Gisbert Hasenjaeger

In October 1942, after starting work at OKW/Chi, Hasenjaeger was trained in cryptology, given by the mathematician, Erich Hüttenhain, who was widely considered the most important German cryptologist of his time. Hasenjaeger was put into a newly formed department, whose principal responsibility was the defensive testing and security control of their own methods and devices.[6][9] Hasenjaeger was ordered, by the mathematician Karl Stein who was also conscripted at OKW/Chi, to examine the Enigma machine for cryptologic weaknesses, while Stein was to examine the Siemens and Halske T52 and the Lorenz SZ-42.[9] The Enigma machine that Hasenjaeger examined was a variation that worked with 3 rotors and had no plugboard. Germany sold this version to neutral countries to accrue foreign exchange. Hasenjaeger was presented with a 100 character encrypted message for analysis and found a weakness which enabled the identification of the correct wiring rotors and also the appropriate rotor positions, to decrypt the messages. Further success eluded him, however. He crucially failed to identify the most important weakness of the Enigma machine: the lack of fixed points (letters encrypting to themselves) due to the reflector. Hasenjaeger could take some comfort from the fact that even Alan Turing missed this weakness. Instead, the honour was attributed to Gordon Welchman, who used the knowledge to decrypt several hundred thousand Enigma messages during the war.[6][9] In fact fixed points were earlier used by Polish codebreaker, Henryk Zygalski, as the basis for his method of attack on Enigma cipher, referred to by the Poles as "Zygalski sheets" (Zygalski sheets) (płachty Zygalskiego) and by the British as the "Netz method".

It was while Hasenjaeger was working at Westfälische Wilhelms-Universität University in Münster in the period between 1946 and 1953 that Hasenjaeger made a most amazing discovery - a proof of Kurt Gödel's Gödel's completeness theorem for full predicate logic with identity and function symbols.[3] Gödel's proof of 1930 for predicate logic did not automatically establish a procedure for the general case. When he had solved the problem in late 1949, he was frustrated to find that a young American mathematician Leon Henkin, had also created a proof.[3] Both construct from extension of a term model, which is then the model for the initial theory. Although the Henkin proof was considered by Hasenjaeger and his peers to more flexible, Hasenjaeger' is considered simpler and more transparent.[3]

Hasenjaeger continued to refine his proof through to 1953 when he made a breakthrough. According to the mathematicians Alfred Tarski, Stephen Cole Kleene and Andrzej Mostowski, the Arithmetical hierarchy of formulas is the set of arithmetical propositions that are true in the standard model, but not arithmetically definable. So, what does the concept of truth for the term model mean, the results for the recursively axiomatized Peano arithmetic from the Hasenjaeger method? The result was the truth predicate is well arithmetically, it is even ${\displaystyle \Delta _{2}^{0}}$.[3] So far down in the arithmetic hierarchy, and that goes for any recursively axiomatized (countable, consistent) theories. Even if you are true in all the natural numbers ${\displaystyle \Pi _{1}^{0}}$ formulas to the axioms.

This classic proof is a very early, original application of the arithmetic hierarchy theory to a general-logical problem. It appeared in 1953 in the Journal of Symbolic Logic.[10]

It was only in the 1970s that Hasenjaeger learned that the Enigma Machine had been so comprehensively broken.[6] It impressed him that Alan Turing himself, considered one of the greatest mathematicians of the 20th century, had worked on breaking the device. The fact that the Germans had so comprehensively underestimated the weaknesses of the device, in contrast to Turing and Welchman's work, was seen by Hasenjaeger today as entirely positive. Hasenjaeger stated:

Would it not been so, then the war would have lasted probably longer and the first atomic bomb had not fallen on Japan, but on Germany.[6]