"Robert Low" wrote in message ...
Schoenfeld wrote:

If you consider commutative rings (e.g. integers) or ordered fields
(e.g. reals) there is the additive identity axiom:
There exists y such that for all x, x + y = x
This is entirely insufficient to imply a single unique additive
identity y, although this seems to be the universal interpretation.

The axiom as stated there does not immediately state that
the additive identity is unique; that is a consequence of
this together with other axioms. It is easy to see that
commutativity of addition implies that the additive inverse
is unique.

For suppose that y is an additive identity and
y' is an additive identity.

Then since y is an additive identity,

y' + y = y'

(adding y to anything leaves that thing unchanged)

and, since y' is an additive identity,

y + y' = y

(adding y' to anything leaves that thing unchanged).

Now we wheel out the commutativity of addition.

y' = y' + y = y + y' = y

So any two objects which have the defining property
of an additive identity must be equal. Or, to put
that in plainer language, the additive identity
is unique.

(You might even see presentations with uniqueness
in there are part of the definition. It's a matter
of taste.)

He seems not to be prepared to admit that he understand this.
He's trying to avoid losing his face, which must be a painful
process for someone who already lost it ;-)

Dirk Vdm