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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.
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.
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 di's.
You may set up that ∀x({(y,1)|y∈x}∈Ax) ∈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∈AS)∧({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∪{Ad=A(d)}∪{d"∉d}∪{1∈d}∪…∪{n∈d} has a model Mn for which
Mn |= (d={1,2,…n})∧(d"=n+1) and is consistent for ∀n∈N.
So X∪{Ad=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∈Ad=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.[]
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.
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 di's.
You may set up that ∀x({(y,1)|y∈x}∈Ax) ∈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∈AS)∧({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∪{Ad=A(d)}∪{d"∉d}∪{1∈d}∪…∪{n∈d} has a model Mn for which
Mn |= (d={1,2,…n})∧(d"=n+1) and is consistent for ∀n∈N.
So X∪{Ad=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∈Ad=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.