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Neue Grundlagen der Logik, Arithmetik, und Mengenlehre
Gyula König died in April 1913 when he was still working on the final chapter of his book Neue Grundlagen der Logik, Arithmetik, und Mengenlehre. The publication was completed by Gyula König's son Denes König. We give below an English translation of Gyula König's Preface and the Preface of Denes König written in October 1913.
1. Gyula König's Preface.
Synthetic logic is intended to provide a scientific picture of the thought processes examined in logic, arithmetic, and set theory, just as the mechanics of the heavens provides a scientific picture of planetary motion.
This primarily involves reaching an agreement on those first facts that we wish to recognise as binding for all thought, and on this basis, constructing or synthesising the concept of a thing that appears in the mathematical and logical sciences. In particular, collective concepts (set concepts) occupy a prominent position in this investigation. Experience shows that such collective concepts can differ not only in the elements included in the collection, but also in the way in which these elements are contained within that collective concept, in the set, which is the natural explanation for Russell's "contradiction." It is not a restriction, but a clarification and the associated generalisation of the set concept that makes the so-called antinomies of set theory disappear, fulfils the strict conditions for our mathematical and logical thinking demanded by Poincaré, and yet preserves the classical results of Cantor's set theory
A deeper formulation of logical deduction proves to be a necessary foundation. The law of non-contradiction cannot be substantiated by ontological and metaphysical considerations; indeed, it has always been said that where the validity of this proposition ceases, our logical thinking must also cease. Its validity must become "evident"; and this occurs exclusively through appeal to immediate intuition, above all to the undeniable fact that the same things cannot be both different and not different at the same time. Just as Hilbert strengthens the consistency of his "geometries" by reducing them to the realm of number intuition, so too does the consistency of logical thinking become evident through reduction to that aforementioned realm of intuition.
Our belief in the reliability of logical reasoning is thereby significantly deepened, and at the same time a central method is created that solves the problem of consistency for the areas under consideration here. Translated into more familiar terms, this means that the consistency of certain groups of axioms is confirmed. This, in turn, establishes a "new" foundation for arithmetic and set theory, the details of which are not insignificant and cannot be discussed in this brief report, this means that the consistency of certain groups of axioms is confirmed. This, in turn, establishes a "new" foundation for arithmetic and set theory, the details of which are not insignificant and cannot be discussed in this brief report.
Just one more thing, let it be mentioned, that in this way we also arrive at Zermelo's principle of selection and the related general principle of well-order. No one can be "forced" to accept the intuition postulates contained therein, but it can be shown that these assumptions never lead to a contradiction. This, however, is what can actually be demanded, for a group of propositions that never leads to contradictions through logical deduction is no longer a hypothesis; it possesses that logical existence which alone can be demanded for "free creations of our minds."
Thus, I believe I have incorporated these interesting and hotly debated areas into our definitive, in a certain sense of the word, scientific knowledge.
2. Denes König's Preface.
With these words, my father, who passed away on April 8th of this year, characterised the aim and results of the book on which he had worked for eight years until the last day of his life and which is hereby presented to the public. He was granted the opportunity to bring his work to a conclusion in its essence, even though the final chapter can only be considered a fragment. Only a few pages were missing. These would have contained, above all, the exposition of Zermelo's much-debated proof for the well-order theorem, corresponding to his definitively presented view in this book. He had already recognised this proof as binding for several years - at least for what he called Cantor's sets (Chapter VI, Article 6), which also include the continuum. Furthermore, his only remaining intention was to include a proof of the theorem "" in his book. As is well known, proofs for this theorem by Cantor were published by Hessenberg and Jourdain. These discussions would not have presented any new fundamental difficulties. The author even stated, after completing Chapter VII, and has repeatedly expressed this view, that anyone who has thoroughly studied and understood these seven chapters could easily follow up on everything else.
The book appears here without additions or supplements, exactly as the author left the manuscript. The changes I found unavoidable during my review of the manuscript and the corrections are so minor that it seems unnecessary to list them individually. I was fortunate enough to discuss them all with Professor Kürschák, who, like all his other scholars, maintained his interest in my father's work, even in the last book of his teacher and friend. If it wasn't too noticeable that the final revision of the manuscript could not be carried out by the author himself, this is primarily due to his cooperation. My sincere thanks also go to Professor F Hausdorff. By undertaking the arduous and thankless task of proofreading the entire book, he fulfilled one of my father's last wishes. Thanks are also due to the publishing house Veit & Co. I am deeply grateful for their kindness in readily accommodating all my requests regarding the book's production, some of which were even made by my father himself, and for prefacing the book with the author's portrait.
Last Updated March 2026