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    $\begingroup$ I just wrote a long answer to this question, but it was closed just as I was about to click submit. Can we re-open please? I think that there are a number of very interesting issues here. $\endgroup$ Commented Oct 29, 2010 at 12:48
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    $\begingroup$ Meta thread: web.archive.org/web/20131023075657/http://tea.mathoverflow.net/… . I have also voted to reopen. $\endgroup$ Commented Oct 29, 2010 at 13:07
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    $\begingroup$ I disagree with the continuing votes to close. The topic of definability is mathematically rich and forms the basis of huge parts of model theory, particularly where it connects with algebra and algebraic geometry, such as in the deep work of o-minimality. In the set-theoretic context, various technical meta-mathematical issues become prominent. The question is well-motivated, sincere and has mathematically interesting answers. $\endgroup$ Commented Oct 30, 2010 at 22:43
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    $\begingroup$ In particular, I can imagine further technical answers arguing the line that in a model of $V=HOD$, the definable objects indeed form an elementary substructure of the universe, fulfilling the OPs observation that statements of analysis can be viewed as ultimately about definable objects. $\endgroup$ Commented Oct 30, 2010 at 22:43
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    $\begingroup$ Similar issues with definability lay at the heart of this MO question: mathoverflow.net/questions/34710/… $\endgroup$ Commented Oct 30, 2010 at 23:27