My discovery about mathematics

Naoto Meguro : Amateur. MSC 2020: 03B10.

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The predicate logic has defect. You must correct it a little.

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Theorem 1. The predicate logic or the set theory isn't consistent.
Proof. Assume that predicate logic and the set theory are consistent.
Let c and d be individual symbols. Let X={c∈d,d∉d} be an axiom system.
X has a model in which c=∅ and d={∅} and is consistent.
Let a be a free variable. Let p(t) be a predicate symbol. Define p(t)=(X |- t∉d).
p(a)=(X |- a∉d)⇔(X |- ∀x(x∉d))⇒(X |- c∉d)=p(c). p(a)→p(c)) is true.
¬p(c)→(¬p(a)) is true. ¬p(c)→∀x(¬p(x)) should be true. X |- c∈d and X is consistent.
So ¬p(c) is true.∀x(¬p(x)) should be true. ¬p(d) should be true. But p(d) is true.♦
Wrong one is quatification principle q→q(a). ⇒ q→∀xq(x).
Let's construct the predicate logic without using the free variables.
Set up that T(q₁,…,q_n) is a logical axiom when T(p₁,…,p_n) is a tautology and the q_i's
are closed logical formulas. (The p_i's are propositional variables.)
Set up that ∀xq(x)→q(t) is a logical axiom when t is a closed term and ∀xq(x) is a
closed logical formula.
Set up that ∀xq(x) is a logical axiom when q(c) is a logical axiom and c is an individual
symbol.
Set up that ∀x(q(x)→r(x))→(∀yq(y)→∀z r(z)) is a logical axiom when ∀xq(x) and
∀xr(x) are closed logical formulas. The rule of inference is only deduction principle.
Theorem 2. If the axiom system Y doesn't include the individual symbol c,
Y |- q(c) ⇒ Y |- ∀xq(x).
Proof. The case that q(c) is an axiom is trivial. If Y |- q(c) by Y |- r(c)→q(c) and Y |- r(c),
you may assume that Y |- ∀x(r(x)→q(x)) and Y |- ∀xr(x) by induction.
Y |- ∀x(r(x)→q(x))→(∀xr(x)→∀xq(x)). Y |- ∀xr(x)→∀xq(x). Y |- ∀xq(x).♦
The following theorems are formed in the above predicate logic treating only closed logical
formulas.
Theorem3. Y∪{q} |- r ⇒ Y |- q→r.
Theorem 4. If Y∪{q} isn't consistent, Y |- ¬q.
Theorem 5. If Y is consistent and doesn't include the individual symbol c,
Y∪{q(c)→∀xq(x)} is consistent.
You can get the completeness theorem without using the free variables.

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Known theorems won't be lost by the above correction though new theorems won't be
proved.