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Naoto Meguro :Amateur. MSC2020:13A02.
The theory of the polynomial ring isn't consistent in ZFC.
ZFC proves every conjecture of the algebraic geometry with its denial.
Let R be a ring. R[x]=R⊕xR[x]. For ∀p(x)∈R[x], p(x)+0R[x]=p(x).
xp(x)+x0R[x]=xp(x). x0R[x]∈xR[x]. x0R[x]=0xR[x}. If x∈R[x], x0R[x]=0R[x]. 0R[x]=0xR[x].
Theorem 1. The ring R[x] which satisfies R[x]=R⊕xR[x] and x∈R[x] doesn't exist in ZFC.
Proof. 0R[x]=(0R,0xR[x])=(0R,0R[x])={{0R,0R},{0R,0R[x]}}∋{0R,0R[x]}∋0R[x].
This is a contradiction by the axiom of regularity and the axiom of choice. ♦
The theory of R[[x]] and the theory of the integral functions aren't consistent in ZFC too.
Theorem 2. The set theory treating the set N isn't consistent.
Proof. Such a theory defines Z={(m,n)|m,n∈N}/~ ((m,n)~(m",n")⇔m+n"=m"+n)
and treats ZN and Z[[x]] by the mapping ZN∋(a,b,c,…)→a+x(b+x(c+…∈Z[[x]]
and the axiom of replacement and leads a contradiction in ZFC like theorem 1. ♦
The proposition including N may be nonsense.
For example, RH=∀n((n∈N)∧(n≥5041)→(∑d|nd ≤eγn log log n)) may be nonsense.
If you set up that the object domain includes only natural numbers, you can evolve
the natural number theory without using the set N. So Fermat conjectire isn't
nonsense and may be still unsolved for Taniyama-Weil conjecture may be nonsense.
You need the set N to define the real numbers. So the standard real number theory
isn't consistent by theorem 2. RH is nonsense in it.
Many conjectures may be back to the starting points or may become nonsense.
1
The theory of the polynomial ring isn't consistent in ZFC.
ZFC proves every conjecture of the algebraic geometry with its denial.
2
Let R be a ring. R[x]=R⊕xR[x]. For ∀p(x)∈R[x], p(x)+0R[x]=p(x).
xp(x)+x0R[x]=xp(x). x0R[x]∈xR[x]. x0R[x]=0xR[x}. If x∈R[x], x0R[x]=0R[x]. 0R[x]=0xR[x].
Theorem 1. The ring R[x] which satisfies R[x]=R⊕xR[x] and x∈R[x] doesn't exist in ZFC.
Proof. 0R[x]=(0R,0xR[x])=(0R,0R[x])={{0R,0R},{0R,0R[x]}}∋{0R,0R[x]}∋0R[x].
This is a contradiction by the axiom of regularity and the axiom of choice. ♦
The theory of R[[x]] and the theory of the integral functions aren't consistent in ZFC too.
Theorem 2. The set theory treating the set N isn't consistent.
Proof. Such a theory defines Z={(m,n)|m,n∈N}/~ ((m,n)~(m",n")⇔m+n"=m"+n)
and treats ZN and Z[[x]] by the mapping ZN∋(a,b,c,…)→a+x(b+x(c+…∈Z[[x]]
and the axiom of replacement and leads a contradiction in ZFC like theorem 1. ♦
The proposition including N may be nonsense.
For example, RH=∀n((n∈N)∧(n≥5041)→(∑d|nd ≤eγn log log n)) may be nonsense.
If you set up that the object domain includes only natural numbers, you can evolve
the natural number theory without using the set N. So Fermat conjectire isn't
nonsense and may be still unsolved for Taniyama-Weil conjecture may be nonsense.
You need the set N to define the real numbers. So the standard real number theory
isn't consistent by theorem 2. RH is nonsense in it.
3
Many conjectures may be back to the starting points or may become nonsense.