Finite sets of numbers have obvious problems, well explored in computer science (if you integer has be be expressed in 4 bytes, you can express only a finite set of integers. Or real numbers, for that matter).
Trivially, if your finite set ends at n, what is the sum of (n-1) + (n-2)? A plausible answer is “you can’t do that”, but that’s also an answer to any paradox whatsoever.
How do you feel about imaginary numbers, by the way?
A plausible answer is “you can’t do that”, but that’s also an answer to any paradox whatsoever.
No, this is not true. The only answer we have to paradoxes is—change you axiom system!
How do you feel about imaginary numbers, by the way?
There are no paradoxes there. At least no paradoxes which wouldn’t be present in the real numbers already.
SQRT(-1)=i is no paradox whatsoever. People might often say—“oh, well that’s paradoxical”.
No, it’s not. No A & ~ A here. It might be weird to somebody, that you can calculate square roots from negative numbers. I really don’t see why, but it might.
Weirdness and paradoxicality are two different things.
You can eliminate paradoxes by declaring operations which lead to them “illegal”
Precisely! You narrow your axiomatic system, hoping that one of contradictors will fall out. They’ve eliminated self containing sets from the Naive Set Theory and Russell’s paradox vanished. Hopefully.
Is i a number? If not, what is it?
It is not “what is it”, it is “how does it behave”. Whatever behaves as a number, is a number.
Does it have a sign?
If it behaves as it had a sign … And it does. So yes, it has a sign!
Not only arithmetic, but an arithmetic with arbitrary large numbers is required for the Godel’s Incompleteness Theorem.
You need at least “all integers”, that is an infinite set.
Maybe some incompleteness can be proved for the finite sets as well. But it’s not known to be so. Nobody proved that yet.
Finite sets of numbers have obvious problems, well explored in computer science (if you integer has be be expressed in 4 bytes, you can express only a finite set of integers. Or real numbers, for that matter).
Trivially, if your finite set ends at n, what is the sum of (n-1) + (n-2)? A plausible answer is “you can’t do that”, but that’s also an answer to any paradox whatsoever.
How do you feel about imaginary numbers, by the way?
No, this is not true. The only answer we have to paradoxes is—change you axiom system!
There are no paradoxes there. At least no paradoxes which wouldn’t be present in the real numbers already.
SQRT(-1)=i is no paradox whatsoever. People might often say—“oh, well that’s paradoxical”.
No, it’s not. No A & ~ A here. It might be weird to somebody, that you can calculate square roots from negative numbers. I really don’t see why, but it might.
Weirdness and paradoxicality are two different things.
I don’t mind weirdness, I hate paradoxicality.
You can eliminate paradoxes by declaring operations which lead to them “illegal” (or their outcomes to be undefined).
Is i a number? If not, what is it? Does it have a sign?
Precisely! You narrow your axiomatic system, hoping that one of contradictors will fall out. They’ve eliminated self containing sets from the Naive Set Theory and Russell’s paradox vanished. Hopefully.
It is not “what is it”, it is “how does it behave”. Whatever behaves as a number, is a number.
If it behaves as it had a sign … And it does. So yes, it has a sign!
Which is what?
Everything which behaves like a sign.