>>Ordered fields must obey a few certain properties. There are several reasons why C is not an ordered field, but basically the problem is that i is not defined in R (though i**2 is).
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>I can't follow you here. What's the connection between the posibility to order C and i?
(See below)
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Actually, I'm just paraphrasing my advanced calaculus book. They say that C is not ordered, and there are several ways to show this (but they don't explain further). But all ordered number fields must obey 4 properties of inequalities, as I said:

>>One of the properties for an ordered field that results from order axioms is that for any number, say x, in a set, one of these statements is true: 'x = 0', 'x is > 0', '-x > 0'. Clearly, this is not true for i. Thus C is not an ordered field.

>When I say "order relation", it has nothing to do with the usual order in R. An order relation is a relation (on a set) that is reflexive, tranzitive and antisymetric. This is the definition. The usual order in R is just an example of order relation.
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We're talking in 2 different contexts, I think. Abstract algebra and advanced calculus use different, though somewhat analagous, languages. But if you look at the definitions of isomorphism and order in each area, they do appear quite different. Calculus is explained in a numerical context while algebra is explained in a general abstract set context. (Why I always prefer calculus) (s)
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>You can't use the order from R in C. According to my order on C, (0,0) <= (0,1). Which means 0 <= i. You can replace here (and in the definition I gave) "<=" with "M". Thus, "z1 M z2" means "z1 is in the relation M with z2", where M is my order relation. "<=" is just a notation, is not the same relation as the usual order in R.
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>Which means 0 <= i. How is this?
You are calling this your order, but in calculus i is expressly defined as sqrt(-1), which has no way to be ordered relative to 0. You can't arbitrarily assign your own ordering in calculus context, we use a real context.

>I never said that C and R are isomorphic. I said RxR is isomorphic with C. (An element in RxR is an order pair: (x,y).)
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I don't quite follow. What does C plane look like here?