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Inconsistency of the real number theory

2017-07-11 05:07:24 | Mathematics
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] 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. The mathematical axioms in X don't include the di's. Put d=d1.
Set up that ∀x∀y((x≥y)∧(y≥x)↔(x=y))∈X and ∀x∀y((x≥y)→(1/2[y]≥1/2[x]))∈X
and ∀x(1/2[x]≥0)∈X and ∀x∀y∀z((x≥y)∧(y≥z)→(x≥z))∈X.
Theorem 1. X has a model M for which M |= (1/2[d]≠0)∧(d≥n) for ∀n∈N.
Proof. X∪{1/2[d]≠0}∪{d≥1}∪…∪{d≥n} has a model Mn for which
Mn |= (d=n) and is consistent for ∀n∈N.
So X∪{1/2[d]≠0}∪{d≥1}∪{d≥2}…(infinitely) is consistent and has a model M
M is a model of X and M |= (1/2[d]≠0)∧(d≥n) 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.
Q∋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]=|1/2[d]|≤1/2n for ∀n∈N.)
So M |= 1/2[d]=0. But M |= 1/2[d]≠0. 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.[]
                               3

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.

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