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== Does S have to be a set of natural numbers? ==
Why not just say that a set ''S'' (of natural numbers, teacups, or whatever) is r.e. if <insert favored def. here>? Why (as is done in the article) restrict S to a set of nat. num's and attempt to accommodate other sets via goedel numbering? Infinite sets of natural numbers don't necessarily have godel numbers. Consider the set ''C'' of those that do. Now consider one that doesn't--of course, it's not possible to actually ''specify'' one--and call it ''s''. ''C'' U {''s''} is r.e., but not (not entirely) due to goedel numbering. In any case, it seems to me that the best method would be to present the general definition (where the nature of the elements of ''S'' is left unspecified) and then later, if necessary, restrict attention to r.e. sets of natural numbers.
...or more simply: ''C'' U {teacup} is r.e., but goedel numbering won't tell which is the ''x'' s.t. <math>f(x) = teacup</math> <small>—The preceding [[Wikipedia:Sign your posts on talk pages|unsigned]] comment was added by [[User:Futonchild|Futonchild]] ([[User talk:Futonchild|talk]] • [[Special:Contributions/Futonchild|contribs]]) 06:25, 28 February 2007 (UTC).</small><!-- HagermanBot Auto-Unsigned --> :The problem is that there is no definition of an r.e. set of teacups, because no computer has ever been seen to output a teacup. [[User:CMummert|CMummert]] · <small>[[User talk:CMummert|talk]]</small> 13:14, 28 February 2007 (UTC)
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