Abstract
Abstract In this paper I discuss Ernst Zermelo’s ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences. The presentation of Zermelo’s ideas is accompanied with some remarks concerning the development of infinitary logic. I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1
Highlights
I am going to present here Zermelo’s ideas behind his project of infinitary logic developed over eighty years ago
Maticians of the twentieth century. He is known primarily as the author of the first axiomatization of set theory. Other topics he covered include the calculus of variations, applications of mathematics in physics, navigation problems, and the application of set theory to the game of chess
One must remember that Zermelo’s ideas concerning infinitary logic appeared at the time of interregnum in mathematical logic and foundations of mathematics, that is between the paradigms of Principia Mathematica and Grundlagen der Mathematik, and they did not belong to the main stream of investigations in these domains
Summary
I am going to present here Zermelo’s ideas behind his project of infinitary logic developed over eighty years ago. Systematic research on infinitary logic was only to begin two decades later. My considerations are based on Zermelo’s original papers (including his Nachlaß) as well as the discussion of Zermelo’s ideas published by other authors. The complete works of Ernst Zermelo were published quite recently in German, with an accompanying English translation; see Ebbinghaus, Fraser and Kanamori 2010. A few years ago I translated Zermelo’s papers on the foundations of mathematics (from German to Polish). The translation, under the title Matematyka jest logiką nieskończonego. Ernst Zermelo’s works on the foundations of mathematics) remains unpublished
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