>What does this have to do with adding axioms? You don't get the complex numbers from the real numbers by adding axioms. For example, the following is provable for the reals, but not for complex numbers:
> ∀x,y . x < y ∨ x > y ∨ x = y
Since R is a subset of C, you can write C as R with additional axioms. See for example:
>I'm not looking for handwaving. Formal mathematics is math that can be done mechanically. Write down the axioms for undefined. Again -- I'm not saying it hasn't been done, but it's certainly not part of standard first-order logic, and it is unnecessary.
You are trying to define the undefined in a formal system. Undefined is just the absence of a definition. Not 0, 3 , 2pi
> No. You can write 1/x < 5 ∨ x > 20.
Yes, those are valid mathematical statements. 5 ∨ TRUE, or undefined V TRUE, I doubt it.
> There is no such thing as "doesn't make sense" in formal math, even though there is such a thing in informal math. >That's the whole point. Either the expression is ill-formed, i.e. not in the language or a "syntax error", or it must make some sense.
Well in language, in which I am corresponding with you, there is such a thing as 'makes no sense'.
You are somehow trying to capture everything in your logical system only to try to prove that 1/0 = 0 is part of a field, which it isn't. I've made my point here, it was nice talking to you.
> Since R is a subset of C, you can write C as R with additional axioms.
That's not even remotely how it works. You may want to read up on logical theories and models[1].
> Well in language, in which I am corresponding with you, there is such a thing as 'makes no sense'.
Yes, and for the same reason we can have such a thing as "undefined" (that means more than merely 'not specified') in informal mathematics -- because both English and informal math are informal. But we are talking about formal languages[2], which do not have such a thing.
> ∀x,y . x < y ∨ x > y ∨ x = y
Since R is a subset of C, you can write C as R with additional axioms. See for example:
http://www.math.mcgill.ca/gantumur/math249w15/numbers.pdf
>I'm not looking for handwaving. Formal mathematics is math that can be done mechanically. Write down the axioms for undefined. Again -- I'm not saying it hasn't been done, but it's certainly not part of standard first-order logic, and it is unnecessary.
Undefined is just undefined, that is no handwaving, it is a primitive notion. See https://en.wikipedia.org/wiki/Primitive_notion
You are trying to define the undefined in a formal system. Undefined is just the absence of a definition. Not 0, 3 , 2pi
> No. You can write 1/x < 5 ∨ x > 20. Yes, those are valid mathematical statements. 5 ∨ TRUE, or undefined V TRUE, I doubt it.
> There is no such thing as "doesn't make sense" in formal math, even though there is such a thing in informal math. >That's the whole point. Either the expression is ill-formed, i.e. not in the language or a "syntax error", or it must make some sense.
Well in language, in which I am corresponding with you, there is such a thing as 'makes no sense'.
You are somehow trying to capture everything in your logical system only to try to prove that 1/0 = 0 is part of a field, which it isn't. I've made my point here, it was nice talking to you.