My discovery about mathematics

                            1

I prove that the real number theory isn't consistent. It's led by contradiction by using infinite
symbols. As similar one,I prove that you must not use the countable union of sets. You will see
defect of the homological algebra by it. You must examine many theorems.

                            2

Set up that F={0",1",2",3"}≅Z/4Z, O=2F={0",2"}. Define A+B={a+b|a∈A,b∈B} for A,B∈F/O.
F/O is a module then. O is the zero of the module.
Theorem 1. O+O+…(infinitely)≠O
Proof. O+O+…(infinitely)={n₁+n₂+…(infinitely)|n₁∈O,n₂∈O,…}∋e=2"+2"+…(infinitely).
(2"+2"+2")=2"+(2"+2"),(2"+2"+2"+2")=2"+(2"+2"+2"),… Repeating this infinitely,you get e=2"+e.
0"≠2"+0" So e≠0". Similarly e≠2". e∉O. So O+O+…(infinitely)≠O. ♦
Theorem 2. If c_i is a set for ∀i∈N, c₁∪c₂∪…(infinitely) isn't a set.
Proof. Put C=c₁∪c₂∪…(infinitely). If C is a set,c_i∈2^C for ∀i∈N. Define S(c)=O for ∀c∈2^C.
For c,c"∈2^C,S(c∪c")=O=O+O=S(c)+S(c"). So S(c₁∪c₂)=S(c₁)+S(c₂),
S(c₁∪c₂∪c₃)=S(c₁)+S(c₂)+S(c₃),…,S(c₁∪…∪c_m)=S(c₁)+…+S(c_m),….
Repeating this infinitely,you get S(c₁∪c₂∪…(infinitely))=S(c₁)+S(c₂)+…(infinitely).
S(C)=O+O+…≠O. But S(C)=O for C∈2^C. This is contradiction.♦
You cannot evolve the measure theory.
For the set s, s∪s∪…(infinitely) isn't a set. s≠s∪s∪…(infinitely)
though ∀x((x∈s)↔(x∈s∪s∪…)).You will have to use only finite symbols to express the set.
Let A be a commutative ring.
Theorem 3. For the infinite set S, A^(S) isn't a set.
Proof. A^(S)={0}^S∪L₁∪L₂∪…(infinitely) (L_m={f|(f∈A^S)∧∃s₁…∃s_m∀s((f(s)≠0)↔(s=s₁)∨…∨(s=s_m))})
Every L_m is a set. A^(S) isn't a set by theorem 2.♦
You cannot get the projective resolution of the infinite A module M by A^(M)→M→0 (exact).
Theorem 4. For the A modules M and N,M⊗_AN isn't a set if M or N isn't a finite set.
Proof. If M⊗_AN is a set,∪(M⊗_AN)=A^(M×N)(or =Z^(M×N)) is a set. This is contradiction by theorem 3
if M×N isn't a finite set.♦
M,N,A must be finite to define Tor^A_n(M,N) (n≥1).
Theorem 5. N={1}∪{2}∪{3}∪…(infinitely) isn't a set.
Proof.1={∅},2=1∪{1},3=2∪{2},… are finite sets. {1},{2},{3},… are sets too.
N isn't a set by theorem 2.♦
Theorem 6. The standard real number theory isn't consistent.
Proof. Put r(c)=O for ∀c∈R. For c,c"∈R, r(c+c")=O=O+O=r(c)+r(c").
Assume that a_i=0 or 1 for ∀i∈N. r(a₁/2+a₂/2²)=r(a₁/2)+r(a₂/2²),
r(a₁/2+a₂/2²+a₃/2³)=r(a₁/2)+r(a₂/2²)+r(a₃/2³),…, r(a₁/2+…+a_m/2^m)=r(a₁/2)+…+r(a_m/2^m),…
Repeating this infinitely,you get
r(a₁/2+a₂/2²+…(infinitely))=r(a₁/2)+r(a₂/2²)+…(infinitely)=O+O+…(infinitely)≠O.
a₁/2+a₂/2²+…(infinitely)∉R. This is contradiction.♦
You will have to use only finite symbols to express the number. But you need infinite symbols to
express the irrational numbers.

                          3

The analytics and theory of the manifold aren't consistent by theorem 6. The millennium problems
except P≠NP problem are nonsense. The physics isn't consistent too. every phenomenon
like free energy is possible.This is an answer to the 6th problem of Hilbert.

Supplement
Let X be a countable and consistent axiom system of the standard mathematics. Let d be a free
individual symbol. The mathematical axioms in X don't include d.
theorem 2". Let c_i be a set in the mathematics by X for ∀i∈N. If c_m-(c₁∪…,∪c_m-1)≠∅ for ∀m∈N,
there isn't a set c for which ∀x(M |=(x∈c)⇔((M|= x∈c₁)∨(M|= x∈c₂)∨…(infinitely))).
M is an arbitrary model of X.
Proof. Assume that there is such a set. X ∪{d∈c}∪{d∉c₁}∪…∪{d∉c_m} has a model M_m for which
M_m |= d∈c_m+1-(c₁∪…∪c_m) and is consistent.
So X∪{d∈c}∪{d∉c₁}∪{d∉c₂}∪…(infinitely) is consistent and has a model M". M" is a
model of X. (M"|= d∈c)⇔(M"|= d∈c₁)∨(M" |=d∈c₂)∨…(infinitely)
But (M" |= d∉c₁)∧(M" |= d∉c₂)…(infinitely). This is contradiction.♦
Theorem 2" leads theorem 3,theorem 4,theorem 5 too.
If　Z=N×N/～ ((a,b)～(c,d)⇔a+d=b+c) is a set, N×N=∪Z is a set.
N={y|∃x((x∈N×N)∧∃z(x=(y,z)))} is a set. By theorem 5,Z isn't a set. Similarly,Q isn't a set.
If the real number r={a,b} defined by Dedekind cut is a set, ∪r=a∪b=Q is a set. The real numbers
aren't sets which are the elements of the object domain of the standard mathematics.