My discovery about mathematics

Naoto Meguro :Amateur. MSC2020:13B25. e-mail :

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The theory of the polynomial ring isn't consistent if you assume the axioms of ZF.
The algebraic geometry treating the polynomial ring isn't consistent and proves every
conjecture with its denial if you assume the axioms of ZF.
The theory of the integral functions isn't consistent if you assume the axioms of ZF.
Riemann hypothesis(RH) is nonsense then. ZF is harmful.

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Let's express the polynomial ring by the direct sum. (a,b)+(c,d)=(a+c,b+d). x(a,b)=(xa,xb).
Let R be a ring. R[x]=R⊕xR[x]={(r₁,x(r₂,x(r₃,…|r_i∈R, r_i=0_R except finite ones.}.
Theorem 1. The theory treating R[x] isn't consistent if you assume the axioms of ZF.
Proof. (a,b)={{a,a},{a,b}}∋{a,b}∋b. xⁿR[x]=xⁿR⊕xⁿ⁺¹R[x].
0_xⁿR[x]=(0_xⁿR,0_{xⁿ⁺¹R[x]})∋{0_xⁿR,0_{xⁿ⁺¹R[x]}}∋0_{xⁿ⁺¹R[x]}.
T={0_xⁿR[x]| n∈ω} and U={{0_xⁿR,0_{xⁿ⁺¹R[x]}}| n∈ω} are sets by the axiom of replacement.
T∋0_xⁿR[x]∋{0_xⁿR,0_{xⁿ⁺¹R[x]}}∈U. U∋{0_xⁿR,0_{xⁿ⁺¹R[x]}}∋0_{xⁿ⁺¹R[x]}∈T.
Put S=T∪U. (S≠∅)∧∀y((y∈S)→(y∩S≠∅)).
This is a contradiction by the axiom of regularity. ♦
The theory treating R[x₁]…[x_n] isn't consistent if you assume the axioms of ZF.
When the x_i's are indeterminates, the algebraic geometry treating V=R[x₁]…[x_n]/P"
treats ∪V=R[x₁]…[x_n] and isn't consistent if you assume the axioms of ZF.
Let's express the module of Maclaurin series of the integral functions by H=C⊕xH.
0_xⁿH∋{xⁿ0,0_xⁿ⁺¹H}∋0_xⁿ⁺¹H for ∀n∈N. This leads a contradiction by the axioms of ZF.
The complex function theory treating H isn't consistent if you assume the axioms of ZF.
(x-1)ζ(x) is an integral function. You can write (x-1)ζ(x)=∑_i∈Nb_ix^i-1 (b_i∈C) and can put
b=(b₁,x(b₂,x(b₃,…∈H. (x-1)ζ(x)=(x-c)∑_i∈Nc_ix^i-1 ⇒b=(0,x(c₁,x(c₂,x(c₃,… -c(c₁,x(c₂,x(c₃,….
RH=∀c((c∈C)∧∃h((h∈H)∧(b=(0,xh)-ch))→(Re(c)=2^-1)∨(Im(c)=0)) and ¬RH are proved
in the complex function theory if you assume the axioms of ZF. RH is nonsense then.
p(c,d)=0 ⇔ p(x,y)=(x-c)q+(y-d)r. Set up that R=Q[y] and p∈R[x] and
P=∀c∀d((c∈Q)∧(d∈Q)∧∃q∃r((q∈R[x])∧(r∈R)∧(p=(0_R,xq)-cq+((y-d)r,0_R[x])))→(cd=0)).
¬P is a denial of Fermat conjecture if p=(yⁿ⁺²-1,(0_R,(,…,(0_R,(xⁿ⁺²,0_R[x])…)=xⁿ⁺²+yⁿ⁺²-1.
P and ¬P are proved then if you assume the axioms of ZF by theorem 1.

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If R[x₁]…[x_n] is a ghastly fake as Gauss said, the algebra after Abel is nonsense.
You couldn't admit it and must correct the set theory.
The algebraic geometry,the complex function theory and RH are nonsense until it.