Re: Godel proved maths inconsistent not incompleteness theorem

On Mar 29, 11:57 am, William Hale <h...@xxxxxxxxxx> wrote:
In article
<7f6e70ce-1110-4b32-a182-8592f6754...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

[cut]

It should be a standard that a person give the result here
(some scintilla of evidence) if challenged.

Would you accept the following as some evidence that the branch of
abstract algebra at least uses the axioms of ZFC (not ruling out your
claim that abstract algebra might also use axioms that are not part of
ZFC):

Suppose the proof of some statement in abstract algebra has a subsection
of the following form:

     1) Let J be the set of algebraic integers such that ....
     2) Let K be the set of algebraic integer such that ....
     3) Then, such and such.
        Hence, (Ax)(x in J implies x in K)
     4) Furthermore, such and such.
        Hence, (Ax)(x in K implies x in J)
     5) Thus, (Ax)(x in J iff x in K)
     6) Therefore, J = K by the Axiom of Extensionality.

That is, would you agree with the claim that at least one of the axioms
used by Abstract Algebra is the ZFC Axiom of Extensionality?

So if a proof makes reference to the fact that A==B is equivalent to
(A=>B)^(B=>A) then it's based on ZFC? Boy, I can see how that sure
makes things easy.

C-B
.

Relevant Pages
Quantcast