Once you’ve got N, 0 shows up in Z either way. The construction’s pretty standard, so you should have no trouble finding it in google.
I used to teach college level Math…
This thread reminds me of the dreadfully named ‘imaginary numbers’. Stupid, stupid name. Because of the name, students would think they didn’t exist.
Particularly curious, smart and devious 
students would ask me to ‘point at 2i since you insist it exists’.
I would then ask them if they thought -2 exists. The response would yes. I would then ask them to point to -2 of something.
One student brought in his mortgage statement the next day but no dice. I wanted him to point to -2 in the world and not symbols on paper 
This exercise worked well.
Sometimes students would ask why (-2)*(-2)=4. I would respond something like:
If the temperature is rising 2 degrees a day how much higher will the temperature be in 2 days? 2*2=4
How much higher(lower) was the temperature 2 days ago? 2*(-2)=-4
If the temperature is falling 2 degrees a day, how much higher/lower was the temperature 2 days ago? (-2)*(-2)=4
It worked well. Gave em something to think about anyway.
Um, but then, shouldn’t it be:
2[sup]2[/sup] = 1/4 ?
Actually, 1/4, but I see what you’re saying. There would be no way to tell if the individual components were negative, or if the entire expression were negative.
No, (-2)^(-2) = 1/4. It’s equal to (-1/2)^2, which is the same as (1/2)^2.
:smack: I was confused by my own notation. How embarassing.
Goedel’s theorem certainly does not require that 0 be a natural number. I suppose some proof might, although I don’t recall that. Whether or not you call 0 a natural number is just a matter of convention and it is slightly, but only slightly more convenient to do so.
The (usual) formal construction of the rational numbers proceeds like this: Take pairs of integers call them a/b using a slash instead of a comma. Of course you want b non-zero. Say that a/b = c/d if ad = bc. You can do the same for constructing the integers from the natural numbers. Take pairs of natural numbers, call them a-b, using this horizontal line instead of a slash. Say a-b = c-d if a + d = b + c. You add such pairs by a-b + c-d = (a+c)-(b+d). The parens are there for grouping only. Then 0 = 1-1 = 2-2 = 3-3 = … and -2 is just a sign for 1-3 = 2-4 = 3-5 = … . See the parallel.
If you don’t use 0 as a natural number, then probably the easiest way to proceed is to first construct the positive rationals and then the positive reals and then 0 and the negative numbers as above, since you don’t have to worry about excluding division by 0 and it is easier to deal with multiplication if you don’t have negatives till the end. But it is only a minor improvement.
Mendelson’s statement of Godel’s theorem seems to require it, at least if you want to use the theorem on a theory that only talks about natural numbers. If you have integers at hand, then no, you certainly don’t need 0 to be a natural.
Avoiding negative sign confusion is simple. Simply don’t use negative signs. Instead of writing:
4 + -2 or -5 - 8 or -2^2
write them like this:
4 - 2 or 0 - 5 - 8 or 0 - 2^2
So much easier that way.