Difference between Gentzen and Hilbert Calculi What is the difference between Gentzen and Hilbert Calculi?
From  my understanding of Rautenberg's Concise Introduction to Mathematical Logic, Gentzen calculus is based on sequents and Hilbert calculus, on tautologies.
But isn't every Gentzen sequent a tautological modus ponens?
For instance, the sequent $X\vdash a \wedge b |X \vdash a,b$ can be written as the tautology $\forall X \forall a \forall b\:( X\vdash a \wedge b \rightarrow X \vdash a,b$), can't it?

Actually, Gentzen introduced two families of calculi, of which the more important perhaps are the natural deduction calculi (which Bruno Bentzen is referring to in his answer). Sequent calculi are rather different from both natural deduction calculi and Hilbert calculi. The latter are both ways of constructing arguments from propositions (wffs in the relevant logical language) to propositions; sequent calculi instead manipulate sequents (which can be read as metalinguistic claims about what follows from what in the relevant language). 
What you really need to get to understand is, first, the relation between Hilbert calculi and Gentzen natural deduction calculi: see e.g. this handout of mine on various proof styles.
And then second, you need to understand the relation between natural deduction calculi and sequent calculi: see Ch. 1 of the splendid Structural Proof Theory by Sara Negri and Jan von Plato, available here.
