Detail of Diagonalization Lemma

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SUMMARY

The discussion centers on the Diagonalization Lemma as presented in C. Smorynski's "Self-Reference and Modal Logic" (Springer, 1985). Participants seek clarification on the relationship between coded terms and their names, specifically questioning whether the name of a coded term is equivalent to the coded term itself. The consensus establishes that while a coded term and its name may appear identical, they serve distinct functions: the coded term represents a value, while the name denotes the symbol used to reference that value.

PREREQUISITES
  • Understanding of Gödel numbering and its application in formal logic.
  • Familiarity with the concepts of coded terms and their names in mathematical logic.
  • Knowledge of the Diagonalization Lemma and its implications in self-reference.
  • Basic comprehension of modal logic as discussed in Smorynski's work.
NEXT STEPS
  • Study Gödel numbering in detail to grasp its role in formal systems.
  • Explore the implications of the Diagonalization Lemma in various logical frameworks.
  • Investigate the differences between coded terms and their names in formal logic.
  • Review modal logic principles to understand their application in self-reference contexts.
USEFUL FOR

Logicians, mathematicians, and philosophy students interested in formal systems, self-reference, and the nuances of coded terms in logic will find this discussion beneficial.

nomadreid
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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.

1756204018859.webp

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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nomadreid said:
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.
 

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