By A. Ehrenfeucht, V.W. Marek, M. Srebrny
Andrzej Mostowski used to be one of many prime twentieth century logicians. His legacy is tested during this quantity of papers committed either to his impressive medical background and to the reminiscence of him as a superb researcher, instructor, organizer of technology and individual. Professor Mostowski pioneered and mastered many parts of mathematical good judgment. His contributions spanned set concept, recursion conception, and version conception - the spine of the rules of arithmetic. the whole exact bibliography of Mostowski's writings is incorporated. for a few years after WWII and particularly within the overdue sixties and until eventually his premature dying in 1975, Warsaw, the place he led the centre of foundational reports, used to be a spot the place many prime logicians visited, studied, and commenced their profession. Their stories shape a huge a part of this quantity, trying to convey again the intense achievements and character of Mostowski.
Anyone desirous to see the roots of present purposes of common sense, be it in arithmetic, computing device technology or philosophy, will locate during this number of articles lots of precious, stimulating and encouraging fabric either in technological know-how and concerning the social surroundings during which technological know-how is made.
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Extra info for Andrzej Mostowski and foundational studies
Logistics can contribute to the development of other sciences by discovering new forms of reasoning and by training thinking, but this is only a possibility and would be its by-product. The direct aim of logistics in application to other sciences can only consist in explaining their logical structure. This is an interesting passage, because it shows that Janiszewski, who studied in France, did not accept skepticism about logic, very characteristic of French mathematicians, and explicitly rejected Poincaré’s objections against logic as pointless.
Let me first discuss second-order arithmetic. Second-order arithmetic is, in fact, a theory of natural numbers and their subsets. It can be formalized in first-order logic. When we do so, it is sometimes called analysis (Hilbert used this term). There are fundamental reasons why it is worthwhile to study second-order arithmetic. First, via definable mappings with reals, it gives us deep insights into the properties of real numbers. But there are deeper and even more profound reasons. When we limit second-order arithmetic to some parts (too technical to be discussed here), it turns out that we capture some of the most fundamental theorems of mathematics (for instance the König theorem on finitely splitting trees or the Baire Category Theorem).
As common in such situations, forcing at the beginning appeared mysterious. The technique worked, we just did not know why. Many groups of logicians rushed into the area (and many people migrated into foundations in the process). Numerous proposals for explanation and an improved presentation were made. Most exotic tools (for instance, sheafs and other tools from algebraic geometry) were proposed first. Poland was not absent from the quest for explanation – there was a proposal by Professor Ryll-Nardzewski (he presented it in Warsaw), and then Professor Mostowski wrote a book with his presentation of Cohen’s work.