Before, the concept of imaginary numbers, the number i=√−1 was shown to have no solution among the numbers that we had. So we declared i to be a new type of number. How come we don’t do the same for other “impossible” equations, such as
x=x+1, or x=1/0?
OK, a lot of people have said that a number x such that x=x+1 would break the rule that 0≠1. However, let’s look at the extension from whole numbers to include negative numbers (yes, I said that I wasn’t going to include this) by defining −1 to be the number such that −1+1=0. Note that this breaks the “rule” that “if x≤y, then ax≤ay“, which was true for all a,x,y before the introduction of negative numbers. So I’m not convinced that “That would break some obvious truth about all numbers” is necessarily an argument against this sort of thing.
Here’s one key difference between the cases.
Suppose we add to the reals an element i such that i2=−1, and then include everything else you can get from i by applying addition and multiplication, while still preserving the usual rules of addition and multiplication. Expanding the reals to the complex numbers in this way does not enable us to prove new equations among the original reals that are inconsistent with previously established equations.
Suppose by contrast we add to the reals a new element k postulated to be such that k+1=k and then also add every further element you can get by applying addition and multiplication to the reals and this new element k. Then we have, for example, k+1+1=k+1. Hence — assuming that old and new elements together still obey the usual rules of arithmetic — we can cheerfully subtract k from each side to “prove” 2=1. Ooops! Adding the postulated element k enables us to prove new equations flatly inconsistent what we already know. Very bad news!
Now, we can in fact add an element like k consistently if we are prepared to alter the usual rules of addition. That is to say, if we not only add new elements but also change the rules of arithmetic at the same time, then we can stay safe. This is, for example, exactly what happens when we augment the finite ordinals with infinite ordinals. We get a consistent theory at the cost e.g. of having cases such as ω+1≠1+ω and 1+1+ω=1+ω.