My discovery about mathematics

Naoto Meguro. Amateur.
2010MSC. Primary 03C62;Secondary 03C55.
Key Words and Phrases. The direct sum, non-standard model.
The abstract. Theory of the direct sum isn't consistent.

                               1

I indicate that theory of the direct sum isn't consistent by Gödel's theorem.
You must reexamine the homological algebra and theory of the tensor product.
Fermat conjecture may be still unsolved.

                               2

Assume that theory of the direct sum is consistent. Put A={0,1}.
Let X be a countable and consistent axiom system of theory of the direct sum.
Let d₁,d₂,… be the free individual symbols. The mathematical axioms in X don't include the d_i's.
You may set up that ∀x({(y,1)|y∈x}∈A^x) ∈X and ∀S∀f((f∈A^(S))→(∑_m∈S f(m)≠1+∑_m∈S f(m))) ∈X and
∀x∀y((x∉y)→(∑_m∈y∪{x} 1=(∑_m∈y 1)+1))∈X.
These axioms are formed for the standard model for which
A^(S)={f|(f∈A^S)∧({m|(m∈S)∧(f(m)≠0)} is a finite set.)}. Put d=d₁, d"=d₂.
Theorem 1. Theory of the direct sum isn't consistent.
Proof. X∪{A^d=A^(d)}∪{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∪{A^d=A^(d)}∪{d"∉d}∪{1∈d}∪{2∈d}∪…(infinitely) is consistent and has a countable model M.
d becomes a countable set then. Let D be the object domain of M. D is the set of the closed terms.
Put e={(x,1)|x∈d}. d∈D. e∈D. M |= e∈A^d=A^(d). So M |= e"≠1+e" (e"=∑_m∈d e(m)).
For the countable set {c(1)}∪{c(2)}∪,…(infinitely) (c(i)≠c(k) for i≠k),
∑_m∈{c(1)} 1=1,∑_{m∈{c(1)}∪{c(2)}} 1=1+1,…,∑_{m∈{c(1)}∪{c(2)}∪…(infinitely)} 1=1+1+…(infinitely).
For M,d and d∪{d"} are countable sets and ∑_m∈d 1=∑_m∈d∪{d"} 1=1+1+…(infinitely).
M |= e"=∑_m∈d 1=∑_m∈d∪{d"} 1=(∑_m∈d 1)+1=1+e". This is contradiction.[] 

                                    3

Generally, you cannot get the projective resolution of the module.
You cannot define Ext and Tor then. You cannot define the tensor product by direct sums. You must
rewrite the proofs of theorems gotten by using ⊕ or ⊗ like Fermat conjecture, Mordell conjecture,
Serre's problem etc.. Theorem 1 means that you can prove any conjectures by using x^(y) or ⊗.
Tate conjecture and Hodge conjecture are provable nonsense.

Naoto Meguro. Amateur.
MSC 2010. Primary 03C62; Secondary 03C55.
Key Words and Phrases. The rational number theory treating the repeating binary fractions,
the real number theory,Gödel's theorem.
The abstract. Theory of the field including the repeating binary fractions isn't consistent.

                             1

I indicate that the rational number theory treating the repeating binary fractions isn't
consistent by Gödel's theorem.
The mathematics treating the real number field isn't consistent too.

                             2

Assume that the rational number theory treating the repeating binary fractions is consistent.
Let X be a countable and consistent axiom system of the rational number theory treating
the repeating binary fractions.
Let's treat the functions x+y,x-y,xy,x/y,|x|,[x] and x^[y]. [ ] is the symbol of Gauss.
[x]=0 for x∈[0,1) and [x+1]=[x]+1.
Let d₁,d₂,… be the free individual symbols. X doesn't include the d_i's. Put d=d₁.
Theorem 1. X has a model M for which M |= (1/2^[d]≠0)∧(1/2ⁿ≥|1/2^[d]|) for ∀n∈N.
Proof. X∪{1/2^[d]≠0}∪{1/2¹≥|1/2^[d]|}∪…∪{1/2ⁿ≥|1/2^[d]|} has a model M_n for which
M_n |= (d=n) and is consistent for ∀n∈N.
So X∪{1/2^[d]≠0}∪{1/2¹≥|1/2^[d]|}∪{1/2²≥|1/2^[d]|}…(infinitely) is consistent and has a model M
M is a model of X and M |= (1/2^[d]≠0)∧(1/2ⁿ≥|1/2^[d]|) for ∀n∈N[]

Theorem 2. The rational number theory treating the repeating binary fractions isn't consistent.
Proof. 1-1/2ⁿ=1/2+…+1/2ⁿ→1/2+1/2²+…(infinitely)=s (n→∞,n∈N) s(=0.1) is a repeating binary fraction
and is an element of the rational number field. s=1/2+s/2. s=1. 1/2ⁿ→0 (n→∞,n∈N).
If 1/2^[d]≠0 and n∈N, 1/2ⁿ≤|1/2^[d]| except finite n's.
This doesn't occur for M and d in theorem 1. (M |= 1/2^[d]≠0, M |= |1/2^[d]|≤1/2ⁿ for ∀n∈N.)
This is contradiction.[]
Theorem 3. The mathematics treating the real number field isn't consistent.
Proof. The mathematics treating the real number field includes the rational number
theory treating the repeating binary fractions and isn't consistent by theorem2.[]

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The theorems and the conjectures treating the real number field like the millennium problems
except P vs NP problem are provable nonsense by theorem 3.