Different definitions of Natural Numbers and Proof by Contradiction I finished discrete math and now I am in foundations of Computer Science class and I have 2 questions. In discrete math the teacher taught us that the set of Natural Numbers starts at 1, but in Foundations of Computer Science the teacher is saying that the set of Natural Numbers starts at 0. How can that be?
Another question is about the contradiction method of proofs. 
In discrete math they taught us for $P \Rightarrow Q$

assume ${\sim} P$
therefore $Q$
therefore ${\sim}Q$
hence $Q \, \& \, {\sim}Q$ a contradiction
thus, $P$

Now in my foundations of computer science class the teacher is saying that you never start by negating $P$ and you always start by negating $Q$ and then you come up with the contradiction ${\sim}P \,\&\, P$.
So why are these 2 things different the beginning of the set of natural numbers and the way to do proof by contradiction? Thank you for your time.
 A: For the first question, it depends who you ask. Some textbooks include $0$ as a natural number, and some do not. This is why you sometimes see the notation $\mathbb{N}_0$ and $\mathbb{N}$.
For the second question, there might be a bit of confusion between proof by contradiction and proof by contrapositive. To prove $P \implies Q$, you can prove the contrapositive (just the logical negation) $\sim Q \implies \sim P$. So you assume $\sim Q$ and try to prove $\sim P$. This is equivalent to $P\implies Q$, and is called the contrapositve. It is not uncommon to see people write a proof by contradiction that is actually a proof by contrapositive. 
A proof by contradiction would assume the result you want to prove is false, and then derive a contradiction to some previously established fact (like obtaining $1=0$). In this setting you would assume $P \implies Q$ is false, and then try to derive a contradiction. Assuming $P \implies Q$ is false means you assume $P$ and $\sim Q$ both hold. It's not really a proof by contradiction unless the contradiction is of a previously established fact (not something you assumed at the start of the proof).
