We argue internally in the effective topos. The argument is a bit long but it uses only standard techniques.

We need some notation first. Let ξ₀, ξ₁, ξ₂, ξ_3, ... be a standard enumeration of the sentences of Peano arithmetic. Let Prf(k, n) be the arithmetical statement coding “n is a Gödel code of a proof of ξₖ” and Rft(k, n) be the arithmetical statement coding “n is a Gödel code of a proof of ¬ξₖ”.

If a ∈ {0,1}* is a finite binary sequence, let |a| be its length. Define the binary tree

𝔾 := { a ∈ {0,1}* | ∀ k, n < |a| . (Prf(k, n) ⇒ aₖ = 1) ∧ (Rft(k, n) ⇒ aₖ = 0) }

In words, a ∈ 𝔾 when the sequence a approximates the decidability of the first |a| sentences of Peano arithmetic.

The tree 𝔾 is obviously infinite: to get a ∈ 𝔾 of any desired length m, consider the sequence a of length m defined by aₖ := 1 ⇔ ∃ n < m, Prf(k, n), noting that the condition is decidable and that Prf(k, n) and Rft(k, n) cannot both hold simultaneously.

Any infinite path α : ℕ → {0,1} through 𝔾 is computable by formal Church's thesis (valid in the effective topos), i.e., there is x ∈ ℕ such that ∀ k ∃ j . T(x, k, j) ∧ U(j) = αₖ, where T and U are Kleene's predicates (which are primitive recursive and so expressible in Peano arithmetic). Recall that T(x, k, j) expresses “j encodes a terminating computation of the x-th Turing machine on input k”. For every x and k there is at most one such j, and Peano arithmetic proves this fact.

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