The arbitrary in induction:
Induction step: We assume that P(k) is true and then we need to show that P(k+1) is true as well.
k is arbitrary and means "any one"
Consider this example of arbitrary used in the rule of Universal Instantiation
Rule:
(FOR ALL x) P(x) => P(c) where c is some arbitrary element of the universe.
c is arbitrary and essentially means "all".
My issue here is, how can the value c represent all values at once? The argument is probably because c is arbitrary, that's how it can do it. But if c is arbitrary, notice that k in the induction type of problem is also arbitrary and it clearly cannot represent "all".
Are there two different definitions of arbitrary? What is wrong here? Am I misunderstanding something?
You have $\forall x P(x)$ which means that $P(x)$ is true for every element in your space. This implies, of course, that $P(c)$ is true for any $c$. I do not see the problem in interpretation. When you use induction you want to prove $P(k) \Rightarrow P(k+1)$ for every $k$.
Arbitrary means you can pick whatever element you want from your space. If you talk about $k \in \Bbb{N}$ arbitrary then $k$ does not represent all natural numbers but one natural number. At least that's how I understand it.