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## 2007/10/07

### 南轅北轍

《Gödel, Escher, Bach》p.94 如是說：

[...] Consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it.

Repeat after me: Consistency is a property of a deductive system; it has nothing to do with semantics!

But now let us say exactly what is meant by consistency of a formal system (together with an interpretation): that every theorem, when interpreted, becomes a true statement.

A system would be internally inconsistent if it contained two or more theorems whose interpretations were incompatible with one another, and internally consistent if all interpreted theorems were compatible with one another.

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