The Berry paradox

This famous paradox is the idea of "defining" a natural number n as, for example "the smallest number not definable in less than twenty words". This would define in 10 words a number... not definable in less than 20 words.
By lack of mathematical precision, this reasoning does not bring a contradiction in mathematics. Let us formalize this paradox into the form of a proof of the weak Truth Undefinability theorem (using definitions from there). For convenience we shall modify the reasoning to use not the concept of number n defined by a formula, but that of number n defined as the smallest k such that F(k).
Let M⊨T in which we shall implicitly interpret T.
Instead of using Godel's hard work of developing 1T from Z₁ (arithmetic), we only need the easier fact that Z₁ can be developed from 1T and thus from T as follows:

First developing from 1T a description of Z₁, let N₀ be defined in this description as the set of "closed terms of Z₁" (using no other symbols than 0,1,+,⋅). Then a suitably defined binary relation ≈ on N₀ intuitively meant as "these terms designate the same number" lets such terms play the role of the numbers they designate. So N₀ plays the role of a model N of Z₁ once ≈ is used in guise of equality symbol in the axioms of Z₁ extended by the axioms of equality.

Let N₀ be the set of closed terms of the so developed Z₁. So ∀t∈N₀, "t"∈N₀ and the image of "t" in N is the value of t.
Let H be the notion in T playing the role of range (thus contain the values of quotes) of "formulas of T with only one free variable with type N".
Let h denote some "size measure" functor from H to N, so that

∀m∈ℕ, {F∈H | h(F) ≤ "m"} is finite (it has a number of elements in ℕ).

(If the language of T was finite, some other definition of h than "size" can be used to make things work.)
Let C∈S₁ with the following quality requirement (otherwise more complicated tricks could still make things work...):

For any two "statements" A, A'∈S obtained from an a∈H by respectively replacing its free variable by t, t'∈N₀ (i.e. they are a(t), a(t') whose use of parenthesis is interpreted), we have
t ≈ t' ⇒ (C(A)⇔C(A'))
Thus we can use C as a binary relation C(a,n)⇔C(A) with arguments a∈H and n∈N where t represents n.

Let G be the formula of T with free variables n, m of type N defined as

∀F∈H, h(F) ≤ m ⇒ (C(F,n) ⇒ ∃k<n, C(F,k))

As any big number can be expressed by some arithmetical term whose size is significantly smaller,

∃t∈N₀, h("G(n,t)") ≤ (value of t).

Fixing such a t interpreted in ℕ as m and in N as m, let B the formula G(n,t) with free variable n.
As (C(F,n) ⇒ ∃k<n, C(F,k)) means that n is not the smallest k such that C(F,k), by the above finiteness property of h we have ∃n∈ℕ, B("n") .
Let n be the smallest of them:

B("n")∧∀k<n, ¬B("k").

As moreover "B"∈H and h("B") ≤ m we have

C("B","n")) ⇒ ∃k<n, C("B","k")
(B("n")∧¬C("B","n"))∨(∃k<n, C("B","k")∧ ¬B("k"))
∃k≤n, B("k") ⇎ C("B","k") ⇔ C("B("k")")
∃A∈S, M⊨(A ⇎ C("A")).∎