New Foundations

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New Foundations (NF) is the name of an axiomatic set theory by Willard Van Orman Quine , named after its attachment New Foundations for Mathematical Logic ( New Fundamentals of mathematical logic ) of 1937. NF tends in many ways to type theory and used to amount formation stratified or layered Expressions. In addition to the axiom of extensionality, NF contains a comprehension scheme which stands for an infinite number of individual axioms and is used for the construction of sets. Theodore Hailperin showed in 1944 that NF is finitely axiomatizable, i.e. that the comprehension scheme can be replaced by a finite number of individual axioms (9 axioms). NF differs from the Zermelo-Fraenkel set theory in many respects: The axiom of foundation does not apply here, since there are sets in NF that are elements of themselves, such as the universal set; this is in contradiction to conclusions from the axiom of foundation. The axiom of choice cannot be added as an axiom to NF either - Ernst Specker proved in 1949 that it was incompatible with the other axioms of NF. Furthermore, one cannot prove the existence of the set of natural numbers. The NF set concept differs greatly from the established set concept, so that NF can be viewed more as a theory of real classes .

Mathematical Logic

In his book Mathematical Logic ( Mathematical Logic ) Quine extended 1,940 NF to ML. He added an axiom of class existence to the axioms of NF, a scheme that stands for an infinite number of individual axioms, which groups sets together. However, in the set formation axiom of the 1st edition of his book, he allowed the use of class variables as parameters that could not be quantified . In 1942 John Barkley Rosser showed that the Burali-Forti paradox can be derived from this system of axioms . Hao Wang blocked this antinomy from the path to ML indicated by Rosser, in which he also used the set existence scheme from NF in ML in his article from 1950. Quine then used it in later revised editions of his Mathematical Logic as well. Hence the axiom of set existence is finitely axiomatizable as in NF, while for the axiom of class existence it is still an open question.

literature

  • Hailperin, Theodore: A new set of axioms for logic , in: Journal of Symbolic Logic Vol. 9 (1944), pp. 1-19.
  • Quine, Willard Van Orman: New Foundations for Mathematical Logic , in: Am. Math. Monthly 44 (1937), pp. 70-80.
  • Quine, Willard Van Orman: From a logical point of view . 9 logico-philosophical essays, Harvard UP, Cambridge, Mass. 1953, pp. 80-101.
  • Quine, Willard Van Orman: Mathematical Logic . Harvard University Press, 4th edition 1981, 1st edition 1940
  • Rosser, John Barkley: Burali-Forti paradox , in: Journal of Symbolic Logic, Vol. 7 (1942), pp. 1-17
  • Specker, Ernst Paul: The axiom of choice in Quine's New Foundations for Mathematical Logic , in: Journal of Symbolic Logic, Vol. 14 (1949), pp. 145-158
  • Wang, Hao: A formal system for logic , in: Journal of Symbolic Logic, Vol. 15 (1950), pp. 25-32

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