My discovery about mathematics

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I indicate that the set theory isn't consistent by Gödel's theorem. You can prove any conjectures
by using the set theory.

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Theorem 1. The set theory isn't consistent.
Proof. Assume that the set theory is consistent. Let X be a countable and consistent axiom sytem
of the set theory like ZFC. Let's write ∅=0, x∪{x}=x+1. You may set up that
(n(∅)=0)∈X and ∀x∀y((x∉y)→(n(y∪{x})=n(y)+1)) ∈X and ∀x∀y((x=y)↔∀z((z∈x)↔(z∈y))) ∈X.
n(x) is the number of the elements of the set x.
Let d_i (i∈N) be the free individual symbols. The mathematical axioms in X don't include the d_i's.
Put d=d₁,d"=d₂. Let D be the countable set of the closed terms.
X∪{d"∉d,n(d)≠n({d"}∪d),1∈d,…n∈d} has a model M_n for which
M_n |= (d={1,2,…,n})∧(d"=n+1) and is consistent for ∀n∈N.
So X∪{d"∉d, n(d)≠n({d"}∪d),1∈d,2∈d,…(infinitely)} is consistent and has a countable model M
whose object domain is D by Gödel's theorem. M |= n(d)≠n({d"}∪d) M |= n∈d for ∀n∈N
Put E(c)={z|(z∈D)∧(M |= z∈c)} for c∈D. Define M" |= c=E(c) for c∈D.
For c∈D and c"∈D,M |= c∈c" ⇔ c∈E(c") ⇔ M" |= c∈E(c")=c".
M |= (c=c")⇔ ∀u(M |=(u∈c)⇔ M |= (u∈c")) ⇔∀u((u∈E(c))⇔(u∈E(c")))⇔E(c)=E(c")
⇔ M" |= c=c" for c∈D and c"∈D. M" |= n(d)≠n({d"}∪d) and M" |= n∈d for ∀n∈N.
N⊂E(d)⊂D E(d) is a countable set. You can set up that
M" |= d=E(d)={c₁}∪{c₂}∪…(infinitely) (c_i∈D, M |= c_i∈d,M |= c_i≠c_k for i≠k).
M" |= n(c₁)=1, M" |= n({c₁}∪{c₂})=1+1,…,M" |= n({c₁}∪{c₂}∪…(infinitely))=1+1+…(infinitely)
={0}∪{1}∪{2}∪…={0}∪N. M" |= n(d)={0}∪N.
Similarly,M" |= n({d"}∪d)=n({d"}∪{c₁}∪{c₂}∪…(infinitely))=1+1+…(infinitely)={0}∪N.
M" |= n(d)=n({d"}∪d) This is contradiction.

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You may have to treat only finite ones as Gauss insisted. The set theory may be a ghastly fake.