My discovery about mathematics

Naoto Meguro : Amateur. MSC 2020: 03B10.

                              1

The predicate logic using the quantification principle (QP) has defect.
It cannot treat the commutative algebra and the homological algebra.
It proves Riemann hypothesis(RH) plausibly.

                              2

Let a be a free variable. Let p(s) and h(s) be predicate symbols. Let t be a closed term.
Let U be a consistent axiom system. Define p(s)=¬(U |- q(s)) and h(s)=(U |- ¬q(s)).
q(s) doesn't include a.
Theorem 1. If U |- ¬q(t)), ¬(U |- q(s)) for every term s if you use QP.
Proof. ¬p(a)⇒ U |- ∀xq(x) ⇒ ¬p(t). p(t)→p(a) is true. p(t)→∀xp(x) is true. h(t) is true.
h(t)→p(t) is true. p(t) is true. So ∀xp(x) is true. ♦
Assume that U is the axiom system of the theory of the commutative ring A and t=1
and q(s)=¬(∃u((u∈A)∧(us=1))). ¬(U |- q(s)) for ∀s∈A if you use QP. But U |- q(0).
This is contradiction. U has a model in which A=Z/2Z and is consistent.
Let V be the set of the theorems of the metamathematics treating p(s),h(s),U and QP.
Theorem 2. The axiom system V isn't consistent if you use QP.
Proof. ¬p(a)→(¬p(1)) ∈V. p(1)→p(a) ∈V. h(1)→p(1) ∈V. h(1 )∈V. p(1) ∈V.
p(a) ∈V. V |- p(a). V |- ∀xp(x). V |- p(0). ¬p(0) ∈V. V |- ¬p(0). V |- p(0)∧(¬p(0)).
V isn't consistent.♦
The metamathematics about the commutative algebra isn't consistent and proves
that the logical formulas are provable if you use QP. For example,
V |- (U |- ∃x∃y∃z((x∈A)∧(y∈A)∧(z∈A)∧(xyz≠0)∧(xⁿ+yⁿ=zⁿ))) for ∀n∈N if you use QP.
Assume that U is the axiom system of the theory of ⊗ and t=Z and q(s)=Z⊗s≅{0}.
U |- ¬q( Z). So ¬(U |- q(s)) for every term s if you use QP. But U |- q({0}).
The metamathematics about ⊗ isn't consistent if you use QP and U is consistent.
Assume that U is the set of the theorems of the complex function theory and t=2 and
q(s)=(s∈C)∧(ζ(s)=0)∧(Re(s)≥0)∧(Re(s)≠2^-1). If you use QP, ¬(U |- q(s)) for ∀s∈C.
This means that you cannot find a counterexample of RH in the consistent complex
function theory if you use QP.

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You must not use QP to evolve the consistent mathematics though it isn't natural.