Topic

To diagnose a fundamental crisis in mathematics, as the author states in his book "Nothing", Crisis and reEvolution of the foundations of mathematics, seems very questionable at first glance. Mathematics achieves great success in describing nature, and pure mathematics, detached from reality, accomplishes admirable results.
When formulating the foundations of mathematics, the axioms of set theory and mathematical logic, reality also was not the focus, which turned out to be the reason for a fundamental crisis. Because set theory and logic not only have to satisfy abstract quantities, but also "apples, pears and apricots", as was rightly stated but not achieved adequately.

Georg Cantor had developed his set theory, which had not yet been founded axiomatically, at the end of the 19th century. The core theme was a new conception of the infinite, the transfinite, as Cantor called it. Antinomies arose that initiated a foundational crisis in mathematics, so set theory was subsequently founded on axioms. Primarily the elimination of inconsistencies by avoiding the antinomies was targeted.

Ernst Zermelo and Abraham Fraenkel developed the axiom system ZFC, which should fulfill these conditions. However, David Hilbert stated that the transfinite "neither exists in nature nor is permissible as a basis for our intellectual thinking". But he perceived an “idea .... which surpasses all experience", however comprising a "complete calculus", i.e. a consistent theory. A compelling counter-argument has to be adduced, a theory that contradicts reality must necessarily be wrong. The foundational crisis was not overcome, it still continues.

The articles present the core themes of the topic of this website. The inconsistenies of Cantors argumentation and of the axioms of set theory are part of them. Not only the idea of the "transfinite", but also the "number 0", the "empty set" and the "infinite subdivision" with converging infinite sequences and limits contradict reality. In the foundations of mathematics the discrepancy between rationalism and empiricism, that was transcended by Immanuel Kant, is not overcome.

But mathematical logic also shows problems. The asserted insignificance of the syntactic signs and formulas is not tenable.

The logician Kurt Gödel besides demands a fundamental restriction of the proof by his incompleteness theorems. He constructs propositions that are neither provable nor disprovable, i.e. undecidable, but demonstrably true. Further axioms should also not be able to eliminate undecidability. This statement of Gödel is refuted, the incompleteness theorems are surmounted. The crisis started a long time ago. In the 16th century the true meaning of the 0, "nothing" of numbers, i.e. "no number", was contradictorily inverted into its opposite, "something" and "number". Reservations about the metaphysical Nothing in philosophy and Christian religion were responsible for the fact that the "nothing" of mathematics was suppressed. In fact, unrecognized, it plays a significant role as is demonstrated in the book “Nothing”.
Consistent axioms in agreement with reality are formulated.