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'' The pieces and the squares of the board correspond
to the elementary signs of the calculus; the permitted configurations of
pieces on the board correspond to the axioms or initial formulas of the
calculus; the subsequent configurations of pieces on board correspond to
formulas derived from the axioms (i.e.,to the theorems); and the rules of
the game correspond to the rules of derivation for the calculus.  A
meta-chess statement may assert, for example, that there are 20 possible
opening moves for White, or that, given a certain configuration of pieces
on the board with White to move, Black is mate in three moves. It is
pertinent to note, moreover, that general meta-chess theorems can be
established, whose proof involves the consideration of only a finite
number of permissable configurations on the board. The meta-chess theorem
about the number of possible opening moves for White can be established in this way; and so can the meta-chess theorem that if White has only two
Knights and the King, and Black only his king, it is impossible for White
to force mate against Black. These and other meta-chess theorems can thus
be proved by finitary methods of reasoning, consisting in the examination
in turn of each of a finite number of configuratiosn that can occur under
stated conditions. The aim of Hilbert's theory of proof, similarly, was
to demonstrate by such finitary methods the impossibility of deriving
certain formulas in a calculus.  --Ernest Nagel and James R. Newman
''Godel's Proof'' 1956