1 Interpretations , According to Tarski

The notion of interpretation was first carefully defined and developed in the book [TMR53]. The notion of interpretation is absolutely fundamental to mathematical logic and the foundations of mathematics. It is also crucial for the foundations and philosophy of science-although here some crucial conditions generally need to be imposed; e.g., " the interpretation leaves the mathematical concepts unchanged ". The most obvious and direct use of interpretations is for relative consistency. PROPOSTION 1.1. Suppose S is interpretable in T. If T is consistent then S is consistent. But the notion was mostly used by Tarski to show that various formal systems are undecidable in the sense that there is no algorithm for determining whether a sentence in its language is provable. PROPOSITION 1.2. Suppose S is interpretable in T. If T has a consistent decidable extension in its own language, then S has a consistent decidable extension in its own language. I.e., if S is essentially undecidable then T is essentially undecidable.