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1
I give examples of mathematical logic which are wrong in the predicate logic.
When you assume existence of an element x, fix it and continue deduction,
it becomes a wrong logical formula ∃xP(x)→Q(x).Such logic is used in
mathematics as right one. If you don't admit such logic for strict formalism,
you must reexamine many theorems. It will be an important problem to foundations.
2
A(∋1) is a commutative ring in the following.
(1) The next proof of a1/s1=a2/s2, a2/s2=a3/s3⇒a1/s1=a3/s3 in a ring of quotients AS.
∃s4((s4∈S)∧(s4(s2a1-s1a2)=0))∧∃s5((s5∈S)∧(s5(s3a2-s2a3)=0))→(s2s4s5∈S)∧
(s2s4s5(s3a1-s1a3)=0)
This is a wrong logical formula for s4 and s5 appear out of ∃s4( ) and ∃s5( ).
You can rewrite this proof by right logical formulas in this case.
You can get ∀s4∀s5((s4(s2a1-s1a2)=0)∧(s5(s3a2-s2a3)=0)→(s2s4s5(s3a1-s1a3)=0)) arithmetically.
And ∀s4∀s5((s4∈S)∧(s5∈S)→(s2s4s5∈S))
By these and the tautology
∀x∀x"(P(x,x")→Q(x,x"))→(∀y∀y"(R(y,y")→S(y,y"))→
∀z∀z"(P(z,z")∧R(z,z")→Q((z,z")∧S(z,z"))),you get
∀s4∀s5((s4∈S)∧(s4(s2a1-s1a2)=0)∧(s5∈S)∧(s5(s3a2-s2a3)=0)→
(s2s4s5∈S)∧(s2s4s5(s3a1-s1a3)=0)).
By this and the tautology
∀x∀x"(P(x,x")→Q(x,x"))→(∃y∃y"P(y,y")→∃z∃z"Q(z,z")),
you get
∃s4∃s5((s4S)∧(s4(s2a1-ssa2)=0)∧(s5∈S)∧(s5(s3a2-s2a3)=0))→
∃s"4∃s"5((s2s"4s"5∈S)∧(s2s"4s"5(s3a1-s1a3)=0)).
(2) A proof of the theorem that P= ⊕ Pk is a projective A-module
k∈I
if every Pk (k∈I) is a projective A-module.
Let M and N be A-modules.
Let f (∈HomA(M,N)) be a surjection.
Let jk (∈HomA(Pk,P) be the canonical injection. (k∈I)
And h∈HomA(P,N). hk=hjk (k∈I).
Then, (∀i(i∈I)→∃gi((gi∈HomA(Pi,M)∧(fgi=hi)))
→∃g((g∈HomA(P,M))∧(∀k((k∈I)→(gjk=gk)))) …(*).
fg=h then.
(*) is a wrong logical formula for gi appears out of ∃gi( )
Let's think the case that I is a finite set.
When I={1,2},∀g1∀g2(J(g1,g2)→∃gK(g,g1,g2))
(J(g1,g2)=(g1∈HomA(P1,M)∧(g2∈HomA(P2,M)),
K(g,g1,g2)=(g∈HomA(P,M)∧(gj1=g1)∧(gj2=g2))
is an axiom about the direct sum.
By this and the tautology
∀x∀x"(P(x,x")→Q(x,x"))→(∃y∃y"(P(y,y")∧R(y,y"))→
∃z∃z"(P(z,z")∧Q(z,z")∧R(z,z"))), you get
∃g1∃g2(J(g1,g2)∧(fg1=h1)∧(fg2=h2))→∃g"1∃g"2∃g(J(g"1,g"2)∧(fg"1=h1)∧(fg"2=h2)
∧K(g,g"1,g"2)). fg=h then.
You can prove the theorem in the case that I is a concrete finite set.
You could get projective resolutions, Tor and Ext of modules
which are finitely generated over a Noetherian local ring.
In general cases, the axiom (or a theorem) about the direct sum
∀((i∈I)→∃gi(gi∈HomA(Pi,M)))
→∃g(g∈HomA(P,M)∧(∀k((k∈I)→(gjk=gk))))
is a wrong logical formula.
(3) Let F and U= ∪ Ui be a sheaf and a union of open sets.
i∈I
Assume that Ui≠Uj (i≠j). Then,
∀i((i∈I)→∃xi(xi∈F(Ui)))→∃x((x∈F(U)∧∀((k∈I)→(resUUk(x)=xk))).
This is a wrong logical formula for xi appears out of ∃xi( ).
(3)