goo

The real number theory

2017-07-11 05:07:24 | Mathematics
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I indicate that the real number theory isn't consistent by Gödel's theorem.
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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.
Set up that ∀x∀y((x≥y)∧(y≥x)↔(x=y))∈X and ∀x∀y((x≥y)→(1/2y≥1/2x))∈X
and ∀x((x≥0)→(1/2x≥0))∈X and ∀x∀y∀z((x≥y)∧(y≥z)→(x≥z))∈X.
Theorem 1. X has a model M for which M |= (1/2d≠0)∧(d≥n) for ∀n∈N.
Proof. X∪{1/2d≠0}∪{d≥1}∪…∪{d≥n} has a model Mn for which
Mn |= (d=n) amd is consistent for ∀n∈N.
So X∪{1/2d≠0}∪{d≥1}∪{d≥2}…(infinitely) is consistent and has a model M
M is a model of X and M |= (1/2d≠0)∧(d≥n) for ∀n∈N

Theorem 2. The real number theory isn't consistent.
Proof. Let D be the object domain of M in theorem 1. Define x≥y ⇔ M |= x≥y for x∈D and y∈D.
x=y ⇔ (x≥y)∧(y≥x) ⇔ M |= x=y for x∈D and y∈D then. 1/2d≠0. d≥n for ∀n∈N.
Assume that the real number theory is consistent.
1-1/2n=1/2+…+1/2n→1/2+1/22+…(infinitely)=∑m∈N 1/2m=s (n→∞,n∈N)
R∋s=1/2+s/2. s=1. 1/2n→0 (n→∞,n∈N). 0≤1/2d≤1/2n for ∀n∈N. 1/2d→0 (n→∞, n∈N)
1/2d=0. But 1/2d≠0. This is contradiction.♦
You could prove that the p-adic number theory isn't consistent too.
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Theorems and conjectures treating the real numbers (the millennium problems except
P vs NP problem etc.) are provable nonsense in the formalism.
The physics isn't consistent too. Every phenomenon like the free energy or the free productivity
is possible theoretically.
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