Abstract
Two parts of Volume I of Alfred Pringsheim’s long-awaited Lectures on Number Theory and the Theory of Functions have now appeared: Part One is subtitled ‘Real Numbers and Number Sequences’, part Two ‘Infinite Series with Real Members’; a third part will contain an introduction to complex numbers, the completion of the theory of series which this necessitates, and the theory of complex numbers and continued fractions. The second volume is to contain “an introduction to the theory of one-valued analytic functions of a complex variable and the simplest many-valued inverse functions on the basis of Weierstrass’s methods and their further development, particularly with respect to the theory of integral transcendental functions and analytic progressions.”
First published in Göttingische Gelehrte Anzeigen, 1919.
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Notes
A. N. Whitehead and B. Russell, Principia Mathematica, Cambridge.
G. Peano, Arithmetices principia nova methodo exposita, Turin 1889.
This system is also found in appendix II of A. Genocchi and Peano, The Differential Calculus and the Principles of the Integral Calculus [Calcolo differenziale &c (Rome, 1884) G. T. Differentialrechnung &c. (Berlin, 1889)].
Symbolic logic does not distinguish between the mark characteristic of all things of a class and the class itself. And since it employs the words ‘set’ and ‘class’ synonymously, it used to define the potency of class A simply as the class of all classes equivalent to A. This definition has been abandoned because the concept of the class of all classes equivalent to A proved to be self-contradictory. I believe that this definition can be retained if we make a minor modification in it. Let us start by supposing that we are given a domain D of things. Let us designate collections of these things as sets and lay it down, as a fundamental logical law, that a set is not a thing in D. Let us now extend domain D to D′ by adding sets of improper things. We can now form sets of things in D′; to distinguish them from the previous sets, let us call them ‘second-level sets’ and the previous ones ‘first-level sets’. The definition of a potency will then read: The potency of a first-level set A is the second-level set of all first-level sets equivalent to A. As far as I can see, this reading no longer gives rise to contradictions. Cf. M. Pasch, Grundlagen der Analysis, p. 94.
Cf. B. Russell, The Principles of Mathematics, p. 128.
Encyklopädie der mathematischen Wissenschaften, Vol. I, Part 1, p. 11.
Revenue de mathématique 8 (1903)
The general theory of the extension of a system of magnitudes, as presented by O. Stolz and J. A. Gmeiner in Theoretische Arithmetik, Section 3, Paragraph 7 (p. 67), could be reformulated in exactly the same way.
Die realistische Weltansicht und die Lehre vom Raume, Chapter V, p. 81 ff.
These ‘axioms’ are therefore far from being ‘fundament proposition’ in the sense mentioned above; nor is the requirement that a particular connection be associative and commutative a ‘fundamental proposition’ when it is part of study of the most general associative and commutative connections in a system of magnitudes (as conducted, e.g., by Stolz and Gmeiner in their Theoretische Arithmetik, 2nd ed., p. 50ff.).
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© 1980 D. Reidel Publishing Company, Dordrecht, Holland
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Hahn, H. (1980). Review of Alfred Pringsheim, Vorlesungen über Zahlen- und Funktionenlehre, Vol. I, Parts I and II, Leipzig and Berlin 1916. In: McGuinness, B. (eds) Empiricism, Logic and Mathematics. Vienna Circle Collection, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8982-5_6
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