ZFC vs NBG: A Comparison of Mathematical Axiom Systems

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Which one do you prefer? Which do you think is better?
 
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Neither one is better than the other, nor is one preferable to the other.
 
Er...ZFC, because it matches my curtains.
This doesn't really strike me as a "preference issue". You can prove the same things in both, so, as a fan of mathematics, I really see no reason to pick a side, since it's unlikely that I'll ever encounter a level of argument "low" enough that the difference is relevant.
 
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Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.
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