention that $s$ can't be zero.



Computability Theory Notation For Entering A Set At A Stage
(I'm going to write "$V$" for the arbitrary c.e. set, since I'm rather wedded to the convention that "$W\_e$" refers to the domain of the $e$th partial computable function.)




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The "$\searrow\_s$" notationis what I've seen before - I think it's in Soare's old book, although I don't have my copy handy to check. In my opinion the automorphism topic is sufficiently narrow that if you're writing a paper not in that context, you can use the arrow notation freely. And anyways, there's no actual notation ambiguity anyways because of the different types of object (and the lack of subscript in the automorphism usage). **So that's what I'd recommend**.


If you are writing in that context, though, and don't want to use two $\searrow$s, I don't think there's another notation established so you'll just have to pick something and tell the reader. Luckily, though, I think there are a few good choices here:


* You could attach the stage to the "$\in$"-relation: "$x\in\_s V$" means that $x$ enters $V$ at stage $s$. *If I recall correctly I've actually seen this one used before.*
* There's also the "inclusion-arrow" $\hookrightarrow$: you could write "$x\hookrightarrow\_s V$," and I think that's actually quite readable. It does clash with the general usage of $\hookrightarrow$ elsewhere, but I actually don't recall seeing that symbol in computability theory (except for category-theory-oriented computability). *Disclaimer: this is actually what I wrote in my old notes way back when, I really like how it looks.*
* Finally, you could introduce new notation for the sets $V\_s\setminus V\_{s-1}$ - something like "$V\_s^!$" or "$V\_{=s}$," perhaps.


Of course if you go this route you'll have to explain that usage - but honestly it's worth reminding the reader of it anyways.


