Are Axioms and Inference Rules Sufficient to Find any True Statement in Science? Gödel, a Great Logician with Psychological Problems Gave the Answer

Until 1931 eminent intellectuals, like Russell, Ackermann and Hilbert were trying to find formal systems based of a finite number of axioms and generally accepted rules of inference in which every statement could be proved or disproved, independently from the objects considered. However, to apply this in an endless variety of objects they had to consider them as mere signs drained from any content. This had as a result a statement to be proved or disproved only syntactically, i.e. only formally, without caring if is true or not. Kurt Gödel in 1930 made a distinction between syntactic provability and semantic truth, the latter resulting from the consideration of properties coming from the content of the objects. Finally, in 1931 Gödel, with his two incompleteness theorems, proved that there are true statements in mathematics including elementary arithmetic of natural numbers [0,1,2…] which are impossible to be proved or disproved, demolishing the optimistic opinion prevailing until then. The consequences of these theorems puzzle logicians, mathematicians and physicists ever since. Although Gödel was a leading logician, ironically he was tormented by lifelong illogical fears which finally caused his death.