"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.


[repost - small typo corrected]

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.

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

Dirk Vdm