How is Mahlo cardinal used? I would like to know how Mahlo cardinals are used - as such examples may help me understand why they were created and so on. 
 A: Here is a neat use of Mahlo cardinals that I have recently learned about:

Denote, for an uncountable cardinal $\kappa$, by $\square_\kappa$ the statement asserting the existence of a sequence $\langle C_\alpha\mid\alpha<\kappa^+\rangle$ with the following properties:
  
  
*
  
*$C_\alpha\subseteq\alpha$ is a club in $\alpha$.
  
*The order type of $C_\alpha$ is less or equal to $\kappa$. (Sometimes you find the requirement that if the equality may occur only when necessary, i.e. $\operatorname{cf}(\alpha)=\kappa$.)
  
*If $\beta$ is a limit point of $C_\alpha$, then $C_\alpha\cap\beta=C_\beta$.
  

Jensen proved that $\square_\kappa$ holds in $L$ for every uncountable $\kappa$. He also proved that if $V\models\lnot\square_\kappa$, then $(\kappa^+)^V$ is Mahlo in $L$. Moreover, Solovay proved that if $\kappa$ is Mahlo, and $\lambda<\kappa$ is regular then there is a forcing extension in which $\kappa=\lambda^+$ and $\lnot\square_\lambda$ fails.
Therefore Mahlo cardinals are used to negate squares on regular cardinals.

One point to make is that the definition of a Mahlo cardinal predates these proofs by three or four decades. This has little to do with the history and the original definitions of a Mahlo cardinal, but it does point out an interesting use for them.
