My discovery about mathematics

The mathematics whose object domain includes ∑_i∈Ny_i isn't consistent.
Let N(x) be a predicate symbol. Set up that N(1),∀x(N(x)→N(x+1)) and
p(1)∧∀x(p(x)→p(x+1))→∀y(N(y)→p(y)) (p(x) is an arbitrary propositional function.).
For the standard model, N(x) ⇔ (x=1)∨(x=2)∨…(infinitely).
You can evolve the natural number theory without using the individual symbol N. If there isn't the set
N,you need not think ∑_i∈Ny_i and can avoid the contradiction.
The countable sets don't exist by the axiom of replacement then.
So Spec(Z)=Spec(Z[√-5])=Spec(Q[x,y])={{0}} then.
Let A be a ring. Let S be an infinite set.
Put L={x | (x∈A^S)∧∃y((y∈S)∧∀z((z∈S)∧(z≠y)→(x(z)=0)))},
L₁=L, L_i+1=L_i+L. Then, A^(S)=∪_i∈NL_i=∪{L₁,L₂,…}.
You cannot define A^(S) for {L₁,L₂,…} is a countable set.
Generally,you cannot get
Another way is to abandan the function symbol s(x,y)=∑_i∈xy_i (y:x∋i→y_i).
You cannot evolve theory of the tensor product then. You cannot get the projective
resolution
0←M←A^(M)←… (exact) . The free A modules may not be projective A modules. etc..