**Wilhelm Ackermann** was a mathematical logician who worked with David Hilbert in Göttingen but spent his career as a high school teacher.

Ackermann entered the University of Göttingen in 1914 to study mathematics, physics and philosophy. However soon after he began his studies Word War I started. Ackermann was drafted into the army in 1915 and continued to serve until 1919 when he was able to return to his studies in Göttingen. He received his doctoral degree in 1925 with a thesis *Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit* Ⓣ written under David Hilbert's supervision. It provided a proof of the consistency of arithmetic without induction. It was intended to be a consistency proof for elementary analysis although this proof contained significant errors. Richard Zach explains the background to the thesis in [6]:-

Ackermann's1924dissertation is of particular interest since it is the first non-trivial example of what Hilbert considered to be a finitistic consistency proof. Von Neumann's paper of1927, the only other major contribution to proof theory in the1920s, does not entirely fit into the tradition of the Hilbert school, and we have no evidence of the extent of Hilbert's involvement in its writing. Later consistency proofs, in particular those by Gentzen and Kalmár, were written after Gödel's incomplete ness results were already well-known and their implications understood by proof theorists. Ackermann's work, on the other hand, arose entirely out of Hilbert's research project, and there is ample evidence that Hilbert was aware of the range and details of the proof.

After submitting his dissertation, Ackermann went to Cambridge, England, where he spent the first half of 1925. He was awarded a Rockefeller Scholarship to support this trip and Hilbert wrote in support of his application showing his high opinion of Ackerman's work [6]:-

In his thesis "Foundation of the 'tertium non datur' using Hilbert's theory of consistency," Ackermann has shown in the most general case that the use of the words "all" and "there is," of the "tertium non datur," is free from contradiction. The proof uses exclusively primitive and finite inference methods. Everything is demonstrated, as it were, directly on the mathematical formalism. Ackermann has here surmounted considerable mathematical difficulties and solved a problem which is of first importance to the modern efforts directed at providing a new foundation for mathematics.

Ackermann was also the main contributor to the development of the logical system known as the epsilon calculus, originally due to Hilbert. This formalism formed the basis of Bourbaki's logic and set theory.

From 1929 until 1948 he taught as a teacher at the Arnoldinum Gymnasium in Burgsteinfurt and in Luedenscheid. He was corresponding member of the Akademie der Wissenschaften in Göttingen, and was honorary professor at the Universität Münster.

In 1928, Ackermann observed that *A*(*x*, *y*, *z*), the *z*-fold iterated exponentiation of *x* with *y*, is an example of a recursive function which is not primitive recursive. *A*(*x*, *y*, *z*) was simplified to a function *P*(*x*, *y*) of two variables by Rozsa Peter whose initial condition was simplified by Raphael Robinson. It is the latter which occurs as Ackermann's function in today's textbooks. Also in 1928 the often reprinted book *Grundzüge der Theoretischen Logik* Ⓣ by Hilbert and Ackermann appeared.

Among Ackermann's later work are consistency proofs for set theory (1937), full arithmetic (1940) and type free logic (1952). Further there was a new axiomatization of set theory (1956), and a book *Solvable cases of the decision problem* (North Holland, 1954. Second edition, 1962).

The new axiomatization of set theory was presented by Ackermann in *Zur Axiomatik der Mengenlehre* Ⓣ (1956). Dana Scott writes [5]:-

A remarkably simple axiomatization of a system of set theory is presented which the reviewer feels deserves serious consideration. The system is formalized in an applied first-order calculus with identity using a binary predicate e(membership)and a singulary predicate M(being a set). It is quite essential for the consistency of the system that M is not definable in terms of E. The axiom of extensionality is assumed so that all the individuals can be considered as collections of individuals, but it is easily proved from the axioms that there are collections that are not sets and even contain non-sets. The necessity for the existence of such improper collections in the theory makes comparison with the standard systems of set theory somewhat difficult.

Rudolf Grewe, who wrote his doctoral dissertation on Ackermann's set theory in 1966, gives models for the theory in [2]. Several other authors have studied this system and references are given in [2].

In 1957 Ackermann published *Philosophische Bemerkungen zur mathematischen Logik und zur mathematischen Grundlagenforschung* Ⓣ and its English translation *Philosophical observations on mathematical logic and on investigations into the foundations of mathematics*. This paper, written for non-experts in the subject, gives an excellent overview of how Ackermann viewed mathematical logic. John Van Heijenoort writes [3]:-

An objection to mathematical logic is that it is not the same as the philosophical logic which forms the foundation of our thought and which alone is necessary for thinking. Ackermann remarks that the traditional modes of inference are included in mathematical logic, besides many others, like the statement logic or the logic of relations. One has the illusion of getting by with "Aristotelian logic" in mathematics just as long as mathematical reasonings are insufficiently analyzed.

A further objection to mathematical logic is that it is incomplete, in the sense that, by Gödel's result of1931 (the text says1932), intuitively correct theorems cannot be proved within a given system. But intuitive thinking is not consistent; paradoxes arise on the basis of modes of inference which from the naive intuitive point of view have to be considered correct. If we restrict intuitive inference so that paradoxes are eliminated, we come to a formal system, and incompleteness is the price we have to pay for consistency.

Mathematical logic is of use in clarifying the concepts of "necessity" and "possibility," the distinction between "analytic" and "synthetic." Discussing the "triviality" of logic, the author presents the decision problem.

Mathematics is today viewed as an investigation cf structures; but the obviousness connected with the concept of natural number is independent of any axiomatically introduced structure. Ackermann presents intuitionism, which constructs a mathematics with a minimum of logic, and the Frege-Russell analysis of the number concept. He defends this analysis against certain objections(circularity, necessity of an axiom of infinity). He remarks that Brouwer reduced the number intuition to two concepts: unity and possibility of repeatedly distinguishing a unity from another. In these concepts "there is nothing foreign to logic." He concludes that "in the theory of natural numbers we have a domain which is capable of an intuitive foundation with a minimum of logic in the sense of[Brouwer]and also of attainment purely by logical definitions if we presuppose an extensive logic."

...

The paper ends with remarks on geometry, as a science conveying knowledge of the external world. Beyond any axiomatic treatment of geometry there are intuitive geometrical impressions which force themselves upon us with compelling power. In order to obtain a systematic and complete whole we may add principles in which free constructions occur. But, although limits may be difficult to trace, there remains a nucleus of obligatory intuitive elements. Thus the author sketches a position distinct, on the one hand, from a comprehensive apriorism in the sense of Kant and, on the other hand, from a thoroughgoing conventionalism.

**Article by:** *J J O'Connor*, *E F Robertson, and Walter Felscher*, Tübingen

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