Wim Benthem wrote:
On 10 Sep 2005 05:35:47 -0700, "Schoenfeld"
wrote:


Dirk Van de moortel wrote:
"Schoenfeld" wrote in message oups.com...

Dirk Van de moortel wrote:
"Schoenfeld" wrote in message ps.com...

As others have said, it is a question of defining things. In the part
of the world in which I live, Bourbaki is the standard. So my
statement that "zero is usually taken to be both positive and
negative" is correct - again, in the part of the world in which I
happen to live.

Is there any definition other than the one given by the additive
identity axiom for rings or fields? The remainder of this posts assumes
no.

AXIOM: Additive Identity
There exists y such that for all x, x + y = x

Does that axiom imply that there exists only 1 additive identity? Based
on my understanding of the existential quantifier, it does not.

Indeed, it does not.
So we suppose there are (at least) two such identities, let's not call them
George and Freddy, but Y1 and Y2.

Fine.

Then we have, thanks to your axiom:
for all x: x + Y1 = x [1]
for all x: x + Y2 = x [2]

That looks reasonable, but there is a subtely here which I can't deduce
from the axioms.

The axiom stated:
There exists y such that for all x, x + y = x

Does this mean that,
STATEMENT 1:
For all y in set of additive identities, for all x in Z, x + y = x.

You make this assumption when you state that:
x + Y1 = x
x + Y2 = x

Dirk doesn't use an axiom here, just the definition of an additive
identity: y is an additive identity iff for all x, x+y=x

Yes, that wasn't immediately obvious to me because I mixed up the order
of the existential quantifers in that axiom (when i was thinking about
it, that is).

Dirk's malicious claims of me somehow lying to not admit this error are
characteristically false. However, they do serve to explicitly
demonstrate that he is the perfect candidate for the OP's award. Not
necessarily for the error he made (and won't admit to), but rather for
his malicious arrogance and transitive ignorance.


--
Wim Benthem