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Bernays & Hilbert

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Bernays Project: Text No. 12 Foundations of Mathematics Vol. 1 (1934) David Hilbert/Paul Bernays (Grundlagen der Mathematik, Vol. 1) Translation by: Ian Mueller Comments: V olker Pec khaus, p ar. 1, Mark Rav aglia § 1. The Problem of consistency in axiomatics as a logical deci- sion problem. The state of research in the ﬁeld of foundations of mathematics, to which our presentation is related, is characterized by three kinds of investigations: 1. the development of the axiomatic method, especially with the help of the foundations of geometry, 2. the founding of analysis by today’s rigorous methods through the re- duction of the theory of magnitudes to the theory of numbers and sets of numbers, 3. investigations in the foundations of number theory and set theory. A deeper set of tasks, linked to the standpoint reached through these investigations, arises on the basis of methods subjected to stricter demands; 1

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