Conflict of Frege and Hilbert; The Right Way to Deal with Meta-Theorems in Axiomatic Systems

In 1899¡ David Hilbert offers an articulated axiomatic system for Euclidean geometry and¡ demonstrating conditionally the meta-theorems of compatibility and independence for this system¡ proposes a solution to one of the enduring problems of mathematics (known as the problem of parallel lines). Gottlob Frege¡ the founder of new formal logic¡ fundamentally disagreed with Hilbert’s formalistic approach and his proofs for the meta-theorems of compatibility and independence. The reasons for the opposition show that Frege''s view on formality of logic and meta-theorems of compatibility and independence is very different from today''s point of view. In this paper¡ after briefly discussing Hilbert’s method in demonstrating meta-theorems of compatibility and independence¡ and also the main Frege’s objections toward it¡ I will indicate to Frege’s own method dealing with these issues¡ and then discuss why eventually mathematicians and logicians¡ following Hilbert¡ ignored Frege’s remarks and modern logic¡ proposing a model theory¡ stepped on a road which was for Frege a wrong way.