My discovery about mathematics

Naoto Meguro. Amateur.
MSC 2010. Primary 03E20 ; Secondary 11S99 .
Key Words and Phrases. p-adic numbers, countable sets.
The abstract. Existence of a countable set leads existence of an infinite sum leading
contradiction.

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Existence of a countable set leads existence of an infinite sum leading contradiction.
Perhaps the countable set and the continuum don't exist in the set theory.
You must correct the mathematics using the set theory then.

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Theorem 1. The countable set and the continuum don't exist in the consistent mathematics.
Proof. Assume that the standard mathematics treating the countable sets is consistent.
The p-adic number theory is consistent then.
Let p(≥3) be a prime number. Put L=Z_p and F=Z.
Let F(r) be the element of L/F for which r∈F(r) for r∈L.
You can define F(r)+F(s)=F(r+s) and can set up that ∀x((x∈L)→(∑_r∈{x}F(r)=F(x)))
and ∀x∀y((x∈L)∧(y⊂L)∧(x∉y)→(∑_r∈y∪{x}F(r)=(∑_r∈yF(r))+F(x)).
∑_r∈{1}F(r)=F(1),∑_r∈{1,p}F(r)=F(1)+F(p)=F(1+p),∑_{r∈{1,p,p²}}F(r)=F(1)+F(p)+F(p²)=F(1+p)+F(p²)=F(1+p+p²),…
∑_{r∈{1,p,…,p^m}}F(r)=F(1)+F(p)+…+F(p^m)=F(1+p+…+p^m).
If the countable set {1,p,p²,…(infinitely)}(⊂L) exists,you can repeat the above infinitely and
are to define ∑_{r∈{1,p,p²,…(infinitely)}}F(r)=F(1)+F(p)+F(p²)+…(infinitely)=F(1+p+p²+…(infinitely))
These definitions are intuitive and use infinite symbols for the definition of {1,p,p²,…(infinitely)} is intuitive and
uses infinite symbols.
Similarly, ∑_{r∈{2,2p,2p²,…(infinitely)}}F(r)=F(2)+F(2p)+F(2p²)+…(infinitely)=F(2+2p+2p²+…(infinitely))
if the set {2,2p,2p²,…(infinitely)} exists.
pⁿ∈F(0) and 2pⁿ∈F(0). F(pⁿ)=F(2pⁿ)=F(0) for ∀n∈N. Put c=1+p+p²+…(infinitely)∈L.
F(c)=F(1)+F(p)+F(p²)+…=F(0)+F(0)+…=F(2)+F(2p)+F(2p²)+…=F(2c). c=2c-c∈F=Z. This is contradiction.
{1,p,p²,…} or {2,2p,2p²,…} doesn't exist. The countable sets don't exist by the axiom of replacement.
∪(2^N)=N. 2^N and the continua don't exist. ♦
By the axiom of replacement, the set S doesn't exist when a countable one C and a surjection S∋s→n(s)∈C
exist. Put C={0}∪N.
Example 1. k[x] and k[[x]] don't exist by the surjections k[x]∋f→deg(f)∈C and k[[x]]∋f→ord(f)∈C.
Example 2. Let A be a commutative ring. Let S be an infinite set. A^(S)=⊕_s∈S A doesn't exist
by the surjection A^(S)∋a→ Card{s|(s∈S)∧(a(s)≠0)}∈C.
Generally,you cannot get the projective resolution →A^(M)→M→0 (exact) and the tensor product
M⊗_AN=A^(M×N)/0⊗_A0 for the infinite A-module M.

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You must not treat the continuum Z_p. Fermat conjecture may be still unsolved.
Theory treating R isn't consistent. Conjectures about manifolds like Poincaré conjecture and
Hodge conjecture are provable nonsense. Gauss and Kronecker might be right.