On Frege’s True Way Out, K50-Set8b
On Frege’s true way out
The lecture introduces a new system of set theory called FNM (=Formalized Naive Mengenlehre). In the usual naive set theory, the (extensional) Principle of Abstraction (PoA) allows the contextual replacement of the set operator. In FNM the set operator is a basic notion and the (PoA) [… refer the PDF sheet] is replaced by the new (metatheoretical or intensional) True Abstraction Principle (TAP) [… refer the PDF sheet] where the additional term […refer the PDF sheet], the so called Zusatz, is a metalinguistic abbreviation for a long formula of the (set theoretical) object language. The Zusatz is constructed in such a way that it is satisfied automatically if we want to form „normal“ sets […refer the PDF sheet], e.g. sets belonging to the commulative hierarchy of ZF. Only by using pathological (not „well-founded“) predicates, the Zusatz is becoming false and prevents that way the generation of inconsistencies.
A New Approach to the Formalization of Naive Mengenlehre?
This paper introduces a new system of set theory called FNM (= Formalized Naive Mengenlehre) which is on example par exelance that only philosophy-guided investigation can solve the essential problems in mathematics. This was Kurt Gödel’s conviction, which in particular led him to his famous results. However it is in opposition to the super-technical research of today. The author solved the 90 year old wish of Georg Cantor and Gottlob Frege of how to formalize (consistently) naive mengenlehre. I could establish the system FNM without a sophisticated technical apparatus by having the philosophical insight about how to solve Frege’s problem? FNM can also be considered as justification for the ignorance of the Bourbaki school which ignored the Zermelo-Fraenkel system ZF up to recently, arguing that mathematicians – guided by their intuition – in practice never fail. But from the logical point of view, also, naive mengenlehre turns out to be a sufficiently solid basis for mathematics, as Bourbaki always wanted. This is shown by formalizing naive mengenlehre and establishing the system FNM which works in practice like naive set theory.