My discovery about mathematics

The next theorem is wrong.
Theorem (?). If F(x) is a number-thoretic logical formula,
∃n,G(F(n))=n. (G(L) is Gödel number of L.)
A counter example. F(x)=(x=1).
s(1)=2, s(2)=3, … s(n)=n+1,…
G(1)≥1. G(n)≥ n→ G(n+1)=G(s(n))=2^G(s)3^G(n)≥2¹3ⁿ>2ⁿ⁺¹>n+1.
∴ G(n)≥n for ∀n(∈N).
G(F(n))=2^G(n)3^G(=)5^G(1)≥2ⁿ3¹5¹>n.
G(F(n))>n for ∀n(∈N).
Remark that you may not be able to write
F(G(P_x(x)))=((x=1)→F(G(P₁(1))))∧((x=2)→F(G(P₂(2))))∧… (infinitely)
as a logical formula.

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For Q(x), put L(x)=∃y∃z(sub_F"(1,x,y,z)∧N(x,z)∧Q(y)) and m=G(L(x)) and L=L(m).
Then, M |= L ⇔ M |= Q(G(L)).
M |= sub_F"(i,b,c,G(t)) ⇔ ∃r(a_i), M |= G(r(a_i))=b, M |= G(r(t))=c.
M |= N(b,c) ⇔ ∃m∈N, M |= m=b, M |= G(m)=c.