"Schoenfeld" wrote in message ups.com...

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

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

Schoenfeld wrote:
Nth Complexity wrote:
Dirk Van de moortel wrote:
By the way, zero is usually taken to be both positive and
negative.

Hahahahahahahahahahahahahahahahahahahahahahahahaha hahahahahahahaha!
And you expect to teach OTHERS?!


[IGNORE PREVIOUS POST - SYMBOLS GOT MIXED UP DUE TO CARELESS EDITING]


I'm not certain Dirk is wrong. Most websites (like Wolframs) imply that
0 is neither positive or negative, but I don't think it's possible to
prove this (at least I can't, perhaps someone else can comment).

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

No. That was not what the axiom stated.
It stated:
AXIOM: Additive Identity
There exists y such that for all x, x + y = x
This way, a thing y that satisfies the above condition, is an
additive identity by definition.


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

I did not state that.
When you are doing mathematics, try to be precise.
I stated that if Y1 and Y2 are additive identities, then
for all x: x + Y1 = x [1]
for all x: x + Y2 = x [2]
because that is how the axiom defines additive identities.
For every AdditiveIdentity the axiom allows us to say:
for all x: x + Additive Identity = x.
So your statement is trivially true - by definition.
The axiom says that the set of additive identities is not empty.
Below I prove that this set can only have one element.



so we also have when we apply [1] to our number Y2:
Y2 + Y1 = Y2 [3]
and likewise, when we apply [2] to our number Y1:
Y1 + Y2 = Y1 [4]

With the axiom of commutativity
for all x, y: x + y = y + x
we then have when we apply it to Y1 and Y2
Y1 + Y2 = Y1 + Y2

So you didn't even spot the typo.
That should be
Y1 + Y2 = Y2 + Y1

so, with [4] and [3] we conclude
Y1 = Y2.

So, using the commutativy axiom, there is only one additive identity.

[snip]

If I am wrong then I would highly appreciate you pointing out the exact
error in my reasoning.

hope this helps.

Everything you said relies on "STATEMENT 1" being true. Can show you
this statement true? :-)

Yes, but not to an imbecile.
Did my "pointing out the exact error in your reasoning" help you,
or are you an imbecile, or perhaps merely pretending to be one?

Dirk Vdm