"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.



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. 'y'
is usually called zero and given symbol 0.

Considering the integers Z, you can split Z into two sets P and N such
that:
1. For all x in Z, P contains x iff N contains -x; AND
2. P is closed under addition and multiplication.

Trivially, P is the set of positive integers and N is the set of
negative integers.

Where does 0 lie in here?

If your phrase "split Z into two sets P and N" implictly means
that the union of P and N is Z and that the intersection of P and N
is the empty set, then indeed by design, 0 is not in P and not in N.
So unless you explicitly specify what you mean, you can choose,
and the conditions you gave are not sufficient to decide more than
something like
P contains 0 == N contains -0 == N contains 0
i.o.w. 0 is in both or in neither.


Well if it lies in BOTH P and N there are no contradictions at all. But
this implies that 0 occurs twice in the integers (otherwise it couldn't
be placed in any of P or N).

That depends on what you want the phrase "split Z into two sets"
to mean. As soon as you decide on that, you have your answer.


Again, this is not strictly prohibited by
additive identity axiom. You could say there are two additive
identities 0+ and 0-, positive and negative respectively. A cursory
analysis their arithmetic reveals no axiomatic contradictions.

Based on this, I would need to say that 0 by itself does not exist.
Rather you have positive 0+ and negative 0-.

Is there an error here?


REMARK: The widely used terminology related to all this is:
"Positive integers" = 1,2,3, ...
"nonnegative integers" = 0, 1, 2, 3, ...
"negative integers" = -1, -2, -3, ...
"nonpositive integers" = 0, -1, -2, -3, ....

In my part of the world:
"Positive integers" = { 0, 1, 2, 3, ... }
"Strictly positive integers" = "Nonnegative integers" = { 1, 2, 3, ... }
"Negative integers" = { 0, -1, -2, -3, ... }
"Strictly negative integers" = "Nonpositive integers" = { -1, -2, -3, ... }

Gee.... what a silly non-issue.
Frenchman: "Ceci est une chaise."
Englishman: "No, this is a chair."
Frenchman: "Non, c'est une chaise!"
Englishman: "No, idiot, it is a chair!"
Frenchman: "Mais non, imbecile, je vous dis que c'est une chaise!!"
Englishman: "Get lost, idiot, it is a chair!!"
....

Dirk Vdm