Undergrad Detail of Diagonalization Lemma

[nomadreid]

TL;DR

Is the name of a codification the same as the codification?

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.

 

[Gavran]

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.

The question is more general than it looks and there is a difference between the coded term and the name of the coded term.

For example,
The coded term is 1.
The name of the coded term is 1.
In the first case 1 presents the coded term while in the second case 1 presents the symbol which is used to denote the coded term.