TITLE: Parameters in arithmetic induction axiom schemas
QUESTION [9 upvotes]: The induction schema of Peano Arithmetic is standardly given as the universal closure of $\phi(0)\land \forall x (\phi(x)\rightarrow \phi(x+1)) \rightarrow \forall x\phi(x)$.  However, since the language of arithmetic has a name for every standard number, it is not obvious (to a beginner like me) why parameters are necessary in the induction schema; why not restrict to the case where $x$ is the only free variable in $\phi$?  

Does having parameters in the induction schema really make the system stronger, and, if so, how is that proven?  
Are there natural theorems that can only or most easily be proven using the stronger system?  
Is the weaker system of any interest?

REPLY [13 votes]: $\def\fii{\varphi}$While the other answers resolve the question, a simple way of deriving induction for a formula $\fii(x,\vec p)$ with parameters $\vec p$ is to use parameter-free induction on the formula
$$\psi(x)=\forall\vec p\,[\fii(0,\vec p)\land\forall y\,(\fii(y,\vec p)\to\fii(y+1,\vec p))\to\fii(x,\vec p)].$$
In fact, this derives the induction schema with parameters from the (parameter-free) induction rule, since $\psi(0)$ and $\psi(x)\to\psi(x+1)$ are provable in Q without any assumptions on $\fii$.