The Topic:
Dividing by (x-y) when x=y

The Question:

First of all, mathematical induction requires a countable set, so this inductive proof works, the best you can do is the rational numbers, though you can just add a set of measure zero to them, this does not work for all real numbers though, since they are uncountable. Essentially the proof works on a set of cardinality aleph null, but not aleph one. This is also assuming that you are redefining division by zero to be legal. OK. What are you defining 1/0 to be? Whatever you define it to be had better be equal to one when multiplied by zero, since if X=1/0, then X*0=1. Say you choose the extended real number system, putting a point at infinity and negative infinity. OK. 1/0=infinity. So does 2/0. Then 0*infinity=1 and 0*infinity=2, implying that 1=2. Read Goedel for further fun with these implications.

 Sam the Eagle

You cannot redefine division by zero to be legal!

 Sam the Eagle

We have laws against such here in America!

 Sam the Eagle

Do it again and I will have to inform the authorities.

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