"Schoenfeld" writes:

Dik T. Winter wrote:
In article .com "Schoenfeld" writes:
Dik T. Winter wrote:
...
The statement,
Axiom: Additive Identity
"for all x there exists y such that x + y = x"
defines an additive identity for all x.

No it does not. Consider the following addition table:
+ a b c
a a c b
b c b a
c b a c

For each 'x' there is an 'y' such that 'x + y = x'. But there is not
an additive identity.

It is an additive identity by definition, the definition was for
each x there is y.

What is the additive identity in the addition table above? In what
way does the addition table above fail the requirement: "for all x
there exists y such that x + y = x"?

Your table has no relation to what I said. In fact, it doesn't even
have a relation to how additive identities are defined for
rings/groups/fields (another elementary error you've just made).

I'll repeat again, the AXIOM:
"Axiom: Additive Identity, for all x there exists y, x + y = x"

You got quantifier dyslexia, right?

guarantees that an object 'y' exists for every single object 'x',
however the object 'y' is not an number.

The word "number" as you use it is meaningless. Anyway, additive
identities are not something dependent on x. You got your quantifiers
wrong.

If you were able to read a simple table, you'd notice the difference.

--
David Kastrup, Kriemhildstr. 15, 44793 Bochum