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Hello can someone explain me exacty how in this fragment of the paper (SELF-REFERENCE AND THE LANGUAGES OF ARITHMETIC, RICHARDG.HECK,JR.):

  • (9) Tr(x) ≡∃y(rhs(x,y)∧¬Tr(y)), where rhs(x,y) is a formula representing the relation: y is (Gödel number of) the right-hand side of the biconditional (whose Gödel number is) x. Diagonalization then yields a formula G such that PAs (Paeno arithmetic , 's' stands for standard language)

proves:

  • (10) G≡[Tr(G)≡∃y(rhs(G,y)∧¬Tr(y))], where, as usual, G abbreviates the numeral denoting the Gödel number of the formula G.

How (10) can be derived from (9) trough Diagonalization?

  • See e.g. Gödel’s Incompleteness Theorems. – Mauro ALLEGRANZA Oct 30 '19 at 13:33
  • I don't see how following the steps of the "diagonalization lemma" referenced there takes one from (9) to (10) – Noname Oct 30 '19 at 13:40
  • But then G in (10) must be equivalent to (9) , how then in (10) is it equivalent to the right-hand side of the equivalence(10)? – Noname Oct 30 '19 at 13:54
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See e.g. Gödel’s Incompleteness Theorems : Diagonalization: it is a general result of FOL that :

Let A(x) be an arbitrary formula of the language of F with only one free variable. Then a sentence D can be mechanically constructed such that F ⊢ D ≡ A(⌈D⌉).

In Heck's paper, page 2, the author applies this general result to formula (9) above (of system PA), which is of "form" A(x) [variable y is bound].

Thus, there is a sentence (closed formula) G such that : PA ⊢ G ≡ A(⌈G⌉), i.e. a formula G such that PA proves:

G ≡ [T (⌈G⌉) ≡ ∃y(rhs(⌈G⌉, y)∧¬T (y))].

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  • Thanks, altough this: G ≡ [T (G) ≡ ∃y(rhs(G, y)∧¬T (y))]. Supposed to be --> G ≡ [T (⌈G⌉) ≡ ∃y(rhs(⌈G⌉,y)∧¬T (y))]. – Noname Oct 30 '19 at 14:43

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