Naoto Meguro. Amateur.
MSC2010. Primary 03B10;Secondary 03B80.
Key Words and Phrases. The complex number theory, an axiom system.
The abstract. The complex number theory isn't consistent.
The known complex number theory isn't consistent. Let's see it.
Let X be an axiom system of the standard mathematics whose object domain is C. Set up that
X⊃{0≠1, ∀x(x0=0.}. Put P=∀x(x=1/(1/x)). 1/0∉C. But x=0 is a removal singular point of 1/(1/x)=x.
P is valid as an axiom of the complex number theory. Set up that P∈X. Let d be a free individual symbol.
X doesn't include d.
Theorem 1. X isn't consistent.
X |- ∀x(x=1/(1/x))→(d=1/(1/d)). X |- d=1/(1/d). X\- ∀x(1≠0=0x). X|- ∀x(1/x≠0)
X |- ∀x(1/x≠0)→(1/(1/d)≠0). X |- 1/(1/d)≠0. X |- d≠0. X doesn't include d. So X |-∀x(x≠0).
X |- 0≠0. X isn't consistent.♦
Theory of the field must treat the function symbol x/y and 1/0. 1/0 isn't an element of the field.
If you admit P, X is an axiom system of the theory of the field like R,Q and Z/pZ.
Theorem 2. The field C doesn't exist in the consistent mathematics.
Proof. If C exists, the standard model of X whose object domain is C exists. (d=0 etc..)
X is consistent then. This is contradiction by theorem 1.♦
The theory treating C isn't consistent.
Physics isn't consistent too. Every phenomenon like the free energy is possible theoretically.
MSC2010. Primary 03B10;Secondary 03B80.
Key Words and Phrases. The complex number theory, an axiom system.
The abstract. The complex number theory isn't consistent.
1
The known complex number theory isn't consistent. Let's see it.
2
Let X be an axiom system of the standard mathematics whose object domain is C. Set up that
X⊃{0≠1, ∀x(x0=0.}. Put P=∀x(x=1/(1/x)). 1/0∉C. But x=0 is a removal singular point of 1/(1/x)=x.
P is valid as an axiom of the complex number theory. Set up that P∈X. Let d be a free individual symbol.
X doesn't include d.
Theorem 1. X isn't consistent.
X |- ∀x(x=1/(1/x))→(d=1/(1/d)). X |- d=1/(1/d). X\- ∀x(1≠0=0x). X|- ∀x(1/x≠0)
X |- ∀x(1/x≠0)→(1/(1/d)≠0). X |- 1/(1/d)≠0. X |- d≠0. X doesn't include d. So X |-∀x(x≠0).
X |- 0≠0. X isn't consistent.♦
Theory of the field must treat the function symbol x/y and 1/0. 1/0 isn't an element of the field.
If you admit P, X is an axiom system of the theory of the field like R,Q and Z/pZ.
Theorem 2. The field C doesn't exist in the consistent mathematics.
Proof. If C exists, the standard model of X whose object domain is C exists. (d=0 etc..)
X is consistent then. This is contradiction by theorem 1.♦
3
The theory treating C isn't consistent.
Physics isn't consistent too. Every phenomenon like the free energy is possible theoretically.