Gödel's [First Incompleteness Theorem](http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem) states

> 
> Any effectively generated theory
>  capable of expressing elementary
>  arithmetic cannot be both consistent
>  and complete. In particular, for any
>  consistent, effectively generated
>  formal theory that proves certain
>  basic arithmetic truths, there is an
>  arithmetical statement that is
>  true, but not provable in the
>  theory (Kleene 1967, p. 250).
> 
> 
> 

What are the philosophical implications of this theorem? In particular, are there possibly more general analogues of this theorem -- necessarily "unprovable truths" within other kinds of formal systems?