Studies in the Philosophy of Mathematics and History of Logic
Part II: Hilbert’s Program vs. Incompleteness Phenomenon
Part II Hilbert’s Program vs. Incompleteness Phenomenon Hilbert’s Program: Incompleteness Theorems vs. Partial Realizations 1. Hilbert’s Program Mathematics on the turn of the 19th century was characterized by the intense development on the one hand and by the appearance of some difficulties in its foundations on the other. Main controversy centered around the problem of the legitimacy of abstract objects. The works of K. Weierstrass have contributed to the clarification of the roˆle of the infinite in calculus. Set theory founded and developed by G. Cantor promised to mathematics new heights of generality, clarity and rigor. Unfortunately paradoxes appeared. Some of them were known already to Cantor (e.g., the paradox of the set of all ordinals and the paradox of the set of all sets1) and they could be removed by appropriate modifications of set theory (cf. Cantor’s distinction between absolut unendliche or inkonsistente Vielheiten and konsistente Vielheiten, i.e., between classes and sets2). Frege’s attempt to realize the idea of the reduction of mathematics to logic (which was in fact a continuation of the idea of the arithmetization of analysis developed among others by Weierstrass) led to a really embarrassing contradiction discovered in Frege’s system by B. Russel and known today as Russell’s antinomy or as the antinomy of nonreflexive classes. This meant a crisis of the foundations of mathematics (called the second crisis the first being the crisis caused by the discovery of incommensurable segments in the ancient Greek mathematics). Various ways of overcoming those difficulties and of...
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