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I understand or rather, I have heard, that Gödel as part of his incompleteness theorem enumerates all statements. But how do you single out those that can be used in a test of provability. You will encounter such statements as “===” etc?

Are these considered true (since they obviously don’t contradict anything) or how is that done?

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"===" is not a statement at all. Part of Gödel's work defines a function for deciding whether a given string of symbols is a syntactically correct (known as "well-formed" in the logic jargon) formula. – Henning Makholm Oct 29 '13 at 22:36

You are correct that the series of symbols $===$ has a number in Gödel's system. His paper has a series of functions and predicates, each building on the earlier ones. He builds up from defining a predicate Prime(n), which is true if $n$ is prime, later Term(n), which is true if $n$ represents a term (a variable or constant), and eventually to wff(n), which is true if $n$ represents a well formed formula. Even later comes axiom(n), which is true if $n$ represents one of the axioms of PA and so on. If $m$ represents ===, wff(m) will be false.

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So the theorem is restricted to well formed formulas with no internal wff identification process. – Mikael Jensen Oct 30 '13 at 9:06
It has an internal wff identification. The point of the theorem is to construct an undecidable wff. Only wffs can have truth values. – Ross Millikan Oct 30 '13 at 13:54

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