Hilbert's axioms
Editions and translations of Grundlagen der Geometrie

The original monograph, based on his own lectures, was organized and written by Hilbert for a memorial address given in 1899. This was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902. This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. In the Preface of this edition Hilbert wrote:

New editions followed the 7th, but the main text was essentially not revised. The modifications in these editions occur in the appendices and in supplements. The changes in the text were large when compared to the original and a new English translation was commissioned by Open Court Publishers, who had published the Townsend translation. So, the 2nd English Edition was translated by Leo Unger from the 10th German edition in 1971. This translation incorporates several revisions and enlargements of the later German editions by Paul Bernays.

The Unger translation differs from the Townsend translation with respect to the axioms in the following ways:

The last two modifications are due to P. Bernays.

Other changes of note are:
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